Technische Universit¨at Mun¨ chen Zentrum Mathematik · Technische Universit¨at Mun¨ chen...

Technische Universitat Munchen

Zentrum Mathematik

Liability Driven InvestmentOptimization

Diplomarbeit

von

Yiying Zheng

Themensteller/in: Prof. Dr. R. Zagst

Betreuer/in: Dr. G. Scheuenstuhl

Abgabetermin: 18.04.2007

Hiermit erklare ich, dass ich die Diplomarbeit selbststandig angefertigt und nur dieangegebenen Quellen verwendet habe.

Garching, den 18.04.2007

i

Acknowledgments

First and foremost, I would like to thank my supervisor at the Technical University Mu-nich, Prof. Dr. Rudi Zagst, for accepting the proposed topic and for brilliant professionalsupervision.

At risklab germany GmbH, Munich, I would like to express my sincere thanks to Dr.Gerhard Scheuenstuhl for his useful comments and his patience. He agreed to be my the-sis supervisor at risklab Germany and gave me the chance to touch such a great topic.I am also grateful to all the other employees from risklab germany GmbH for providingassistance and support. Moreover, I would like to express my thanks to Kai Fachinger. Iappreciate his useful advice and patience during the whole period of my thesis.

Last but not least I would thank my parents who always supported me on the long wayof my life, especially during the study in China and in Germany.

Thank you.

Contents

1 Introduction 1

2 LDI Concept for ALM Approach 42.1 Economic Scenario Generator . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Description of the Benefit Scheme . . . . . . . . . . . . . . . . . . . 102.3.2 Modeling of Defined Benefit Pension Liabilities . . . . . . . . . . . 11

2.4 Balance Sheet Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Contribution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.1 Basic Regulatory Environment . . . . . . . . . . . . . . . . . . . . . 152.5.2 Individual Regulatory Framework . . . . . . . . . . . . . . . . . . . 17

3 Immunization Approaches 203.1 Linear Approaches to Immunization . . . . . . . . . . . . . . . . . . . . . . 203.2 Traditional Duration Matching . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Generalized Duration Vector Model . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Nelson and Siegel Model . . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 Traditional Duration Vector Model . . . . . . . . . . . . . . . . . . 353.3.3 Generalized Duration Vector Model . . . . . . . . . . . . . . . . . . 38

3.4 Key Rate Duration Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5 Principal Component Duration Model . . . . . . . . . . . . . . . . . . . . . 54

3.5.1 Principal Components Analysis . . . . . . . . . . . . . . . . . . . . 543.5.2 Principal Component Duration . . . . . . . . . . . . . . . . . . . . 57

4 Case Study: Dynamic and Stochastic Optimization 654.1 Traditional Duration Model . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2 Generalized Duration Vector Model . . . . . . . . . . . . . . . . . . . . . . 744.3 Key Rate Duration Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4 Principal Component Duration Model . . . . . . . . . . . . . . . . . . . . . 864.5 Markowitz Optimization Using the Resampling Method . . . . . . . . . . . 93

4.5.1 A Description of the Problem . . . . . . . . . . . . . . . . . . . . . 934.5.2 Michaud’s Methodology . . . . . . . . . . . . . . . . . . . . . . . . 944.5.3 Optimal Portfolio of Reampling Optimization . . . . . . . . . . . . 95

ii

CONTENTS iii

5 Result Interpretation 985.1 Plan Asset, Plan Asset before Contributions . . . . . . . . . . . . . . . . . 985.2 Contributions, Present Value of Contributions . . . . . . . . . . . . . . . . 1005.3 Funding Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.4 Surplus Return and Surplus Volatility . . . . . . . . . . . . . . . . . . . . . 1045.5 Comparison with Results of Resampling Optimization . . . . . . . . . . . . 105

5.5.1 Plan Assets before Contributions . . . . . . . . . . . . . . . . . . . 1055.5.2 Contributions and Present Value of Contributions . . . . . . . . . . 1065.5.3 Funding Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.5.4 Surplus Return and Surplus Vola . . . . . . . . . . . . . . . . . . . 108

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Summary and Outlook 111

A List of Symbols 113

B Summary of Immunization Models’ Features 115

C Correlation Matrix of Spot Rates Changes 117

D Mean of Optimal Portfolio Allocation 120

E MVP of Resampling Optimization 123

F Evaluations of Chapter 4 126

G Evaluations of Chapter 5 128

Bibliography 135

List of Figures

2.1 Building Blocks for LDI-Concept . . . . . . . . . . . . . . . . . . . . . . . 42.2 Cascade Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Example of GDP and Inflation . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Example of 1y- and 10y-Treasury Yield . . . . . . . . . . . . . . . . . . . . 72.5 Example of Stock Returns and Dividend Yields . . . . . . . . . . . . . . . 82.6 Evolution of the Different Quantiles of the Liabilities . . . . . . . . . . . . 122.7 An Example of a Contribution Strategy . . . . . . . . . . . . . . . . . . . . 17

3.1 Data of Bonds Payments and Liability Cash Flows . . . . . . . . . . . . . 233.2 Bond Price Change by Interest Rate Movements . . . . . . . . . . . . . . . 263.3 Valuation of Pension Liability . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Cash Flows Comparison for Traditional Duration Matching . . . . . . . . . 293.5 Discount Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6 Cash Flows Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.7 Pension Liabilities and Key Rate Duration I . . . . . . . . . . . . . . . . . 453.8 Correlation Matrix of Spot Rates Shifts . . . . . . . . . . . . . . . . . . . . 473.9 Pension Liabilities and Key Rate Duration II . . . . . . . . . . . . . . . . . 493.10 Key Rate Duration of Liability and Instrument Universe . . . . . . . . . . 513.11 Cash Flows Comparison for KRD Model . . . . . . . . . . . . . . . . . . . 533.12 Impact of the First 3 PCs on the TSIR I . . . . . . . . . . . . . . . . . . . 613.13 Cash Flows Comparison for PCD Model . . . . . . . . . . . . . . . . . . . 64

4.1 Expected Benefits 80 Years . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2 Yield Curve and Discount Curve . . . . . . . . . . . . . . . . . . . . . . . . 674.3 Distribution of Objective Function Values For Traditional Duration Matching 704.4 Distribution of Portfolio Allocation for Traditional Duration Matching . . . 724.5 Distribution of 1-year Bond Weight . . . . . . . . . . . . . . . . . . . . . . 734.6 Allocation Trend for Traditional Duration Matching . . . . . . . . . . . . . 744.7 Squared Deviation of Duration Vector Model . . . . . . . . . . . . . . . . . 764.8 Distribution of Portfolio Allocation along 30 Years . . . . . . . . . . . . . . 794.9 Mean of Allocation Trend for Generalized Duration Vector Model . . . . . 804.10 Distribution of Objective Function Values at Each Time Step . . . . . . . . 824.11 Distribution of Key Rate Duration Corresponding to 8 Key Rates . . . . . 834.12 Allocation Trend of Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.13 Impact of the First 3 PCs on the TSIR II . . . . . . . . . . . . . . . . . . . 88

iv

LIST OF FIGURES v

4.14 Distribution of Squared Deviation of Principal Component Duration . . . . 894.15 Distribution of Bond PCD corresponding to the 3 Components . . . . . . . 914.16 Comparison of Mean Portfolio Allocation Trend . . . . . . . . . . . . . . . 924.17 MVP of Resampling Optimization with 9 Bonds . . . . . . . . . . . . . . . 964.18 Mean-Variance-Frontiers over 30 Years . . . . . . . . . . . . . . . . . . . . 974.19 MVP of Resampling Optimization with 22 Instruments . . . . . . . . . . . 97

5.1 Plan Asset before Contributions and its Volatility . . . . . . . . . . . . . . 995.2 DBO of 80 Years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 Contributions and its Volatility . . . . . . . . . . . . . . . . . . . . . . . . 1015.4 Present Value of Contributions and its Volatility . . . . . . . . . . . . . . . 1015.5 Funding Level and its Volatility . . . . . . . . . . . . . . . . . . . . . . . . 1035.6 Surplus Return and Surplus Volatility . . . . . . . . . . . . . . . . . . . . . 1045.7 bPA Comparison with Markowitz Optimization Results . . . . . . . . . . . 1065.8 Contributions Comparison with Markowitz Optimization Results . . . . . . 1075.9 PV Contributions Comparison with Markowitz Optimization Results . . . 1075.10 Funding Level Comparison with Markowitz Optmization Results . . . . . . 1095.11 Surplus Return Comparison with Markowitz Optimization Results . . . . . 109

C.1 Correlation Matrix of Spot Rates Changes I . . . . . . . . . . . . . . . . . 118C.2 Correlation Matrix of Spot Rates Changes II . . . . . . . . . . . . . . . . . 119

List of Tables

2.1 Summary Table of Asset Classes . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Individual Regulatory Frameworks . . . . . . . . . . . . . . . . . . . . . . 18

3.1 Optimal Portfolio Allocation of Cash Flow Matching . . . . . . . . . . . . 243.2 Duration and Convexity of Bonds . . . . . . . . . . . . . . . . . . . . . . . 283.3 Traditional Duration Vector for Tracking Universe . . . . . . . . . . . . . . 373.4 Generalized Duration Vector of Liability and Bonds . . . . . . . . . . . . . 433.5 Optimal Allocations Comparison between Two Duration Vector Models . . 433.6 Shift Matrix with 7 Key Rates . . . . . . . . . . . . . . . . . . . . . . . . . 493.7 Covariance Matrix of Key Rate Changes . . . . . . . . . . . . . . . . . . . 513.8 Optimal Allocations Comparison . . . . . . . . . . . . . . . . . . . . . . . 523.9 Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 603.10 Height, Slope and Curvature Factors . . . . . . . . . . . . . . . . . . . . . 613.11 PCD of Liability and Bond Instruments . . . . . . . . . . . . . . . . . . . . 623.12 Optimal Allocation of PCD Approach . . . . . . . . . . . . . . . . . . . . . 63

4.1 Median of Liability and Bond Duration . . . . . . . . . . . . . . . . . . . . 714.2 Duration Vector of Each Bond . . . . . . . . . . . . . . . . . . . . . . . . . 774.3 Median of Liability and Portfolio Duration Vector . . . . . . . . . . . . . . 784.4 Covariance Matrix of Key Rate Changes . . . . . . . . . . . . . . . . . . . 814.5 Median of Bonds Key Rate Duration . . . . . . . . . . . . . . . . . . . . . 844.6 Median Value of Liability Key Rate Duration . . . . . . . . . . . . . . . . 854.7 Median of Bonds PCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.8 Median of Liability PCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

B.1 Summary of Advantages and Shortcomings of Immunization Models . . . . 116

D.1 Mean of Optimal Allocation for Traditional Duration Matching Model . . . 121D.2 Mean of Optimal Allocation for Generalized Duration Vector Model . . . . 121D.3 Mean of Optimal Allocation for Key Rate Duration Model . . . . . . . . . 122D.4 Mean of Optimal Allocation for Principal Component Duration Model . . . 122

E.1 MVP of Resampling Optimization with 9 Bonds . . . . . . . . . . . . . . . 124E.2 MVP of Resampling Optimization with 22 Instruments . . . . . . . . . . . 125

F.1 Median Durations and Convexities of Liabilities and Assets . . . . . . . . . 127F.2 Median of Portfolio Key Rate Duration . . . . . . . . . . . . . . . . . . . . 127

vi

LIST OF TABLES vii

F.3 Median of Portfolio PCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

G.1 Plan Assets before Contributions . . . . . . . . . . . . . . . . . . . . . . . 129G.2 bPA Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129G.3 Plan Assets before Contributions . . . . . . . . . . . . . . . . . . . . . . . 130G.4 bPA Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130G.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131G.6 Volatility of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 131G.7 Present Value of Contributions . . . . . . . . . . . . . . . . . . . . . . . . 132G.8 Volatility of Present Value . . . . . . . . . . . . . . . . . . . . . . . . . . . 132G.9 Funding Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133G.10 Volatility of Funding Level . . . . . . . . . . . . . . . . . . . . . . . . . . . 133G.11 Surplus Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134G.12 Surplus Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Chapter 1

Introduction

Management of defined benefit plans has never been more challenging than it is today.Plan sponsors are faced with more stringent accounting rules and regulatory changes justas plan funding is strained by significant declines in interest rates and several years of neg-ative equity market returns. The most effective way to ensure that plan assets are sufficientto satisfy pension obligations is to link these two explicitly. Under and LDI framework, as-set allocation is designed to service liabilities and risk is measured ont only in the contextof asset return volatility, but in the context of how assets perform in relation to liabilities1.

LDI is the shortcut which must be more frequently faced in connection with the topicasset management and also in the context of operational pension scheme. It stands forLiability Driven Investments and denotes a portfolio strategy, which is directly derivedfrom the future development of liabilities. In contrast to the previous investment ap-proaches, which are only focussed on the assets performance, LDI ensures that the port-folio strategy aims directly for the characteristics of the pension liabilities. It is gen-erated from Asset-Liability-Management. However there exists difference between them.Asset-Liability-Management orients that the portfolio strategies aim at the investors’ risk-bearing capacity. LDI takes one step ahead: Here specific risks which are resulted fromthe liabilities’ structure should be reduced by specific hedging strategies. Usually interestrate, inflation and duration risks stand in the foreground. The hedging of liability riskscan generally be done using three strategies: by constituting a bond portfolio, by usingderivatives or by a combination of these two approaches. In the context of this thesis weconcentrate on the first strategy.

Why do LDI surface now? Actually there are some investors who have devoted their effortsto looking for the portfolio strategies focussing on reducing liability risks for a long time.Already in the Nineties some investors, e.g. British life insurer began to hedge unexpectedduration and interest rate risks with specific solutions2. In recent years with increasingamount and heightened complexity of risks many pension funds have caused a necessarytransformation in the environment of insurance companies. This development has led toa genuine desire to adapt accounting (IFRS) and prudential rules (Solvency II), with the

1See DiMaggio, 2006 [21]2See dpn-Dossier, Juli 2005, page 5 [22]

1

CHAPTER 1. INTRODUCTION 2

aim of providing a better perception of all companies, notably with regard to the risksbeing run. This transformation in the environment is one of the most important factorsor reasons with which LDI win always more focusses.

What is IFRS? From 1 January 2005 over 7,000 listed companies in the EU must report us-ing the standards issued by the International Accounting Standards Board (IASB). Theseare either called International Accounting Standards (IAS) or more recently InternationalFinancial Reporting Standards (IFRS). IASB requires an overview of the entity‘s use offinancial instruments, the exposure to risks they create and how those risks are managed.This disclosure offers greater transparency regarding those risks. The adoption of IFRS inEurope is perhaps the most significant and dramatic change in accounting ever seen. Forsome companies the impact on their financial statements will be modest, but for othersalmost all line items in the income statement and balance sheet, and also many of thesubtotals within cash flow statements, are liable to change and this could have a mate-rial impact on key performance metrics. What at the heart of the IFRS structure is theconcept of fair value. The idea of using the fair value principle for assets and liabilities byincluding all of their risk factors is clearly a significant step forward for financial manage-ment in insurance companies. On this account the investor must move their focus fromprevious portfolio strategies on LDI, which aims exactly at hedging of liability risks.

The Solvency II project aims to reform regulatory capital for insurance companies. Insimple terms, the current arbitrary statutory capital that prevails should be replaced bySolvency Capital Requirement, which is set up according to the risk profile of insurancecompanies. The European Commission is due to release the Solvency II directive in 2007.This directive will be applied to national regulations and implemented in 20103. The basicfocus of Solvency II lies on a adequate and verifiable risk orientation of insurance com-pany, which involves not only the quantitative elements of insurance environment but alsothe quality of company individual risk management. The further goal is to raise the risktransparency for public. For this reason Solvency II will have significant consequences onAsset-Liability Management and Asset Management.

With this background of insurance environment this thesis aims at introducing and com-paring different LDI optimization approaches. However our focus lies on looking for theoptimal immunization portfolio. Immunization, simply described, is a strategy wherebyand investor-having an identifiable future financial liability-invests a determinable amountof money in a way that guarantees that the future value of the investment will preciselyequal the value of the liability when the liability comes due. Because most pension planscannot execute a true immunization strategy, what we are really referring to when wetalk about “immunization” for a pension plan is an investment strategy where pensionassets are invested in a fixed income portfolio that is duration-matched to the pensionliabilities4. This thesis is organized as follows:

In chapter 2 we give at first the introduction to the entire working enviroment and frame-

3See http://www.edhec-risk.com/ALM/solvency II [24]4See Hewitt, 2004 [33]

CHAPTER 1. INTRODUCTION 3

work, in which our immunization approaches will be implemented. At first a brief introduc-tion to Risklab Economic Scenario Generator (ESG) will be given since all the importantfuture data are simulated with help of the Risklab Economic Scenario Generator. As astep forward we introduce our assets and liabilities sides, on which our dynamic and sto-chastic implement are based. In order to compare the results of different immunizationapproaches we bring the results into the balance sheet projection. On this account, atthe end of this chapter we describe the structure of the balance sheet projection and thecontribution strategies we used for the implement.

Chapter 3 is one of the most important parts of this thesis. Here the optimization ap-proaches in terms of immunization will be introduced for theoretic purposes. We use asmall example so that the theoretic introduction will be more directly perceived throughthe senses. Beginning with the easiest form of immunization: cash flow matching we in-troduced 5 immunization models totally. The respective features of each models are alsodemonstrated and a summary of it is given in table B.1.

Since we have got to know the immunization models for theoretic purposes, the next stepis to apply them by means of a case study, which is the idea of chapter 4. For somereasons we implement this time only the optimization models with respect to duration,namely the last 4 models. For comparing the results with the results of the minimum-variance-portfolio generated by the Markowitz optimization with resampling method, abrief introduction to this method of optimization will be demonstrated.

At the end of this thesis we bring the optimal portfolios of immunization models and theresampling optimization into the balance sheet projection to calculate some importanttarget values such as: funding level, surplus return, present value of contributions and so on.The purpose is to compare the immunization models and resampling optimization modelsfrom different points of view. In addition, we implement the Markowitz optimization alsowith the expanded assets universe, so that we are able to see how the results will be ifmore assets classes are used.

Chapter 2

LDI Concept for ALM Approach

Before we begin with our immunization optimization approaches, it is necessary to give anintroduction to the entire working environment and framework, in which our immunizationapproaches will be implemented.

Figure 2.1: Building Blocks for LDI-Concept

Source: Risklab Germany, “LDI Concept based on an Integrated ALM Approach”, 2006

The major LDI concept based on an integrated asset liability modeling (ALM) is demon-strated in figure 2.1 made by Risklab Germany. It describes the whole working procedureof the concept. Although our focus is not put on the single step of entire LDI-conceptsuch as stochastic scenario generator, it is anyhow necessary to get a recognition of theLDI-concept. Hence, in this chapter we will merely give a brief description.

4

CHAPTER 2. LDI CONCEPT FOR ALM APPROACH 5

The first and indispensable step for the whole concept is to achieve the market risk factors.For example, GDP, inflation and interest rates are simulated in the first block: EconomicScenario Generator, which is enhanced by Risklab Germany in cooperation with Prof. Dr.Rudi Zagst, Professor of Mathematical Finance at the Munich University of Technology.The natures and future performance of assets and liabilities will be modeled based onthese market risk factors, which happens actually in the second and third steps in figure2.1. According to different regulatory frameworks variety of contribution strategies are tobe set. Then the immunization models will be brought into play. After implementing ofthe models, we get the optimal allocation of each model. The result will be taken into thebalance sheet projection. By means of distinct criteria, e.g. funding level, surplus returnand surplus risk, we are able to compare the results of the immunization models whichwill be introduced in chapter 3.

To give a concrete introduction to the entire LDI-concept, we will describe the processstep by step in the following sections.

2.1 Economic Scenario Generator

As the first step of the entire working process the economic scenario generator plays adecisive role. It aims at generating the valuation of assets and liabilities, whose marketvalue is a function of a set of key risk factors in the most important economic zones, suchas inflation rates and interest rates, which are all random variables.1

The goal of the economic scenario generator is not to explain past movements in interestrates, nor is the model attempting to perfectly predict interest rates in any future periodin order to exploit potential trading profits. Rather, the model purports to depict plausi-ble risk factor scenarios which may be observed at some point in the future2. A scenariois “a description of a possible state of an organization’s future environment, consideringpossible development of relevant interdependent factors of the environment”3. Mulvey(1994)[47] defines a scenario as “a single coherent and plausible set of returns for all ofthe assets, the value of the accompanying economic factors, and the associated presentvalue of the liabilities.”

The stochastic scenario generator developed by Risklab Germany GmbH relies on themathematical paper ”Using Scenario Analysis for Risk Management” by Rudi Zagst(2001)[75]. This paper describes the mathematics of the “heart of the scenario gener-ator” - the simulation of the risk factors. Based on the risk factors, the instruments and

1Liabilities also depend on the risk factor mortality and morbidity, which are however not simulatedby the economic scenario generator. For these values we take the expected values, since both of them arerelatively stable.

2See Ahlgrim, D’Arcy and Gorvett [1]3See Brauers and Weber, 1988[8]


afterwards various portfolio strategies can be calculated. To obtain the consistent andmeaningful scenarios over a long-term horizon, the scenarios are created by Risklab withrespect to a structural “economic cascade”, while the scenario generators in other modelsrely on correlation assumption. The cascade structure is demonstrated in figure 2.2.

Figure 2.2: Cascade Model

As follows we give a short overview about what happens in single step of the cascade andthe mathematical models behind the simulation of the risk factors:

1. Modeling of gross domestic product and inflation rate.

• Dynamics of the gross domestic product (economic wealth ω)

dω(t) = [θω · ω(t)]dt + σωdWω(t)4 (2.1)

• Dynamics of the inflation rate (rI)

drI(t) = [θI − aI · rI(t)]dt + σIdWI(t) (2.2)

2. Modeling of real rate, inflation rate and using GDP and inflation rate from the lastcascade.

• Dynamics of the real rate (rR)

drR(t) = [θR(t) + bRω · ω(t)− aR · rR(t)]dt + σRdWR(t) (2.3)

4Here W denotes a standard Brownian motion


Figure 2.3: Example of GDP and Inflation

(a) GDP (b) Inflation

Source: Risklab Germany

• Dynamics of the nominal rate, which is defined by r = rR + rI

dr(t) =

θI + θR(t)︸︷︷︸θr(t)

+ bRω︸︷︷︸brω

·ω(t) + (aR − aI)︸︷︷︸brI

·rI(t)− aR︸︷︷︸ar

·r(t)

dt

+√

σ2I + σ2

R︸︷︷︸σr

dWr(t) (2.4)

Figure 2.4: Example of 1y- and 10y-Treasury Yield

(a) 1y-Treasury Yield (b) 10y-Treasury Yield


3. Modeling of equity return, which is determined by GDP, inflation and interest rates.

• Dynamics of the return for the equity market index

dRE(t) = (αE + bER · rR(t)− bEI · rI(t) + bEω · ω(t))dt + σEdWE(t) (2.5)

• Dynamics of the dividend yield

dRD(t) =[θD − aD ·RD(t)

]dt + σDdWD(t) (2.6)


Figure 2.5: Example of Stock Returns and Dividend Yields

(a) Stock Returns (b) Dividend Yield


4. Modeling of sector indices and individual stocks through a parent-child relationships(CAPM)

As formulated above, θω, θI , aI , aR, . . . respect to the parameters of drift, volatility andmean reverting level, which are estimated gradually according to the cascade structure.

This integrated approach realized the stepwise estimation of a complex market model,using a dependent chain of different risk factors. In addition, the inclusion of macroeco-nomic processes gives a straightforward interpretability of the framework.

To implement our immunization approaches, we use 1.000 scenarios over a time horizonof 30 years, whose market factors are generated by Risklab’s ESG. Following risk factorsare simulated to realize our implement:

• GDP

• Inflation Rate

• Treasury Yield Curve

• Credit Spreads

• Equity Market Index

• Property Index

• Hedge Fund Index

With the aid of the risk factors above, the market values of assets and liabilities can beobtained, which are characterized in the next two sections.


2.2 Assets

Although we are able to get the optimal allocation based on the immunization models,the performance and future price of asset instruments remains absent, which depends onseveral risk factors. The dependence on risk factors varies over asset classes. For example,the government bonds solely depend on interest rates, while stocks are modeled as riskfactors themselves, that are additionally influenced by other risk factors5.

To ensure diversification, the universe of possible instruments should be manifold. Therisklab asset universe is widely ranged. It consists not only different fixed income prod-ucts such as cash, government bonds, corporate bonds, inflation bonds, European Stocks,but also some alternatives, e.g., real estate, hedge funds and commodities. However, inthis study, to implement our immunization models, we limit our asset universe only ingovernment bonds. As a step forward, we will compare the results of our models with theresult of Markowitz optimization, which is implemented with 22 asset classes includingCorporate bonds, Government bonds, Inflation bonds, European Stocks, Real Estate andabsolute return. In addition, to demonstrate a direct comparison, we have implementedthe Markowitz optimization also with the asset universe of Government bonds, which wehave used for the immunization models.

Since how the performance and the future price of the asset classes are modeled exceedsthe range of this study, a brief summary table is shown as below:

Table 2.1: Summary Table of Asset ClassesAsset Class Benchmark

Stocks MSCI EuropeMid-Term Bonds Lehmann Aggregate (5Y-Duration)Long-Term Bonds Lehmann Aggregate (12Y-Duration)Real Estate GPR IndexAbsolute Return Funds HFR Fund of Funds

See Panten, 2006[54]

2.3 Liability

In this section, we characterize the nature and components of pension liabilities arisingfrom a defined benefit scheme, which could be seen as a loan granted by the employee tothe employer. As a background of the pension liabilities and our models, we will firstlygive an brief introduction to the pension plan and plan member. In the subsection 2.3.2the calculation of the pension liabilities will be demonstrated.

5See Panten, 2006[54]


2.3.1 Description of the Benefit Scheme

Benefit plays an important role not only in the entire pension plan but for our immu-nization models as well. Therefore, we introduce firstly the types of benefits. There aredifferent types of benefits, to which all members of the plans are entitled to:

• Old age pension is paid at the beginning of age 65.

• Disability pension is paid in case of disablement before age 65.

• Widow’s pension is paid in case of death of the spouse.

All benefits are life-long annuities paid on a monthly basis. For the simplicity purpose, weassume that the benefits are paid at the beginning of each year. The annual increase ofan annuity depends on current year inflation, while increasing in productivity and careeradvancement is assumed to be constant.

Respecting the pension plans, there are two different types of pension plans.

• The first pension plan is Final Pay Plan. The benefits are defined regarding thefinal salary. In this pension plan, no new members will be accepted.

• The other pension plan is Career Average Plan. The benefits are defined with regardto the average salary and new members are accepted in the pension plan.

For our immunization model, we use the liability data of the final pay plan and as popu-lations we take 10,000 persons, thereof:

• status:

– 4,000 active members

– 1,000 early leavers

– 3,000 old age pensioners

– 1,000 disabled

– 1,000 widows/widowers

• sex: 50% female, 50% male

• annual salary: 5 classes of employees

– 1% earn 200,000 monetary units (board member)

– 4% earn 100,000 monetary units (upper management)

– 15% earn 80,000 monetary units (middle management)

– 20% earn 60,000 monetary units (lower management)

– 60% earn 40,000 monetary units (blue collar worker)

The projection period is set to be 80 years. By means of economic scenario generator wemodeled 1.000 scenarios, by which our immunization models will be implemented.


2.3.2 Modeling of Defined Benefit Pension Liabilities

After the brief introduction to the benefit scheme we concentrate on the next question,namely which natures and components the pension liabilities have. Since pension liabilityare not exchange-traded, it is not straightforward to observe its market value. Moreover,liabilities are driven by

• Discount rate,

• Inflation,

• Biometric parameters such as mortality or morbidity,

• Dynamics in career advancement,

• Number and fluctuation of employees.

Hence we have to rely on actuarial studies, which makes assumptions about these sto-chastic input parameters.

The calculation of the liabilities can be realized in two steps. The first step is to cap-ture the firm structure in the future, using actuarial methods and taking variable currentstructure as an input. The firm structure is consisted in the number of the employeesand their situation (e.g., salary, age,. . . ). The actuarial methods is based on the mortalitytables, which show a person at each age, how the probability is that he dies before hisnext birthday. Since all these calculations are based on fixed inflation, interest rate, andso on, they are not scenario dependent.

A step forward is to bring the risk factors in the calculation of the liabilities. Since wehave got a possible picture of the firm structure for the next years, we can estimate thepension payments which will be made over the years.

The liabilities are expressed under the form of6:

• Benefits (Benefitsts)7: necessary pension payments (cash out). The employer must

pay the sum of the pension payment Benefitsts in time t to the people who areretired, disable and to the surviving dependants.

• Expected Benefits (eBenefitsts): expected value of the benefit at time step t, whichis calculated using pension actuarial foundations. That is, we use the mortality ta-bles and find out the probability of a given person to become disabled, retired orleaving a widow. Finally we multiply it by the pension payment that we have tomake in this case.

6See Casanovas, 2007[12]7The index ts means that this variable is on time t and scenario s dependent.


Figure 2.6: Evolution of the Different Quantiles of the Liabilities

(a) Benefits (b) DBO

(c) SC (d) IC

(e) eDBO (f) eBenefits



• Defined Benefit Obligation (DBOts): the present value of the expected future pay-ments requiring to be done due to the actual status of the employee.

• Service Cost (SCts): the actuarial estimate of benefits earned by employee servicein the period. In other words, it is the extra money the employer must pay for theDBO because the salary has grown.

• Interest Cost (ICts): Interest cost is the increase in the present value of the oblig-ation as a result that the benefit payments come closer in time and hence are lessdiscounted.

• Expected Direct Benefit Obligation (eDBOts): which is the value of the DBO int − 1 after making an actualization with the expected payments during the nextyear. For example, if inflation has unexpectedly changed and hence future salaryexpectations had to be adjusted, it would lead to a change in the DBO. In case ofright predictions and hence no gains or losses, the DBO develops as follows:

eDBOt = DBOt−1 + SCt + ICt −Benefitt (2.7)

In figure 2.6 we can see the result of different values of liabilities over the entire opti-mization period. In addition, one more parameters is needed during the procedure, theRemaining Working Life:

• the Remaining Working Life (RWL(t)): average time that an employee has to workuntil retirement.

2.4 Balance Sheet Projection

In the previous two sections we have described the modeling of assets and liabilities,which are consistent with the risk factors modeled by the economic scenario generator.Now we bring the values of assets and liabilities into play, that is, we take these valuesas an input into the balance sheet projection, which gives us the evaluation of our targetcriteria in the environment we are working. In order to model the balance sheet itemsprecisely, Risklab Germany have modeled two kinds of liabilities, one with regard to theaccounting (IAS19) and the other regarding the regulations of different states (Germany,Netherlands, Japan,. . . ). Because of the different inflation rates, expected salary trend andother factors the values of these two kinds of liabilities are different. For the immunizationmodels, which will be introduced in next chapters, we implement the models only for thebasic case, that is, we take only the basic regulatory environment into account. Thus, theother frameworks with respect to different state regulations will not be described in thisthesis.8

8For details reference to Panten, 2006 [54]


The balance sheet projection works as follows: Every scenario will be regarded separately.The initial values for the balance sheet items are given at t = 0. In each future periodfrom t to t + 1, the evolution of the assets and liabilities according to the risk factors ineach scenario will be also given as input parameters. Thus, at t = 1, based on the planasset values, PA and DBO values, all the relevant balance sheet items can be obtaineddirectly or through stepwise calculation. At t = 2 the same calculation will be iterated,till the end of the projection period. The most important calculations are the following:

• Calculation of the actuarial gains and losses due to the unexpected movements inliabilities:

agllt = eDBOt −DBOt (2.8)

• Calculation of the actuarial gains and losses due to the unexpected movements inassets:

aglat = ReturnPAt − eReturnPAt (2.9)

where ReturnPA is the return of the portfolio calculated using the Risklab’s sce-nario generator9 and eReturnPA the expected return of the portfolio (which is setbefore starting the simulation)

• Calculation of the total actuarial gains and losses:

aglt = aglat + agllt (2.10)

• Calculation of unrecognized actuarial gains and losses. Every year, new gains orlosses are coming into being. If gains or losses are already realized in the respectiveyear, they will be subtracted from the unrecognized actuarial gains and losses ofthis year:

uaglt = uaglt−1 + aglt − raglt (2.11)

where raglt denotes the recognized actuarial gains and losses.

• Calculation of the corridor, which defines the range, in which the unrealized gainsand losses may move without being booked in the profit and loss account.

corridort = 100% ·maxDBOT ; PAT (2.12)

9Risklab’s scenario generator delivers the values of a set of assets return for each time steps andscenarios


• Calculation of the recognized actuarial gains and losses. The actuarial gains andlosses should be offset one another during the long term, that’s the reason why thefirm is not obliged to recognize them as a balance sheet item. However, they will berecognized if they exceed an amount which depends on the liabilities:

raglt = sign(uaglt−1) ·max|uaglt−1| − corridor; 0

RWLt−1

(2.13)

• Calculation of the plan assets, which is in fact the amount invested in the pensionplan.

PAt = PAt−1 −Benefitst + ReturnPAt + Contributionst (2.14)

where Contributions denotes the cash flow transferred in the pension plan in eachtime period.

• Calculation of the Defined Benefit Liabilities (DBL), which remain in the balancesheet.

DBLt = DBOt + uaglt − PAt (2.15)

• Calculation of the pension expense, which will be shown in the profit and lossaccount.

PE = SCt + ICt − eReturnPAt − raglt (2.16)

2.5 Contribution Strategies

As shown in (2.14), the value of plan asset still depends on the value of contributions,which mean how much will be put in the fund by the firm, in order to continue withthe balance sheet. To calculate the value of the contributions, we need to get the ideaabout our Contribution Strategies. Contribution strategies are predefined decision rulesto calculate the value of contributions. The strategies can be divided into two steps, thefirst one where contributions are made according to a basic strategy, and the second stepwhere possible additional contributions are made according to the regulatory environment.

2.5.1 Basic Regulatory Environment

There is a new and relevant parameter during the process to calculate the contributions:namely Funding Level (FL), which is defined as follows:

FL =Plan Asset before contribution

DBO(2.17)


For our basic regulatory environment, there are no rules for the value of the funding level,and the contribution strategies are defined by three levels (4 intervals):

• Lower level (ll)

• Upper level (ul)

• Withdrawal level (wl)

The concrete contribution strategies are the following:

• If the current funding level is within the interval [ll, ul], then the normal contri-butions will be paid, which actually equals to the amount of service cost.

• If the current funding level is higher than the upper level, but lower than the with-drawal level, no contributions will be paid in this year, which is also called contri-bution holiday.

• If the current funding level even exceeds the withdrawal level, then the restitutionswill be paid such that the funding level after restitutions equals withdrawal level.

• However, if the current funding level is lower than the lower level, a recovery plancomes into effect. The gap of deficit (ll−FLt) ·DBOt will be divided by yrec, whichdenotes the number of years a recovery plan is supposed to be in place. In otherwords, it implies the number of years, so that the funding level goes back to reachthe lower level. This parameter is also predefined with other values, such as lowerlever, upper lever and withdrawal level. In the next year, the deficit will be dividedby yrec − 1, and so forth, in the last year of the recovery plan, the deficit will bedivided by one. As a result, the funding level is pushed to the lower level again.

The whole contribution strategies can be expressed using a step function:

Contributionsbasic(t) =((Withdrawal level − FL(t)) ·DBO(t), FL(t) > Withdrawal level;

0, Upper level < FL(t) < Withdrawal level;SC(t), Lower level < FL(t) < Upper level;

infContr(t), FL(t) < Lower level.

where

infContr(t) = min

(MaxContr(t);max

((Lowerlever − FL(t)) ·DBO(t)

yrec −BaseP lanY ears;SC(t)

))


With

BaseP lanY ears : the number of consecutive years that the firm has beenunderfunded.

MaxContr(t) : Withdrawal level ·DBO(t)− bPA(t),bPA is defined as the plan asset before contributions.

Straightforward we can see an example of contribution strategies from figure 2.7 withll = 90%, ul = 100% and wl = 140%.

Figure 2.7: An Example of a Contribution Strategy

If we keep working within the basic regulatory environment without considering the in-dividual regulatory framework, the funding level after contributions is expressed as:

FL(t) =bPA(t) + Contributions(t)

DBO(t)(2.18)

There is one point left which must be underlined. For our modeling approach, we assumethat the sponsor is not able or willing to pay any amount of contribution. So a upperbound of it is set to be 10% of the DBO. Therefore, the actual contribution at time t isthe minimum of the calculated contribution and 10% of the DBO.

2.5.2 Individual Regulatory Framework

According to different regulatory frameworks the contribution strategies will adjusted,e.g., the values of lower level, upper level, withdrawal level and the length of the recoveryplan. However, the immunization models introduced in this thesis will be implementedonly in the basic regulatory environment, on this account we will only give a brief intro-duction to the regulatory frameworks of individual states in this subsection.

Focussing on three primal points, we compare the contribution strategies among fivestates: UK, Germany, Netherland, Japan and USA. The result of the comparison is shown


Tab

le2.

2:In

div

idual

Reg

ula

tory

Fra

mew

orks

Definit

ion

ofFundin

gLevel

for

Fundin

gR

equir

em

ents

Underf

undin

gO

verf

undin

g

UK

liab

ility=

DB

Oll

=10

0%w

l=

105%

→ad

ditio

nal

contr

ibuti

on,y

rec

=10

liab

ilitie

sas

book

valu

e,w

hic

his

the

-If

fundin

gle

vel<

100%

ul=

120%

DB

Ow

ith

→im

med

iate

additio

nal

contr

ibuti

onw

l=

140%

Germ

any

-a

fixed

inte

rest

rate

of3.

5%so

that

FL

=10

0%-no

futu

resa

lary

/pen

sion

incr

ease

-If

fundin

gle

vel<

104.

5%→

additio

nal

contr

ibuti

on,y

rec

=3

liab

ility=

guar

ante

edpen

sion

righ

ts,

-P

robab

ility,

that

fundin

gle

velst

ill

ul=

130%

whic

his

DB

Obut

>10

5%on

eye

arla

ter,

must

be

Neth

erl

and

-no

futu

resa

lary

/pen

sion

incr

ease

hig

her

than

2.5%

ever

yye

ar.

-If

fundin

gle

vel<

105%

→ad

ditio

nal

contr

ibuti

on,y

rec

=1

liab

ility=

term

inat

ion

liab

ility,

whic

his

ll=

90%

ul=

150%

Japan

DB

Obut

→ad

ditio

nal

contr

ibuti

on,y

rec

=7

-in

tere

stra

te:20

-yea

rgo

vern

men

tbon

d-no

futu

resa

lary

/pen

sion

incr

ease

-cu

rren

tliab

ility

isth

eD

BO

but

-If

asse

ts/c

urr

ent

liab

ility

<90

%,

-If

asse

ts>

•no

futu

resa

lary

/pen

sion

incr

ease

→ad

ditio

nal

contr

ibuti

onnee

ded

max

90%

liab c

urren

t;U

SA

•dis

count

rate

isa

four

year

aver

age

whic

his

calc

ula

ted

asa

per

centa

ge10

0%liab a

ccrued

yie

ldof

30-y

ear

ofunfu

nded

curr

ent

liab

ilit

ies

→co

ntr

ibuti

on-ac

crued

liab

ility

isth

eD

BO

trea

sury

bon

dhol

iday

sSo

urce

:R

iskl

abG

erm

any,

“LD

IC

once

ptba

sed

onan

Inte

grat

edA

LM

App

roac

h”,20

06


in table 2.2. Apparently, there are more details of each regulatory framework than thatshown in this table. As mentioned, what in shown in table 2.2 is just an example.10

10More details reference to Casanovas, 2007[12]

Chapter 3

Immunization Approaches

After becoming acquainted with the entire environment and framework, which we areworking with, let us come to the main part of this study: Immunization Approaches.

In the narrowest sense matching-techniques are understood as the methods which aimat “similar making” of the structure on profit and loss sides. Thus a so-called immu-nization of liability with external inflows should be reached. First and foremost, interestrates are here taken into consideration, since the values of investments (particularly thefixed-interest instruments) and pension liability will change with an unscheduled changeof market interest level. In this chapter we will introduce the matching and immunizationstrategies, which are for static and deterministic purposes. That is, all the future variablesfor profit and loss accounts (e.g. liability cash flows, payments of bonds portfolio and thefuture interest rate) are already known today. With a case study we will simulate the sto-chastic and dynamic optimization in chapter 4. We will give an introduction to 5 differentimmunization approaches in the following sections of this chapter. Using examples willhelp us to performance these approaches and compare the results.

3.1 Linear Approaches to Immunization

Conceptually, the easiest form of immunization is cash flow matching. Cash flow matchingis a process of hedging, which bases on a deterministic analysis of cash flows over a specificperiod of time. That is, for a certain liability cash flows we are seeking after the mosteconomical portfolio, whose regular cash payments can cover the liability cash flows. Thisstrategy is based upon the assumption, that at the present time the cash flows on bothsides are known. Obviously this assumption is unrealistic in almost every supposable sit-uation; however it can be approximated with the expected values if the possible deviationis not far too large.

We begin the cash flow matching with an example. Supposedly, we are given a set ofcoupon bonds j = 1, . . . , J with different maturities ti, for i = 1, . . . , n. These bondscan be purchased at price Pj. To match the liability cash flows we need bonds 1, . . . , Jin quantities n1, . . . , nJ . We are also given the liability cash flows lti , which need to be

20

CHAPTER 3. IMMUNIZATION APPROACHES 21

matched as closely as possible. We formulate all the variables as follows:

P = (P1, P2, · · · , PJ)T (3.1)

l = (lt1 , lt2 , · · · , ltn)T (3.2)

n = (n1, n2, · · · , nJ)T (3.3)

C =

C1, t1 · · · · · · C1, tn

.... . .

......

. . ....

CJ, t1 · · · · · · CJ, tn

(3.4)

where C is the cash flow matrix of the bond portfolio. For simplicity we assume thatliability cash flows and bond payments coincide on date ti. Given a set of bond holdingsallows us to calculate the associated cash flows as well as the corresponding deviation ∆cfrom the required liability payments:

∆c = CT n− l (3.5)

where ∆c = (∆ct1 , ∆ct2 , . . . , ∆ctn)T and ∆cti is the deviation in period ti. Ignoring thatthe excesses will be reinvested the optimization approach can be generated as follows:

minn

P T · n (3.6)

CT · n ≥ L (3.7)

n ≥ 0 (3.8)

However, it is required that portfolio payments must be greater than liability cash flowsduring the whole period. The model is non-applicable in reality because a great numberof excesses have not been considered. In this case the replication cost would be far tooexpensive.

To reduce the replication cost we reinvest the excesses in the next year period with theforward rate rti . For this purpose we introduce a new definition cash balance with thenotation acti :

acti =

∆ct1 : ti = 1

∆cti + ∆cti−1

(1 + rti−1

): ti > 1

(3.9)

Generally ac = (act1 , act2 , . . . , actn)T ≥ 0, otherwise borrowing would be allowed, so thatwe must still consider the credit interest rate. Here we lay it down that borrowing is notallowed, i.e., ac ≥ 0.

To take a step forward we still need the forward rate matrix which arises from the forwardrates of each period. In an arbitrage free market it holds:

d(t, T1, T2) =d(t, T2)

d(t, T1)(3.10)


where d(t, T1, T2) is the forward discount factor of the period [T1, T2], which is measured attime t. With r(t, T1, T2) we denote the interest rate for the time interval [T1, T2] measuredat time t1. According to the forward discount factors the forward rates are said to be:

(1 + r(t, T1, T2))(T2−T1) =

1

d(t, T1, T2)

=d(t, T1)

d(t, T2)

(3.11)

Here we use the simplified notation rti instead of r(0, ti, ti+1). Since the length of eachperiod to be considered here is always 1, we are allowed to formulate rti as follows:

rti =d(0, ti)

d(0, ti+1)− 1 (3.12)

Finally we can define our forward rate matrix R shown as below:

R =

−1 0 · · · · · · 0

1 + rt1 −1...

0 1 + rt2 −1...

.... . . . . . 0

0 · · · 0 1 + rtn−1 −1

(3.13)

With consideration of the reinvestment possibility we can now formulate our cash flowoptimization approach as below:

minn, ac

P T · n (3.14)

CT · n + R · ac = l (3.15)

n ≥ 0, ac ≥ 0 (3.16)

As in (3.14) formulated our objective is to minimize the replication cost under replicationconstraint (3.15) and non-negativity constraints (3.16). The replication constraint (3.15)states that the liability cash flows should be matched with the payments from bondinvestments (CT · n) plus cash carried forward (R · ac). In other words: ∑J

j=1 njCj, t1 − act1 = lt1 : i = 1∑Jj=1 njCj, ti + acti−1

(1 + rti−1)− acti = lti : i ≥ 1

(3.17)

Let us illustrate (3.14) to (3.16) by means of an example. Then we will compare the resultsof the two approaches (with or without reinvestment of excesses) and see the advantage


Figure 3.1: Data of Bonds Payments and Liability Cash Flows

of the second approach.

Suppose we are given J = 8 coupon bonds with maturities ranging from t1 to t8. On theloss account we have to match the liability cash flows for 20 years. We are looking for anoptimal portfolio allocation with respect of the above mentioned optimization approachso that liability cash flows will be covered. Figure 3.1 shows us the data of the plan assetand the liability.

We can see the results with respect to the above introduced optimization approaches ontable 3.1. What we are most interested in is the replication cost. It is obvious that thereplication cost is much lower in the case of reinvestment excesses than in the other one,but the replication cost still remains to be huge, which implies the largest limitation ofcash flow matching. In addition, table 3.1 shows us we should invest 100% capital assetin bond 8, 20 years’ government bond, which is also logic. Because during the last 5 yearsthere will be only 20 years’ government bond available. Without the chance of reinvest-ment we must have enough 20 years’ bonds to cover the liabilities, so that its coupon cashflows during the first 15 years are already enough for the liability demand. On the otherhand, if we consider the possibility to reinvest we can allocate the whole capital asset toevery bond. The replication cost falls obviously down and we have reached even anothergoal: diversification.

1See Zagst, 2002[76]


Without reinvestment With reinvestmentAsset Class Maturity

Weight (in %) Quantity Weight (in %) Quantity

Bond 1 1 0 0 1.00 56,343Bond 2 2 0 0 1.16 64,990Bond 3 3 0 0 3.06 171,841Bond 4 5 0 0 4.09 229,330Bond 5 7 0 0 8.63 484,285Bond 6 10 0 0 24.48 1,371,556Bond 7 15 0 0 44.80 2,504,762Bond 8 20 100 16,380,212 12.78 712,988

Replication Cost 1,649,867,655 561,818,711

Table 3.1: Optimal Portfolio Allocation of Cash Flow matchingThis table shows us the results of optimal allocations of cash flow matching in case of noreinvestment and with reinvestment of excesses. It is obvious that the optimal portfoliowithout reinvestment possibility consists in only the 20-year bond, while the optimal port-folio with reinvestment possibility consists in all the available bonds. At the last row thepresent value of both optimal portfolios are shown.

Although the liability cash flows is perfect to be matched, the replication cost is unbeliev-able high. Because the universe of potential investments is constrained by the specificity oftiming requirements, this tends to be the most expensive of the five immunization strate-gies to implement. Furthermore, in the case of long-term liabilities, e.g. liability cash flowsof more than 30 years, problems arise because instruments with payment streams beyond30 years are relatively scare, which means the liability cash flows can never be perfectlyimmunized2.

3.2 Traditional Duration Matching

Since a certain risk of cash streams always exists, a perfect cash flow matching comes intoquestion usually just with very limited time horizon and special liability. As an alternativewe focus this time on the sensitivity to the interest rates risk. It demands that plan assethas the similar sensitivity to that of the liability, instead of a complete matching of cashflows, because fluctuations in interest rates are the main risk factor affecting the presentvalue of plan asset and liabilities. Duration and convexity are measures of how interestrate changes affect a fixed income portfolio. These same measure can be calculated for aliability stream. Duration is a linear measure of how the price of a bond changes in re-sponse to interest rate changes. As interest rates change, the price is not likely to changelinearly, but instead it would change over some curved function of interest rates. The morecurved the price function of the bond is, the more inaccurate duration is as a measure ofthe interest rate sensitivity. Convexity is a measure of the curvature of how the price of abond changes as the interest rate changes, i.e. how the duration of a bond changes as the

2Pimco [7]


interest rate changes. Specifically, duration can be formulated as the first derivative ofthe price function of the bond with respect to the interest rate. Then the convexity wouldbe the second derivative of the price function with respect to the interest rate. Therefore,both of the risk measures lead us to the most basic and common immunization strategy,the so-called duration matching.

To begin with the introduction of traditional duration matching, let us firstly get a clearidea about duration. There are two types of duration, Macaulay duration and modifiedduration. Macaulay duration, named for Frederick Macaulay who introduced the concept,is the weighted average maturity of a bond where the weights are the correspondingdiscounted cash flows in each period. Here the yield to maturity is used as the discountfactors. That is:

durationmac(y, TB, C) =

∑ni=1 Ti · C(Ti) · (1 + y(TB, C))−Ti

Bond(y, TB, C)

=n∑

i=1

Ti ·C(Ti) · (1 + y(TB, C))−Ti

Bond(y, TB, C)︸︷︷︸weight

(3.18)

where

n number of periodsC(Ti) Cash flow of period Ti

Bond(y, TB, C) Dirty Price of a Bond with maturity TB andcash flows C = (C(T1), . . . , C(Tn))

y(TB, C) Yield to Maturity of bond

Modified duration is an extension of Macaulay duration and is a useful measure of thesensitivity of a bond’s price (the present value of its cash flows) to interest rate movements.It is calculated as shown below:

durationmod(y, TB, C) = −∂∂y

Bond(y, TB, C)

Bond(y, TB, C)

=

∑ni=1 Ti · C(Ti) · (1 + y(TB, C))−Ti

Bond(y, TB, C) · (1 + y(TB, C))

=durationmac(y, TB, C)

1 + y(TB, C)(3.19)

It delivers an estimate how many percents of the liability or portfolio price will fall by 1%interest rate rising, which assumes, that the price is linearly dependent on the return y.For the new immunization approaches we use modified duration.3

3By traditional duration matching and the other duration matching models in this thesis, e.g. principalduration vector model, we use modified duration. Hence we name simply ‘duration’ instead of ‘modifiedduration’.


Figure 3.2: Bond Price Change by Interest Rate Movements

As shown in figure 3.2, the red curve implies plan liability to be immunized. If we justtake duration into consideration, we will get the black tangent line. After interest ratemoves from Y 1 to Y 2, the estimated dirty price of bond P2 lies under the actual valueP4. To shorten the funding gap convexity should also be considered, which brings us toour new optimization approach.

Now let us concretely introduce our new approach: traditional duration matching . Westart with the valuation of pension liability using fair-value calculation from last section.The present value of future liabilities is derived by using market-based discount factorsand can be calculated from figure 3.3.

l = lt1d(0, t1) + lt2d(0, t2) + · · ·+ ltnd(0, tn)

=n∑

i=1

ltid(0, ti)

= 561, 818, 711.42 (3.20)

Then we calculate the yield to maturity y. It is actually the solution of the equation below:

l − l(y) = l − lt11

(1 + y)ti− lt2

1

(1 + y)t2− · · · − ltn

1

(1 + y)tn= 0 (3.21)

The idea of duration replication is to approximate changes in the present value of ourliability stream from a (2nd order) Taylor approximation of l = l(y).4

∆l

l≈ −D∆y +

1

2Con(∆y)2 (3.22)

4See Scherer, 2005[63]


Figure 3.3: Valuation of Pension Liability The table shows sample liabilities from the liabilitycube generated by risklab Germany GmbH.

where D and Con denote the duration and convexity. They are calculated as shown below:

D =δl

δy

1

l=∑

i

lti1

(1+y)ti+1

l· ti = 10.4392 (3.23)

Con =d2l

dy2

1

l=∑

i

lti1

(1+y)ti+2

l· ti(ti + 1) = 150.2808 (3.24)

The value of D = 10.44 means that pension liability are expected to fall by 10.44%, for a100 bp (1%) increase in the discount rate.

In the same way we can calculate duration and convexity of bonds. The duration of aportfolio is composed of the weighted average duration of individual bonds in this port-folio. It means, that the modified duration of individual bonds must be weighted by therelated percentaged portfolio proportion.

Finally we can formulate our traditional duration approach as follows5:

5In some literatures, this approach is called ‘convexity-matching’ instead of ‘duration-matching’


minω

(ConB · ω − ConL)2 (3.25)

DB · ω = DL (3.26)∑i

ωi = 1 (3.27)

0 ≤ ωi ≤ 1 (3.28)

where

ω Weight of portfolioDl Liability duration

Conl Liability convexityDB = (DB1 , DB2 , . . . , DBJ

)T Duration vector of J bondsConB = (ConB1 , ConB2 , . . . , ConBJ

)T Convexity vector of J bonds.

Function (3.25) describes our goal, that we minimize the deviation between bonds portfo-lio convexity and that of liability under the constraint (3.26),(3.27) and (3.28). Actually,the deviation is the distance between P3 and P4 in the figure 3.2. As formulated in (3.26),we search for the optimal allocation among the portfolios, which have the same durationas the liability. This time we get the weight directly from the optimization approach in-stead of the quantities of bonds, which happens in cash-flow-matching. From the table3.2 we observe the entire valuation of the bond portfolio. The optimal allocation of the 8bonds stands in the last column of the table. The result shows us that to get the optimalallocation, we should invest the greatest part into 10 years’ and 20 years’ bonds.

Result Maturity Duaration Convexity Weight

Bond 1 1 0.9699 0.9699 0%Bond 2 2 1.9027 3.7736 0%Bond 3 3 2.7974 8.2610 0%Bond 4 5 4.4779 21.7223 0%Bond 5 7 6.0230 40.3142 6.79%Bond 6 10 8.1101 75.8551 46.63%Bond 7 15 11.0496 149.2852 0.15%Bond 8 20 13.4223 232.5599 46.43%

Table 3.2: Duration and Convexity of BondsIn this table we get the valuation for the bonds. The third and the fourth column tell usduration and convexity of each bond. In the last column we can see optimal allocation ofthe optimal portfolio for the traditional duration matching approach.

The next step is to check the cash flow mismatch. Each mismatched cash flow exposesthe investor to real-world reinvestment risk. We can calculate the portfolio cash flow asfollowing: Since we can merely get the optimal allocation from the optimization model,


we still need an entire amount that we invest in this portfolio. For example, We invest thistime the amount equal to the present value of liability, which will be multiplied by theoptimal allocation. Then we get the amount how much we should invest into each bond.Dividing them by the bonds cash prices allow us to know the quantity of each bond to bepurchased. What is left to do is just to multiply the quantities by the coupon cash flows.Finally we get the portfolio cash flow for 20 years. From figure 3.4 the mismatching canbe observed directly.

Figure 3.4: Liability Cash Flow versus Portfolio Cash Flow

However in reality, the benefit streams are always more than 20 years in any pension plan,e.g. 50 years or even 80 years. Then duration of liability would be much greater. If thereare only coupon bonds with longest maturity of 30 years available, then the above formu-lated optimization approach would be infeasible. In this case we formulate our approachshown as below:

minx

∥∥∥∥ D TB · ω −Dl

Con TB · ω − Conl

∥∥∥∥ (3.29)∑i

ωi = 1 (3.30)

0 ≤ ωi ≤ 1 (3.31)

That is, we minimize the deviation of portfolio duration and convexity against liabilityduration and convexity at the same time. Now we have for example 50 years’ benefits. Theliability duration is getting greater than before. It is simple to imagine that we must investa greater part into 30 years’ bond, which builds a so-called barbell portfolio(investing atthe “ends” of the yield curve, while leaving the medium-term sector)6. The portfolio cashflows in the 30th year will be many times greater than liability cash flow. If rates drop,receiving a huge cash flow and reinvesting it at lower rates will not suffice to cover the



future benefits.

However the whole traditional duration model is based on the assumption of a paralleland infinitesimal shift in a flat yield curve. Not only infinitesimal but also parallel shifts inthe yield curve are inconsistent with arbitrage-free term structure dynamics, such shiftsoccur rarely in the bond markets. Typically the interest rates with short maturity fluc-tuate much more intensive than those with long maturity and the both don’t move inthe same direction. Even under slight violation of the assumption of parallel yield curveshifts, the duration will lose its quality as a risk measure. Also Considering the measure ofconvexity can not help anything this time, because its construction is based on the sameassumption as duration. Therefore, by a non-parallel rate movement convexity will loseits unique character to reflect bonds’ or liability feature correctly.

3.3 Generalized Duration Vector Model

Since the traditional duration model can only be applied properly based on the assump-tion of a parallel and infinitesimal shift in a flat yield, this model has in reality very limitedapplicability. To immunize a liability portfolio with respect to the interest rate risk exposemeasures for non-infinitesimal, nonparallel yield curve changes under non-flat-yield curve,we introduce here a new immunization model:Generalized Duration Vector Model . It iscalled vector model because it needs to calculate a vector of higher order duration mea-sures. The generalized duration vector model is actually an expansion of the traditionalduration vector model, which seems to be more effective in protecting against immuniza-tion risk than the traditional duration vector model, without increasing the vector length.However, we demonstrate the application not only of the traditional duration vector modelbut of the generalized duration vector model as well, when the term structure is estimatedusing the exponential functional form of the Nelson and Siegel (1987) model. Therefore,we will firstly give an introduction to how to estimate the term structure of instantaneousforward rate by using the Nelson and Siegel model. Before introducing the generalizedduration vector model we will in section 3.3.2 introduce the traditional duration vectormodel at first and in subsection 3.3.3 we come to our new and better immunization model:generalized duration vector model.

3.3.1 Nelson and Siegel Model

Because of lack of data, since 1997 the German Central Bank has used a calculationmodel for zero coupon bonds, so that the term structure of interest rates can be esti-mated from the observable price of coupon bonds. The estimation model is based on theadvice of Nelson and Sigel, who used a single exponential functional form over the en-tire maturity range. The starting point of this model is the assumption that there is afunctional relationship between two sides, on one of which are zero coupon rates, forwardrates and discount factors and on the other is the remaining time to maturity. Over more


complicated alternatives, the Nelson-Siegel model has many advantages, for example, itis a so-called parsimonious model, which means that there are only a few parameters toestimate. However the most significant advantage of the model is that, with respect tolong time observation, the Nelson-Siegel model has a better asymptotic behavior of theterm structure than the other models, e.g. cubic-spline model. There is a more complexversion of the Nelson-Siegel model, which was suggested by Svenson (1995), and we willalso give a short introduction to it.

Nelson and Siegel suggest a parameterization of the instantaneous forward rate curvegiven as follows:

f(t) = α1 + α2e− t

β + α3t

βe−

tβ (3.32)

where α = (α1, α2, α3, β)T is the vector of four parameters to be used.

There is a nice relationship between zero-coupon rates s(t) and forward rates f(τ):

(1 + s(t))t =t∏

τ=1

(1 + f(τ)). (3.33)

Now the term structure of zero-coupon rates can be derived from the term structure ofinstantaneous forward rates by using the approximation ln(1 + x) ≈ x:

t · ln(1 + s(t)) =t∑

τ=1

ln(1 + f(τ)) (3.34)

s(t) =1

t

t∑τ=1

f(τ) (3.35)

It implies that the zero-coupon rate for term t is an average of the instantaneous forwardrates beginning from term 0 to term t, which suggests that forward rates should be morevolatile than zero-coupon rates, especially at the longer end. Then it follows:

s(t) =1

t

∫ t

0

f(τ) dτ

=1

t

∫ t

0

α1 + α2e− τ

β + α3τ

βe−

τβ dτ

=1

t

[α1τ + α2 · −β · e−

τβ + α3 ·

τ

β· −β · e−

τβ + α3 · −β · e−

τβ

]t

0

=1

t

[α1t− α2βe−

tβ + α2β − α3te

− tβ − α3βe−

tβ + α3β

]= α1 + α2

[1− e−

tβ

tβ

]+ α3

[1− e−

tβ

tβ

− e−tβ

](3.36)


The advantage of the representation above is that we can easily give economic interpre-tations to the parameters α1, α2, α3 and β:7

• Because limt→0 s(t) = limt→0 f(t) = α1 +α2, α1 +α2 is the instantaneous short rate.

• The loading on α1 is 1, a constant that doesn’t depend on the maturity. Thus α1

affects yields at different maturities equally and hence can be regarded as a levelfactor. Since limt→∞ s(t) = limt→∞ f(t) = α1, α1 is the consol rate and gives alsothe asymptotic value of the term structure of both the zero-coupon rates and theinstantaneous forward rates. Therefore, α1 > 0.

• The loading associated with α2 is (1− etβ )/ t

β, which start at 1 but decays monoton-

ically to 0. Thus α2 affects primarily short term yields and hence changes the slopeof the term structure of zero-coupon rates as well as the term structure of forwardrates; from another point of view, −α2 is the spread between the consol rate and theinstantaneous short rate, which gives another reason why α2 can be interpreted asthe slope of the term structure. Furthermore, from the two points mentioned abovewe can obtain, that α2 < 0 for a normal yield curve.

• α3 has loading (1− e−tβ )/ t

β− e−

tβ , which starts at 0, increases, and then decays to

0. Thus α3 has largest impact on medium-term yields and hence moves the curva-ture of the yield curve. When α3 > 0, the term structure attains a maximum valueleading to a concave shape; and when α3 < 0, the term structure attains minimumvalue leading to a convex shape.

• β > 0, is the speed of convergence of the term structure toward the consol rate. Alower β value accelerates the convergence of the term structure towards the consolrate, while a higher β value moves the hump in the term structure closer to longermaturities.

In this case the discount factor associated with the term structure in (3.36) is given as:

d(t) = e−t·s(t)

= e−α1t−β(α2+α3)(1−e− t

β )+α3te− t

β(3.37)

7Exactly speaking, the representation above was reformed by Diebold and Li (2003) based on theoriginal Nelson-Siegel expression.


To obtain the parameters in equation (3.37) we minimize the squared difference betweenactual market prices and model prices:

minα1, α2, α3, β

∑i

(di − e−α1t−β(α2+α3)(1−e

− tβ )+α3te

− tβ

)2

(3.38)

where di this time denote the market price of zero coupon bonds. Substituting the func-tional form of (3.37) into the pricing formula for a coupon bearing bond allows us to getthe theoretical price of coupon bond with maturity T :

P (T ) =T∑

i=1

CFid(i)

=T∑

i=1

CFie−α1i−β(α2+α3)(1−e

− tβ )+α3ie

− tβ

(3.39)

where P (T ) denotes the theoretical price of coupon bond with maturity T and CFi denotesthe cash flow happening in period i. To optimize the parameter α = (α1, α2, α3, β)T

minimize this time the squared difference between market price of coupon bonds j =1, . . . , J and theoretical price P (T ) in almost the same manner as in (3.38):

minα1, α2, α3, β

J∑j=1

(Pj −

mj∑i=1

CFije−α1i−β(α2+α3)(1−e

− iβ )+α3ie

− iβ

)2

(3.40)

Since the discount factor calculation equation (3.37) and the bond pricing equation (3.38)are nonlinear functions, the four parameters are estimated by using a nonlinear opti-mization technique. To nonlinear optimization problems, the optimal results are usuallysensitive to the starting values of the variables. So it is important to choose reasonableand logical starting values:

• The initial value for the parameter α1, which indicates the asymptotic value of theterm structure of zero coupon rates, may be set as the yield to maturity of thelongest bond in the sample.

• The starting value for the second parameter α2, which is the spread between theinstantaneous rate and the consol rate, may be set as the difference between theyield to maturity of the shortest maturity bond and the longest maturity bond inthe sample, since α2 < 0.

• It is difficult to set initial values for α3 and β, which are usually associated with thecurvature of the term structure and the speed of convergence towards the consolrate. Therefore, it suggests to consider a set of different values. A feasible rangeof α3 for realistic term structure data can be [−10, 10], and β may range from theshortest maturity and the longest maturity in the sample of bonds8.

8See Nawalkha, 2005 [50]


The choice of initial values of parameters is not unique. For example, it also suggests thatthe first parameter α1 may be set as the average yield to maturity of the three bondswith the longest maturities in the sample; and the starting value of α2 may be set as sucha value, that α1 + α2 equals the average yield to maturity of the three bonds with theshortest maturities.

As an example we use the data of discrete interest rates from the example used in lastsection. As the initial value of parameter α1, we set it as the yield of 20-year zero couponbond, which is equal to 0.0417 according to our example. The starting value of α2 willbe set as the difference of the yield between 1-year zero-coupon bond and 20-year zero-coupon bond, which is equal to −0.0107. After trying a set of different values for α3

and β we set the initial values as −0.0005 and 2. Then the initial value of the entireparameter vector α = (α1, α2, α3, β)T is set as (0.0417, −0.0107, −0.0005, 2)T . Figure3.32 shows us the result of the problem (3.38) with the optimal value of parameter vectorα = (0.0421, −0.0134, −0.0001, 1.9997)T .

Figure 3.5: Discount FunctionZero bond price (discount factors) have been fitted according to (3.38). Parameter areα1 = 0.0421, α2 = −0.0134, α3 = −0.0001, β = 1.9997.

Svensson has expanded the model of Nelson and Siegel by using two new parameters andan exponential term in addition. Although using two more parameters usually impliesthat the quality of the result can be improved, it can also cause that calculation cost willbe heavily raised. In this case it should be considered on one hand how better this modelwill be than the model of Nelson and Siegel and on the other hand whether the modelcan be implemented at all, which means it could cause calculation error by some startingvalues and then the results could be unreasonable. Now the term structure of forward


rate is formulated as follows:

f(t) = α1 + α2e− t

β1 + α3

[(

t

β1

)e− t

β1

]+ α4

[(

t

β2

)e− t

β2

](3.41)

where α = (α1, α2, α3, α4, β1, β2)T is the new parameter vector.

From the term structure of forward rate we can get the term structure of zero couponrate again:

s(t) =1

t

∫ t

0

f(τ) d

= α1 + α2

[1− e

− tβ1

tβ1

]+ α3

[1− e

− tβ1

tβ1

− e− t

β1

]

+α4

[1− e

− tβ2

tβ2

− e− t

β2

](3.42)

Although using Svensson model an exacter result can be carried out, due to stability theNelson-Siegel model will be preferred in practice. Since fewer parameters have been usedin Nelson-Siegel model, it seems to be more robust and stable.

3.3.2 Traditional Duration Vector Model

In this section we introduce the new immunization model: traditional duration vectormodel, which captures the interest rate risk under nonparallel and noninfinitesimal shiftin a nonflat yield curve. As mentioned, a vector of risk measures needs to be calculatedfor this model, which capture the riskiness of fixed-income securities. Generally, the firstthree to five duration vector measures are sufficient to capture all of the interest rate risk.

To derive the duration vector model, we start with describing changes in liability values(∆l). Let the continuously compounded term structure of instantaneous forward ratesbe given by f(t) as in the last section. Then an instantaneous shift in the forward ratesfrom f(t) to f ′(t) can be obtained as f ′(t) = f(t) + ∆f(t). The instantaneous percentagechange in the current value of the pension liability is given as:

∆l

l=−Dl

1 [∆f(0)]

−Dl2

[1

2

(∂ (∆f(t))

∂t− (∆f(0))2

)]t=0

−Dl3

[1

3!

(∂2 (∆f(t))

∂t2− 3∆f(0)

∂ (∆f(t))

∂t+ (∆f(0))3

)]t=0

...

−DlM

[1

M !

(∂M−1 (∆f(t))

∂tM−1+ · · ·+ (∆f(0))M

)]t=0

(3.43)


where

Dlm =

∑iwti · tmi , for m = 1, 2, . . . , M, and (3.44)

wti =ltid(ti)∑i ltid(ti)

=ltid(ti)

l(3.45)

Note, that in this framework d(ti) = e−R ti0 f(s) ds. Equation (3.43) expresses that the per-

centage change in liability value as a product of a duration vector and a shift vector. Theelements of the duration vector are defined by (3.44) and (3.45), which are linear in theinteger powers of the maturity of the cash flows of the liability, i.e., t, t2, t3. The first andthe second elements of the duration vector can be regarded as the traditional durationand convexity of the liability, but with considering of instantaneous forward rate changes.To approximate the percentage changes of liability we take the first 3 elements of theduration vector and the shift vector. The first shift vector element describes the changein the level of the forward rate curve for the instantaneous term, given by ∆f(0); thesecond shift vector element describes, as to be see, the difference between the square ofthis change mentioned above and the slope of the change in the forward rate curve, whichis given by ∂f(t)/∂t at t = 0; finally, the third element of the shift vector captures thethird power of the change in the level of the forward rate curve, the interaction betweenthe change in the level and the slope of the forward rate curve, and the curvature of thechange in the forward rate curve, which is given by ∂2∆f(t)/∂t2 at t = 0.

The next question is how to use the traditional duration vector model to find a optimalportfolio which can mimic the liability. Supposing, we are given the liability durationvector with elements Dl

1, Dl2, . . . , Dl

M . To replicate the duration vector of pension liabilityas closely as possible there are k = 1, . . . , K bonds available. The duration vector of bondportfolio is to be obtained as the weighted average of the duration vectors of individualbonds, where the weights are defined as the proportions of investment of the bonds in thetotal portfolio. Thus, a portfolio is matched if the following conditions are satisfied:

DP1 = w1D

B11 + w2D

B21 + · · ·+ wKDK

1 =∑K

k=1 wkDBk1 = Dl

1

DP2 = w1D

B12 + w2D

B22 + · · ·+ wKDK

2 =∑K

k=1 wkDBk2 = Dl

2

· · · = · · ·DP

M = w1DB1M + w2D

B2M + · · ·+ wKDK

M =∑K

k=1 wkDBkM = Dl

M

(3.46)

w1 + w2 + · · ·+ wK = 1 (3.47)

Since equation (3.46) together with equation (3.49) define M + 1 constraints, the opti-mization approach can be formulated depending on different cases as follows:

• K = M + 1. Since the rank of the augmented matrix equals to the number of thevariable, there might exist a unique solution, But it still depends on whether shortsales are allowed or not. If we constrain holdings to be positive (wk > 0), there


might exist no exact solution.

• K < M +1. No solution exists. But this case happens rarely. Usually the number ofthe bonds is larger than the number of the duration vector elements to be considered,since normally three to five elements are already sufficient. However, if this caseoccurs, Scherer (2005) suggests us to define the objective function alternatively thatthe function minimizes the weighted average of duration vector deviations. That is:

minw1,w2,...,wK

M∑m=1

θmΥ(DPm −Dl

m) (3.48)

where Υ(· · · ) denotes such functions like quadratic of absolute deviation and θm

defines the weight given to a particular deviation.

• K > M +1. In this case many solutions exist and this is the most common case. Toselect a unique solution in this case, we optimize the following quadratic function:

minw1,w2,...,wK

K∑k=1

w2k (3.49)

The objective function (3.49) is used for achieving diversification across all bonds,and reduces bond-specific idiosyncratic risks (e.g., liquidity risk) that are not cap-tured by the systematic term structure movements.

To check the adaptability we use the values of the example from the last sections. Previ-ously we have used Nelson and Siegel model to calculate the theoretic values of discountfactors and figure 3.3.3 shows the entire valuations of the liability site. Note, that weconsider here only the first elements of duration vectors. In the same way we can alsocalculate the duration vectors of 9 bonds which can be obtained in table 3.3.

Table 3.3: Traditional Duration Vector for Tracking UniverseBonds Maturity D1 D2 D3 Weight

1 1 1 1 1 4.19%2 2 1.97 3.90 7.77 0.00%3 3 2.90 8.55 25.46 0.00%4 5 4.65 22.53 110.94 0.00%5 7 6.25 41.83 286.23 5.50%6 10 8.41 78.68 760.40 21.39%7 15 11.46 154.63 2197.73 39.73%8 20 13.91 240.53 4464.86 21.19%


Obviously, we have more bonds than the number of the duration vector elements. How-ever, as mentioned, we can formulate the objective function as

∑Kk=1 w2

k and the entireoptimization approach can be captured as follows:

minw1,w2,...,w8

8∑k=1

w2k

w1DB11 + w2D

B21 + · · ·+ w8D

B81 = Dl

1

w1DB12 + w2D

B22 + · · ·+ w8D

B82 = Dl

2 (3.50)

w1DB13 + w2D

B23 + · · ·+ w8D

B83 = Dl

3

w1 + w2 + · · ·+ w8 = 1

0 ≤ wk ≤ 1

The result of the optimal allocation is shown in the last column of table 3.3.

3.3.3 Generalized Duration Vector Model

As we know, increasing the length of duration vectors can improve the immunization per-formance. However, nowadays it tends to produce the portfolio holdings which is producedby the duration vector of a short length. As a result, the portfolio becomes increasinglyexposed to nonsystematic risks and incurs high transaction costs. That is the reason whygeneralized duration vector model will be proposed. Instead of increasing the length of theduration vectors, generalized duration vector model seems to be more effective in protect-ing against immunization risk than the traditional duration vector model introduced inthe last section9.

Firstly, we will give the expression of generalized duration vector model, that is:

− ∆V0

V0

≈ D∗(1)Y ∗1 + D∗(2)Y ∗

2 + · · ·+ D∗(M)Y ∗M (3.51)

where

D∗(M) =

tN∑t=t1

wt · g(t)m, for m = 1, 2, . . . , M (3.52)

and

wt =

[CFt

eR t0 f(s) ds

]/V0 (3.53)

For generalized duration model the risk measures are defined as D∗(1), D∗(2), . . . , D∗(M),whose expressions are similar to the traditional duration risk measures, except that theweighted averages are computed with respect to g(t), g(t)2, g(t)3, and so on, instead oft, t2, t3, and so on. Now we will show how to obtain the expression of the generalized

9See Nawalkha,2003 [50]


duration vector model in (3.51).

We begin with our liability portfolio. Supposing, that at time t = 0 the liability portfoliohas CFt as the payment to maturity t and t = (t1, t2, . . . , tN)T . f(t) denotes the continu-ously compounded term structure of instantaneous forward rates, then an instantaneousshift in the forward rates from f(t) to f ′(t) can be expressed as ∆f(t) = f ′(t)− f(t). Thereturn on the portfolio between t = 0 and t = H is given as:

R(H) =V ′

H − V0

V0

(3.54)

where V0 is the value of the liability portfolio at time 0 before the forward rate shifts:

V0 =

tN∑t=t1

CFte−R t0 f(s) ds (3.55)

and V ′H is the value of the liability portfolio at time H after the forward rate f(t) shifts

to f ′(t):

V ′H =

tN∑t=t1

CFte−R t

H f ′(s) ds (3.56)

Then the equation in (3.54) is equivalent to:

R(H) =eRH0 f(s) ds

[∑tNt=t1

CFt · e−R t0 f(s) ds · k(t)

]− V0

V0

(3.57)

withk(t) = e−

R tH ∆f(s) ds (3.58)

Using a change of variable let the forward rate function f(t) be represented by a chainfunction given as:

f(t) = h (g(t)) (3.59)

where g(t) is a continuous differentiable function of t. We assume that g(t) is monotonicand thus the inverse function of g(t) exists, which is given as:

t = g−1(η) = q(η) (3.60)

Now the instantaneous changes in the forward rate function can be given as:

∆f(t) = ∆h (g(t)) (3.61)


Using equation (3.60) and (3.61), we have:

k(t) = e−R t

H ∆f(s) rmds = e−R g(t)

g(H)p(γ) dγ (3.62)

where

p (γ) = p (g(t)) = ∆h (g(t))∂ (q(η))

∂g(3.63)

Doing a Taylor series expansion of γ around g(H), k(t) can be represented as:

k(t) ≈ 1− [g(t)− g(H)] · [p(g) · φ(g)]g=g(H)

− 1

2[g(t)− g(H)]2 ·

[(∂p(g)

∂g− (p(g))2

)· φ2(g)

]g=g(H)

− 1

3![g(t)− g(H)]3 ·

[((p(g))3 − 3p(g)

∂p(g)

∂g+

∂2p(g)

∂g2

)· φ3(g)

]g=g(H)

(3.64)

· · ·

− 1

M ![g(t)− g(H)]M ·

[((−1)M+1 (p(g))M + · · ·+ ∂M+1 (p(g))

∂gM+1

)· φM(g)

]g=g(H)

where

φ(g) = eR g(H)

g(t)p(γ) dγ = 1 (3.65)

when g(t) = g(H). Thus, equation (3.64) can be written in a simplified form as:

k(t) ≈ 1 +M∑

m=1

[g(t)− g(H)]m · Z∗m (3.66)

where

Z∗1 = −p(g(H))

Z∗2 = −1

2·[∂ p(g)

∂g− [p(g)]2

]g=g(H)

Z∗3 = − 1

3!·[(p(g))3 − 3p(g)

∂ p(g)

∂g+

∂2 p(g)

∂g2

]g=g(H)

(3.67)

...

Z∗M = − 1

M !·[(−1)M+1 (p(g))M + · · ·+ ∂ p(g)

∂gM−1

]g=g(H)


From equation (3.63) we can see that the value of p(g) depends on the change in theforward rate function. Thus, if there is no change of the forward rate, p(g(H)) will be zeroand then Z∗

m = 0 for all m = 1, 2, . . . , M . In this case, k(t) = 1, and the return on theportfolio is riskless, which actually can be expressed as:

RF (H) = eRH0 f(s) ds − 1 (3.68)

If forward rates change, the liability portfolio return can be obtained by substituting(3.66) into equation (3.57), which gives:

R(H) = RF (H) + [1 + RF (H)] ·M∑

m=1

M∗mZ∗m + ε (3.69)

where ε is the error due to higher order Taylor series terms and RF (H) is the risklessreturn defined in (3.68) and M∗m hat the form:

M∗m =

∑tNt=t1

CFt · e−R t0 f(s) ds [g(t)− g(H)]m

V0

(3.70)

for all m = 1, 2, . . . , M .

Now the generalized shift vector elements can be defined for m = 1, 2, . . . , M as:

Y ∗m = (1 + RF (H)) · Z∗

m (3.71)

where

Y ∗1 = − (1 + RF (H)) · p(g(H))

Y ∗2 = −1

2· (1 + RF (H)) ·

[∂ p(g)

∂g− [p(g)]2

]g=g(H)

Y ∗3 = − 1

3!· (1 + RF (H)) ·

[(p(g))3 − 3p(g)

∂ p(g)

∂g+

∂2 p(g)

∂g2

]g=g(H)

(3.72)

...

Y ∗M = − 1

M !· (1 + RF (H)) ·

[(−1)M+1 (p(g))M + · · ·+ ∂ p(g)

∂gM−1

]g=g(H)

If H tends to 0, the equation (3.69) is reduced to the generalized duration vector modelgiven by equation (3.51):

R(0) =∆V0

V0

≈ D∗(1)Y ∗1 + D∗(2)Y ∗

2 + · · ·+ D∗(M)Y ∗M (3.73)


The traditional duration vector introduced previously can also be obtained as a specialcase by setting g(t) = t.

Since we have explained how the expression (3.51) is captured, the next question is howto choose the continuous differentiable function g(t). As a polynomial class of generalizedduration vectors, g(t) = tα seems quite suitable, since it is continuous, differentiable andmonotonic. However, there is a wide choice of the value of α. Not surprisingly, differentvalues of α lead to different portfolio weights. So which value shall we take for our opti-mization approach? Nawalkha, Soto and Zhang (2003) have examined the immunizationperformance of the generalized duration vector model corresponding to g(t) = tα with sixdifferent values of α: 0.25, 0.5, 0.75, 1, 1.25 and 1.5. Different lengths of the generalizedduration vectors (i.e., the value of M ranging from 1 to 5) are used to test which functionalform converges faster. As a result, the lower α generalized duration strategies significantlyoutperform higher α strategies, when three to five risk measures are used. Therefore, weuse α = 0.5 and M = 3 to perform the model with our example.

The optimization approach is similar to that of traditional duration vector model, exceptusing g(t) instead of t, which is shown as follows:

minω1,ω2,...,ωK

K∑k=1

ω2k (3.74)

DPm =

K∑k=1

ωk ·Dkm = Dl

m m = 1, . . . ,M (3.75)∑kωk = 1 (3.76)

0 ≤ ωk ≤ 1 (3.77)

with Dkm =

∑tNt=t1

wt · (tα)m.

Table 3.4 the generalized duration measures of liability and bonds portfolio with α = 0.5:

In order to compare the results of traditional duration vector model and generalizedduration vector model with alpha = 0.5, we set the optimal allocation together in table3.5 and in figure 3.3.3 we compare the cash flows of both models.

3.4 Key Rate Duration Model

Recently, a new class of models, which is called “key rate duration model” has becomepopular among practitioners. Similar to the duration vector models, key rate durationscan manage interest rate risk exposure arising from arbitrary nonparallel shifts in the termstructure of interest rates. The duration vector models hedge against the shape changes in


Duration Vector D1 D2 D3

Liability 3.1294 10.8006 39.5856Bond 1 1 1 1Bond 2 1.4006 1.9671 2.7683Bond 3 1.6960 2.8971 4.9712Bond 4 2.1365 4.6454 10.2010Bond 5 2.4651 6.2520 16.1020Bond 6 2.8374 8.4184 25.5702Bond 7 3.2692 11.4610 41.6745Bond 8 3.5614 13.9103 57.0571

Table 3.4: Generalized Duration Vector of Liability and BondsWe choose α = 0.5 to perform the generalized duration vector duration model and considerthe first three duration vector elements. In the first row is the duration vector of liability,while in the following rows are the duration vectors of 8 bonds we have.

Table 3.5: Optimal Allocations Comparison between Two Duration Vector Models

Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Bond 6 Bond 7 Bond 8Maturity 1 2 3 5 7 10 15 20

Weight (traditional) 4.19% 0% 0% 0% 5.50% 21.39% 39.73% 29.19%Weight (generalized) 2.61% 0% 0% 0.56% 9.70% 21.56% 32.37% 33.19%

the height, slope and curvature, while the key rate durations hedge against the changes ina finite number of key rates that proxy for the shape changes in the entire term structure.10

The basic idea behind the key rate duration model is that any smooth (zero coupon)yield curve change can be modeled as a linear combination of a much smaller set of yieldchanges. These zero rates are also called key rates. The key rate duration model describesthe shifts in the term structure as a discrete vector representing the changes in the keyzero-coupon rates of various maturities. Interest rate changes at other maturity can bederived from these values via linear interpolation. Key rate durations are then defined asthe sensitivity of the portfolio (liability) value to key rates at different points along theterm structure.

However, unlike the duration vector models, where the higher order duration measuresserve as linear as well as nonlinear risk measures, the key rate durations give only the lin-ear exposures to the key rates. To measure nonlinear exposures to the key rates, key rateconvexity measures are required, which are however hardly ever used. On this account weimplement our key rate duration model only with respect to key rate duration.

To introduce the new optimization approach we start with the simplest case, where weassume that we have as many key rates as the liability cash flows we face. Further, weassume that the maturity of key rates and the liability cash flows perfectly coincide. Then,

10See Nawalkha, 2005[50]


Figure 3.6: Cash flows ComparisonWe compare the cash flows of liability, traditional duration vector model and generalizedduration vector model.

the exact present value of corporate liability can be formulated as the sum of n discountedcash flows, which happen at maturities t1, t2, . . . , tn:

l = l(st1 , st2 , . . . , stn)

= lt1d(0, t1) + lt2d(0, t2) + · · ·+ ltnd(0, tn)

=lt1

(1 + st1)t1

+lt2

(1 + st2)t2

+ · · ·+ ltn(1 + stn)tn

(3.78)

Then any smooth change of liability present value can be represented as a linear combi-nation of yield changes in the key rates:

∆l

l≈ −KRDl

1∆s1 −KRDl2∆s2 − · · · −KRDl

tn∆stn

= −∑n

i=1KRDl

ti∆sti (3.79)

KRDlti

= − δl

δsti

1

l= ti

lti(1+sti )

(ti+1)

l(3.80)

where KRDlti

is the corresponding liability key rate duration at maturity ti. Implementingthis simplest case with data we used in the last three sections leads us to the result ofthe key rate duration of the liability side. Note that the sum of all key rate durationsconverge to the traditional duration measure:

∑20

i=1KRDl

ti=

∑20

i=1ti

lti(1+sti )

(ti+1)

l= (0.0472 + 0.0930 + · · ·+ 1.2128) = 10.3780 (3.81)


When the shift in the term structure is non-infinitesimal, the previous framework must beextended to account for the second-order nonlinear effects of the key rate shifts, as (3.79)represents only the first-order approximation of the liability changes. These are given asthe key rate convexities and can be defined as:

∆l

l≈ −

∑n

i=1KRDl

ti∆sti +

1

2

∑n

i=1

∑n

j=1KRC l

titj∆sti∆stj (3.82)

KRC ltitj

=δ2l

δstiδstj

1

l(3.83)

for every pair (i, j) of key rates. The set of key rate convexities can be represented by asymmetric matrix:

KRC =

KRC(t1, t1) KRC(t1, t2) · · · KRC(t1, tn)KRC(t2, t1) KRC(t2, t2) · · · KRC(t2, tn)

......

......

KRC(tn, t1) KRC(tn, t2) · · · KRC(tn, tn)

(3.84)

Anyway, the approximation (3.82) is hardly ever used in practice. For this reason we con-sider by the new model only the key rate duration. The pension liability key rate durationswith respect to the 20 key rates are shown in figure 3.7.

Figure 3.7: Pension liabilities and key rate duration Key rates taken are as manyas pension cash flows .

Imposing the variance operator on (3.79) allows us to model liability risk within a covari-ance framework, familiar from risk of portfolio management applications:

V ar(∆l

l) =V ar(−

∑n

i=1KRDl

ti∆sti)

=∑n

i=1KRDl

tiV ar(∆sti)

+ 2∑n

i=1

∑n

j 6=iKRDl

tiKRDl

tjCov(∆si, ∆sj) (3.85)


Now we can also apply (3.85) for a liability relative risk measure:

V ar(∆l

l− ∆P

P) ≈V ar(

∑n

i=1EKRDti∆sti)

=∑n

i=1EKRDtiV ar(∆sti)

+ 2∑n

i=1

∑n

j 6=iEKRDtiEKRDtjCov(∆sti , ∆stj) (3.86)

where the vector of excess key rate durations describes the difference between liabil-ity key rate duration (KRDl) and portfolio key rate duration (KRDP ) by EKRDti =KRDl

ti−KRDP

ti, ∆P/P measures changes in portfolio return.

However, in reality, we neither model as many key rates as cash flows, nor can we assumethat future cash flows only take place at key rates maturity. So the first step to implementthe model is actually to choose the key rates, which are considered as sensitivity bucketsin the process of hedging. It claims in many literatures that the choice of key rates remainsquite arbitrary. For example, Ho, whoes model was introduced in year 1992 for the firsttime, proposes as many as 11 key rate durations to effectively hedge against interest raterisk. However, the choice of key rates has great impact on the result of optimal portfolioallocation. On one hand, with more key rates the risk management will be brought outas perfectly as possible; on the other hand, the number of key rates may not be far toogreat, because hedging against a large number of key rate durations implies large longand short positions in the portfolio, which can make this approach somewhat expensivein terms of the transaction costs associated with portfolio construction and rebalancing.11

By throwing a glance at the historical price of zero bonds we can find out that the pricechange of zero bonds with different maturities are highly correlated with each other. Sowe decompose the zero interest rate curve into m buckets, where the price changes aremost highly correlated.

Figure 3.8 shows us the correlation matrix of the historical yield curve changes. Accordingto the correlation we decompose the 20-year-yield curve into 7 buckets: [1Y], [2Y,3Y,4Y],[5Y,6Y,7Y], [8Y,9Y,10Y,11Y,12Y], [13Y,14Y,15Y,16Y], [17Y,18Y,19Y] and [20Y]. Thecorresponding key rates are: 1-year, 3-year, 6-year, 10-year, 14-year, 18-year and 20-yearspot rates. Now we can model the sensitivity of pension liabilities with respect to thesekey rates. That is:

∆l

l≈ −KRDl

1∆s1 −KRDl2∆s2 − . . .−KRDl

7∆s7

= −∑7

i=1KRDl

i∆si (3.87)

KRDlk = −

∑20

i=1

lil

(δliδsi

1

li

δsi

δsk

)(3.88)

where li denotes the present value of cash flow lti , which happens at maturity ti as li =lti(1 + sti)

−ti . Equation (3.88) presents a value-weighted (li/l) sum of partial durations

11See Zagst [74]


Fig

ure

3.8:

Cor

rela

tion

Mat

rix

ofSpot

Rat

esShifts

Sou

rce:

Blo

omber

gsw

apra

te(t

ime

hor

izon

:20

01.0

8.17

-200

6.12

.01)


( δliδsi

1l) for each liability cash flow, where each partial duration is adjusted for its sensitivity

with respect to changes in the k-th key rate (δsi/δsk)12. All what remains to be done is to

define these sensitivities. In order to get generalized notations, it is supposed that we haveJ key rates at maturities: tk1 , tk2 , . . . , tkJ

. Then the sensitivities δsi/δskm , for m = 1, . . . , Jare nothing else but a set of step functions:

δsi

δsk1

=

1 : ti < tk1

tk2−ti

tk2−tk1

: tk1 ≤ ti ≤ tk2

0 : ti > tk2

(3.89)

δsi

δskm

=

0 : ti < tkm−1

ti−tkm−1

tkm−tm−1: tkm−1 ≤ ti ≤ tkm

tkm+1−ti

tkm+1−tkm

: tkm ≤ ti ≤ tkm+1

0 : ti > tkm+1

(3.90)

for m = k2, k3, . . . , kJ−1, and

δsi

δskJ

=

0 : ti < tkJ

ti−tkJ−1

tkJ−tkJ−1

: tkJ−1≤ ti ≤ tkJ

1 : ti > tkJ

(3.91)

The entire process can even be represented as a shift matrix, which is shown in the table3.6. Simply we can find out that the sum of each row equals 1, that is, the sensitivity ofcash flow change at each maturity ti can be modelled as a linear combination of changes inboth neighboring key rates and both of the entries in the matrix respond the coefficients.In case it has only one neighboring key rate (either left or right), the sensitivity becomesone. This case implies actually that the interest rate risk at this maturity will be modelledwith a single key rate.

We can now apply (3.88), (3.89), (3.90) and (3.91) to calculate the key rate durations ofpension liability with respect to the chosen 7 key rates. The key rate structure is shown infigure 3.9. It observes directly that the rough shape of the key rate exposure alters greatlyfrom the figure 3.7, where we take as many key rates as liability cash flows. The reason ofthe difference is not that we take fewer key rates this time than before. Even if we takeanother set of 7 key rates, we will get different result of liability key rate durations. Itproves again that the choice of key rates has a deep effect on the immunization result,which we have mentioned before. Note, however, that the sum of key rate durations doesnot change with a particular partition.

The key rate durations of a bond portfolio can be obtained as the weighted average ofthe key rate duration of single bond in the portfolio, where the weights are defined as theproportion of each bond held in the portfolio. In our example the bond portfolio consistsof 8 bonds, then the m-th key rate duration of bond portfolio is defined as:

KRDPm = w1KRDB1

m + w2KRDB2m + · · ·+ w8KRDB8

m (3.92)



Key Rate 1 2 3 4 5 6 7Maturity 1 3 5 8 13 18 24

1 1 0 0 0 0 0 02 0,5 0,5 0 0 0 0 03 0 1 0 0 0 0 04 0 0,67 0,33 0 0 0 05 0 0,33 0,67 0 0 0 06 0 0 1,00 0 0 0 07 0 0 0,75 0,25 0 0 08 0 0 0,50 0,5 0 0 09 0 0 0,25 0,75 0 0 010 0 0 0 1 0 0 011 0 0 0 0,75 0,25 0 012 0 0 0 0,5 0,5 0 013 0 0 0 0,25 0,75 0 014 0 0 0 0 1 0 015 0 0 0 0 0,75 0,25 016 0 0 0 0 0,5 0,50 017 0 0 0 0 0,25 0,75 018 0 0 0 0 0 1 019 0 0 0 0 0 0,50 0,5020 0 0 0 0 0 0 1

Table 3.6: Shift Matrix with 7 Corresponding Key RatesShift Matrix with respect to the key rates: 1-year, 3-year, 6-year, 10-year, 14-year, 18-yearand 20-year spot rates.

Figure 3.9: Pension Liabilities and Key Rate Duration Key rates taken are the1-year, 3-year, 6-year, 10-year, 14-year, 18-year and 20-year spot rates .


where wi is the corresponding weight of each bond and KRDBjm is the m-th key rate

duration of bond j.

Now we have all the necessary values to find the liability tracking portfolio with use ofkey rate durations. Supposedly, we have an M × 1 vector of liability key rate durations,that is, we choose M key rates:

KRDl = (KRDl1, KRDl

2, . . . , KRDlM)T (3.93)

and an M×J matrix of key rate durations for a given bond universe available for liabilityhedging:

KRDB =

KRDB1

1 KRDB21 · · · KRDBJ

1

KRDB12 KRDB2

2 · · · KRDBJ2

......

. . ....

KRDB1M KRDB2

M · · · KRDBJM

(3.94)

where J denotes the number of bonds available for liability hedging, which is actually 8 forour example. In case J = M + 1, ie, the number of bonds equals the number of key ratesexposures plud one and allows short positions, we can find the vector of bond weights was the solution of:(

KRDl

1

)=

(KRDB

1T

)· w (3.95)

KRDl1

KRDl2

...KRDl

M

1

=

KRDB1

1 KRDB21 · · · KRDBJ

1

KRDB12 KRDB2

2 · · · KRDBJ2

......

. . ....

KRDB1M KRDB2

M · · · KRDBJM

1 1 · · · 1

w1

w2...

wJ

(3.96)

In this case we can generate a bond portfolio that has exactly the same key rate durationsas a given liability portfolio, as the rank of the augmented coefficient matrix above inpractice equals the number of components of w. Technically this will bring us a bondportfolio with zero liability relative risk, which is already mentioned in (3.85). In fact,however, we have only immunized those risks that arise from a limited number of keyrates, which in no case means the portfolio is already 100% risk free against the interestrate changes. For this reason our target function for the new optimization approach is tominimize liability relative risk:

minw

σ2

(∆l

l− ∆P

P

)≈σ2

(∑n

i=1EKRDti∆sti

)=∑n

i=1EKRDtiσ

2(∆sti)

+ 2∑n

i=1

∑n

j 6=iEKRDtiEKRDtjCov(∆sti , ∆stj)

= (EKRD)T · Σ · EKRD (3.97)


where

EKRD = KRDl −KRDP

= KRDl −KRDB · w (3.98)

and Σ is the symmetric covariance matrix of key rate changes, namely:

Σ =

cov(∆s1, ∆s1) cov(∆s1, ∆s2) · · · cov(∆s1, ∆sM)cov(∆s2, ∆s1) cov(∆s2, ∆s2) · · · cov(∆s2, ∆sM)

......

. . ....

cov(∆sM , ∆s1) cov(∆sM , ∆s2) · · · cov(∆sM , ∆sM)

(3.99)

Figure 3.10: Key Rate Duration of Liability and Instrument Universe

Table 3.7: Covariance Matrix of Key Rate Changes

KeyrateMaturity

1 3 6 10 14 18 20

1 0,0055 0,0072 0,0061 0,0046 0,0040 0,0034 0,00332 0,0072 0,0117 0,0105 0,0086 0,0077 0,0068 0,00653 0,0061 0,0105 0,0104 0,0090 0,0083 0,0076 0,00744 0,0046 0,0086 0,0090 0,0084 0,0080 0,0076 0,00745 0,0040 0,0077 0,0083 0,0080 0,0079 0,0076 0,00756 0,0034 0,0068 0,0076 0,0076 0,0076 0,0079 0,00757 0,0033 0,0065 0,0074 0,0074 0,0075 0,0075 0,0075

Back to our example, figure 3.10 shows us the key rate durations of liability and the 8bonds we have used and table 3.7 shows us the covariance matrix of key rate changes,namely Σ, which was also calculated from the weekly data of bloomberg in the sametime horizon (2001.08.17-2006.12.01). Then we will compare the results of two differentoptimization approaches: covariance approach and traditional key rate approach without


considering of covariance. These two approaches can be formulated as below:

minw

σ2

(∆l

l− ∆P

P

)= (EKRD)T · Σ · EKRD

1T · w = 1 (3.100)

0 ≤ wi ≤ 1

minw

(KRDl −KRDP

)21T · w = 1 (3.101)

0 ≤ wi ≤ 1

(3.4) is the so-called traditional key rate duration approach. We minimize homely theabsolute deviation between liability key rate duration and that of portfolio.

Table 3.8: Optimal Allocations ComparisonResults Maturity Weight (traditional) Weight (covariance)

Bond 1 1 6.16% 0.00%Bond 2 2 0.00% 2.86%Bond 3 3 4.95% 5.29%Bond 4 5 0.53% 0.59%Bond 5 7 10.96% 1.23%Bond 6 10 14.90% 11.16%Bond 7 15 42.94% 57.90%Bond 8 20 19.56% 20.97%

Table 3.8 shows us the optimal allocations according to two different approaches, whichare formulated in (3.4) and (3.4). The second column is the allocation with respect to thetraditional simple key rate duration approach without considering covariance matrix ofkey rate changes. We can see the result of covariance approach on the last column. Nextstep, we compare the two portfolio cash flows with the pension liability cash flow, whichis shown in figure 3.11.

In reality, the so-called traditional key rate duration approach is also widely used, evenmore popular than the covariance approach. However, some authors assert that this modelis not an efficient model in describing the dynamic of the term structure because historicalvolatilities of interest rates provide useful information about the behavior of the differentsegments of the term structure and the traditional key rate duration model disregards thisinformation13. Therefore, the main advantage of using the covariance approach in com-parison with other methodologies is the ability to calculate liability-relative risks ratherthan exposures alone. The approach fits well into classical mean variance optimization and

13See Hill and Vaysman, 1998[34] and Golub and Tilman, 2000[30]


Figure 3.11: Cash Flows Comparison for KRD Model

hedging, and can therefore be easily integrated into a broader risk management frame-work14. It gains also the efficiency caused by consideration of the history of term structuremovements. Besides the covariance approach we have introduced, there are a number ofother variations of the key rate model that try to deal with the above mentioned undesir-able consequence. For example, Falkenstein and Hanweck (1996)[26] offer an alternativeto the traditional key rate hedging called covariance-consistent key rate hedging that con-sists of finding the portfolio that minimizes the variance of portfolio returns. Reitano(1996)[60] introduces the concept of stochastic immunization as a strategy that instead ofseeking immunization in the traditional sense, searches for the portfolio that minimizes arisk measure defined as a weighted average of the portfolio’s return variance and the worstcase risk. Anyway, both covariance approach and traditional key rate duration matchingprovides the possibility to control the interest rate risk exposure of plan asset more ex-actly. However, limitations still exist in the key rate model, which can be summarizedinto 2 points as follows:

• The choice of the key rates has a strong impact on the immunization results.

• The shapes of the individual key rate shifts are unrealistic.

About the first shortage we have already discussed it previously and the way we havechosen is to select key rate buckets with regard to the correlation matrix of key ratechanges. The second limitation, although the whole set of key rate shifts taken togetherallows for modeling realistic movements in the term structure, each individual key rateshift has a historically implausible shape. Each key rate shock implies the kind of forwardrate saw-tooth shift. To address this shortcoming, a natural choice is to be focussed onthe forward rate curve instead of the zero-coupon curve. Johnson and Meyer (1989) firstproposed this methodology and called it the partial derivative approach or PDA. Accord-ing to the PDA, the forward rate structure is split up to many linear segments and allforward rates within each segment are assumed to change in a parallel way15. More aboutthis methodology will not be introduced here.

14See Scherer, 2005[63]15See Nawalkha , 2005[50]


3.5 Principal Component Duration Model

Over the past few years, a new technique for hedging interest-rate risk, the method ofprincipal component model , has become more popular. It assumes in the principal com-ponent model that the yield curve movements can be summarized by a few compositevariables, which are also called components or factors. These components explain mostof the variation in key rate and are used as the drivers of the yield curve. Applying astatistical technique called principal component analysis (PCA) allows us to constructthese new factors. Therefore, a short introduction on principal components analysis willbe given in section (3.5.1) and with a simple example we will show how to implement thePrincipal Components Analysis.

3.5.1 Principal Components Analysis

In statistics, principal components analysis (PCA) is a technique for simplifying a dataset, by reducing multidimensional data sets to lower dimensions for analysis. Supposedly,that we have observation of data xti at time ti, i = 1, 2, . . . , n and xti is a column vectorwith length m. Σ is the covariance matrix of the data, that is

Σ = cov

xt1,1 xt1,2 · · · xt1,m

xt2,1 xt2,2 · · · xt2,m...

. . ....

vtn,1 xtn,2 · · · xtn,m

(3.102)

Apparently, Σ is symmetric and has the form of m × m. By means of matrix algebraknowledge we know that Σ has m normalized and linearly independent eigenvectors,u1, . . . , um, which correspond to m positive eigenvalues λ1, . . . , λm. Actually, Σ can befactored as follows:

Σ = UT ΛU (3.103)

where Λ is a diagonal matrix with elements λ1, . . . , λm along the diagonal and U is anm × m matrix, whose rows correspond to the vector u1, . . . , um. As U is composed ofnormalized and orthogonal vectors, its inverse matrix equals its transpose:

U−1 = UT (3.104)

Now let us define the components we need. The relationship between the m principalcomponents and the m original variables is given as follows:

c = Ux (3.105)

where c = [c1, . . . , cm]T is the column vector of principal components and x = [x1, . . . , xm]T

is the column vector of the original variables. Obviously, c and x have the same shape of


m × 1 and the matrix U gives the principal component coefficients. In fact, the princi-pal components can be represented as linear combinations of the original variables andthe original variables can, in turn, be expressed as linear combinations of the principalcomponents as follows:

x = U−1c (3.106)

Using (3.104), x can be expressed as follows:

x = UT c (3.107)

We can get the matrix U of principal component coefficients with the help of the meigenvalues of the covariance matrix Σ and the eigenvalues can be obtained by solving thefollowing equation:

Det[Σ− λI] = 0 (3.108)

where I is the identity matrix. Then the eigenvectors are just the nontrivial and normalizedsolutions to the set of equations:

(Σ− λiI)ui = 0 (3.109)

for i = 1, 2, . . . ,m.

What is left to do is to sort the eigenvectors by their corresponding eigenvalues fromthe largest to the smallest. As the eigenvectors with the largest eigenvalues correspondto the dimensions that have the strongest correlation in the data set, the principal com-ponent with the highest eigenvalue is the most important component. Actually, the lasttwo steps: calculating eigenvalues and eigenvectors, and ranking the eigenvalues in orderof their magnitudes, can be realized simultaneously with help of several mathematicalsoftwares, for example, the commands ‘eig’ and ‘eigs’ of Matlab, which is one of themost famous mathematical software developed by MathWorks. So there is only one ques-tion left. Where does the reduction in dimensionality come from? The dimensionality inprincipal component analysis is reduced by disregarding those principal components thatare of minor importance in explaining the variability of the original variables (i.e., thosewith the lowest eigenvalues). In practice, the first 3-4 components can usually alreadyexplain over 95% of the variability of the original variables. Assuming that we retain thefirst k principal components, then the vector x can be expressed as:

x = U∗T c∗ + ε (3.110)

where U∗ refers to the k×m matrix resulting from retaining the first k rows of the matrixU , c∗ refers to the k× 1 matrix composed of the first k principal components and ε refersto the m× 1 matrix, each column of which denotes m error terms caused by disregardingthe left m− k components.

Now let us perform the Principal Component Analysis by a simple example. Supposethat we have the covariance matrix of changes in historical 1-year, 2-year, 3-year, 4-year,


5-year, 7-year, 9-year and 10-year zero-coupon rates:

Σ =

0.0755 0.0749 0.0679 0.0621 0.0565 0.0473 0.0421 0.03810.0749 0.093 0.0923 0.0886 0.0835 0.0729 0.0667 0.06070.0679 0.0923 0.0967 0.0952 0.0911 0.0817 0.0764 0.07070.0621 0.0886 0.0952 0.0954 0.0923 0.0838 0.0792 0.07380.0565 0.0835 0.0911 0.0923 0.0902 0.0827 0.0786 0.07360.0473 0.0729 0.0817 0.0838 0.0827 0.0781 0.0753 0.07170.0421 0.0667 0.0764 0.0792 0.0786 0.0753 0.0738 0.07110.0381 0.0607 0.0707 0.0738 0.0736 0.0717 0.0711 0.0714

(3.111)

With the help of Matlab we get the corresponding eigenvalues and eigenvectors. The 10corresponding eigenvalues are:

0.60510, 0.05741, 0.00902, 0.00137, 0.00066, 0.00042, 0.00014, 0.00002.

With the formula: λiP10j=1 λj

we can calculate the portion of the total variance of the changes

in the 10 interest rates by each principal component, which by the first 3 principal com-ponents are 89.76%, 8.52% and 1,34%. That means the first 3 principal components canalready explain above 99.62% of the spot rate variation. For these 3 eigenvalues namedsimply λ1, λ2, λ3 we get the corresponding eigenvectors, for example, for λ1:

u1 =

0.27050.37170.39560.39500.38230.34970.33180.3125

or u1 =

−0.2705−0.3717−0.3956−0.3950−0.3823−0.3497−0.3318−0.3125

(3.112)

Similarly, for the second eigenvalue, λ2, we can also obtain a set of two correspondingeigenvectors:

u2 =

−0.7011−0.3852−0.11990.02810.12360.25240.33400.3929

or u2 =

0.70110.38520.1199−0.0281−0.1236−0.2524−0.3340−0.3929

(3.113)


and similarly for the third eigenvalue λ3, we obtain:

u3 =

−0.5650−0.2265−0.31510.29560.2430−0.0314−0.2248−0.5764

or u3 =

0.56500.22650.3151−0.2956−0.24300.03140.22480.5764

(3.114)

It is arbitrary which set of solution we take. So suppose that we take the first solution ofeach eigenvector, we get the matrix of U∗ described in (3.110):

U∗ =

0.2705 0.3717 0.3956 0.3950 0.3823 0.3497 0.3318 0.3125−0.7011 −0.3852 −0.1199 0.0281 0.1236 0.2524 0.3340 0.3929−0.5650 0.2265 0.3151 0.2956 0.2430 −0.0314 −0.2248 −0.5764

Using (3.110) the model of interest rate changes can be expressed as:∆s(1)∆s(2)

...∆s(10)

=

0.2705 −0.7011 0.56500.3717 −0.3852 0.2265

......

...0.3125 0.3929 −0.5764

c1

c2

c3

+

ε(1)ε(2)

...ε(10)

3.5.2 Principal Component Duration

We have already got the basic knowledge behind the Principal Component Analysis. Nowwe can begin with our new immunization model:Principal Component Duration Model .

The main purpose of the key rate duration model in the last section is to characterizefixed income risks as exposures to individual zero yield changes and in this section ourpurpose is actually to describe the movement in m = 1, . . . ,M key rates by a smallernumber of factors, the so-called principal components .

Instead of expressing liability risk as in (3.78) and (3.79) we can now express it as:

l = l(c1, c2, c3, . . .) (3.115)

∆l

l= −PCD1∆c1 − PCD2∆c2 − PCD3∆c3 − · · · (3.116)


with

PCDi = − δl

δci

1

l(3.117)

where PCDi are the so-called Principal Component Duration and c = (c1, c2, c3, . . .)T is

the column vector of principal components. Here we use Principal Component Analysisto express changes in key rates by a small number of principal components. Conversely,as we have seen in the last section in the introduction of Principal Component Analysis,any realization of principal components implies a unique change in the key rates. Theprincipal components are linear combinations of key rate changes:

∆cj =m∑

i=1

uji∆s(ti) j = 1, . . . ,m (3.118)

where uji are called principal component coefficients. Actually, (3.118) is consistent with(3.105).

To construct the principal components we need to use the covariance matrix of key ratechanges. Since the covariance matrix is symmetric by construction, it must have m nor-malized and linearly independent eigenvectors, u1, . . . , um, and m corresponding positiveeigenvalues, λ1, . . . , λm. As we have seen in (3.105), the coefficient of the first componentu1 = (u11 u12 · · · u1m)T is actually the eigenvector corresponding to the eigenvalue withthe largest magnitude, the coefficient of the second component u2 = (u21 u22 · · · u2m)T

is given as the eigenvector corresponding to the eigenvalue with the second largest mag-nitude, and so on.

The next step is to reduce the dimensionality in equation (3.118). Now the number ofprincipal components is still m, which is equal to the number of key rate changes. How-ever, as we have mentioned previously, not all the components have the same significance.The first principal component explains the maximum percentage of the total variance ofinterest rate changes. The second component is linearly independent (i.e., orthogonal) ofthe first component and explains the maximum percentage of the remaining variance, thethird components is linearly independent (i.e., orthogonal) of the first two componentsand explain the maximum percentage of the remaining variance, and so on. Usually, 2-3principle components are already enough to explain over 95% of the interest rate changes.

Since any key rate changes can be described by using the m principal components andthe variance of each component is given by the magnitude of its eigenvalues, the totalvariance of the key rate changes can be given as:

m∑j=1

λj (3.119)

and the proportion of the variance explained by the jth principal component is:

λj∑mj=1 λj

(3.120)


Contrarily, since the matrix U of coefficients uji in (3.118) is orthogonal, the inverse ofthe matrix is actually its transpose. Using this feature, the changes in the key rates canbe obtained by inverting equation (3.118) as follows:

∆s(i) =m∑

j=1

uji∆cj i = 1, . . . ,m (3.121)

As we have described previously, the principal components with low eigenvalues makelittle contribution in explaining the interest rate changes, and hence these componentscan be removed without losing significant information. Assuming that we retain the firstk components, the expression (3.121) can be rewritten as:

∆s(ti) =k∑

j=1

uji∆cj + εi i = 1, . . . ,m (3.122)

where εi is the error term which represents the changes not explained by the k principalcomponents. A better approach will be realized by modifying the model to make eachfactor have a unit variance. This can be achieved by multiplying each eigenvector by thesquare root of its eigenvalue, and dividing the principal component by the square root ofthe eigenvalue16. The expression (3.122) then becomes:

∆s(ti) =k∑

j=1

(uji

√λj

)∆c∗j + εi i = 1, . . . ,m (3.123)

where

∆c∗j =∆cj√

λj

(3.124)

The coefficients in the parenthesis of (3.123), which measure the impact of a standarddeviation move in each principal component on each interest rate, are called factor load-ings . Using simpler notation and the first 3 factor loadings (3.123) can be approximatedas follows:

∆s(ti) ≈ li1∆c∗1 + li2∆c∗2 + li3∆c∗3 i = 1, . . . ,m (3.125)

where the factor loadings lij is defined as: lij = uji

√λj.

Now back to (3.116), it can be expressed as:

∆l

l≈ −PCD1∆c1 − PCD2∆c2 − · · · − PCDk∆ck

= −k∑

j=1

PCDj∆cj (3.126)

16Nawalkha, 2005[50]


On the other side, according to (3.87) the percentaged change for liability can also beexpressed as:

∆l

l≈ −

m∑i=1

KRDi∆si

(3.125)≈ −

m∑i=1

KRDi

k∑j=1

lij ·∆cj

= −k∑

j=1

∆cj

m∑i=1

KRDi · lij (3.127)

where KRDi is the ith key rate duration, defined as the (negative) percentage changein the security price resulting from one unit shift in the ith key rate. From (3.126) and(3.127) we can also get the expressions of PCDs, which are actually linear combinationsof the key rate durations:

PCDj =m∑

i=1

KRDi · lij j = 1, . . . , k (3.128)

Before giving our new optimization approach, we will use the values of the example usedpreviously to calculate the principle component duration described above. With respectto the correlation matrix of spot rate changes used in the last section, we choose the samekey rates as before: 1-year, 3-year, 6-year, 10-year, 14-year, 18-year and 20-year spot rates.The covariance matrix of the key rate changes is shown in table 3.7. Table 3.9 shows us theeigenvectors and the eigenvalues of the covariance matrix. The first column contains thecorresponding eigenvector with the largest eigenvalue. The second column contains theeigenvector with the second largest eigenvalue, and so on. The first row is the eigenvaluessorted by their magnitude. Apparently, there is another set of eigenvectors according tothe eigenvalues, which are vectors with negative magnitudes as these ones. It is arbitrarywhich set of eigenvectors we choose, because it has no impact on our immunization result,which will be introduced later.

Table 3.9: Eigenvectors and EigenvaluesEigenvalues 0.05167 0.00628 0.00077 0.00027 0.00017 0.00010 0.00004

0.2505 -0.5401 0.7449 0.1843 0.2317 0.0286 0.04550.4369 0.5140 -0.2573 -0.3897 -0.4581 -0.2648 -0.21670.4400 -0.1753 -0.4153 0.1517 0.1654 0.5656 0.4827

Eigenvectors 0.3982 0.1241 -0.2197 0.3466 0.4321 -0.0693 -0.68280.3779 0.2682 -0.007 0.1947 0.0755 -0.7083 0.48980.3593 0.4031 0.2603 -0.7323 0.2793 0.1618 -0.01230.3497 0.4043 0.3005 0.3113 -0.6624 0.2767 -0.1085


The result, that the first 3 principle components make already the most significant con-tribution in explaining the variance of the interest rate changes, is consistent with thestudy. The first component accounts for λ1/

∑7i=1 λi = 87.1% of the total variance, while

the second and the third components account for 10.6% and 1.3% respectively. In total,the first three components explain 99% of the variability of the data, which indicates thatthese factors are sufficient for describing the changes in the term structure. We multiplyeach eigenvector by the square root of its eigenvalue as we have described previously andget the first 3 vector of factor loadings, which are shown in table 3.10. Figure 3.12 givesus a visualization of the principal components.

Table 3.10: Height, Slope and Curvature FactorsKey Rate PC(h) PC(s) PC(c)

1 0.0569 -0.0428 0.02083 0.0993 -0.0408 -0.00726 0.1000 -0.0139 -0.011610 0.0905 0.0098 -0.006114 0.0859 0.0213 -0.000218 0.0817 0.0320 0.007320 0.0795 0.0321 0.0084

Figure 3.12: Impact of the First 3 PCs on the TSIR

It appears that the first principal component affects all changes in interest rates by roughlythe same amount, which explains why it is usually named the level or the height factor.The second component will lead to a fall in short rates and a rise in long rates. This canbe interpreted as a change in slope, which also explains why it is named slope factor. Thethird principal component is called curvature factor, as it affects both short and long endpositively while it leads to a decrease in intermediate rates.


Table 3.11: PCD of Liability and Bond InstrumentsPCD Maturity PCDh PCDs PCDc

1 0,0552 -0,0415 0,02012 0,1480 -0,0795 0,01343 0,2750 -0,1141 -0,0182

Bonds 5 0,4435 -0,1072 -0,04257 0,5859 -0,0614 -0,058810 0,7392 0,0457 -0,051515 0,9537 0,1974 -0,001320 1,1019 0,3321 0,0671

Liability 20 0,8951 0,1803 0,0087

We have calculated the component loadings, the key rate durations of both bond instru-ments and pension liability can be obtained from the result of the last section. Then theprincipal component durations can also be directly obtained with respect to (3.128). Thevaluations of principal component duration on the instrument site and the liability siteare demonstrated in table 3.11.

Using the second-order Taylor series approximation we can also compute the correspond-ing principal component convexity as follows:

∆l

l= −

∑k

i=1PCD(i) ·∆ci +

1

2

k∑i=1

k∑j=1

PCC(i, j) ·∆ci ·∆cj (3.129)

where

PCC(i, j) =1

l

∂2P

∂ci∂cj

i, j = 1, . . . , k (3.130)

Since the principal components are independent, we can simplify (3.129) by disregardingthe cross effect, which gives:

∆l

l= −

∑k

i=1PCD(i) ·∆ci +

1

2

∑k

i=1PCC(i, i) ·∆c2

i (3.131)

However, during the implement of this model we don’t take the principal component con-vexity into account, since it is hardly used in practice.

Now let us come to the last step: immunization optimization approach. By immunizationwe take only principal component duration into consideration, and the immunization


approach will be as follows modeled:

minω

ωT · ω (3.132)PCDB1

1 PCDB21 · · · PCDB8

1

PCDB12 PCDB2

2 · · · PCDB82

......

. . ....

PCDB1k PCDB2

k · · · PCDB8k

1 1 · · · 1

·

ω1

ω2...

ω8

=

PCDl

1

PCDl2

...PCDl

k

1

(3.133)

0 ≤ ωi ≤ 1 (3.134)

The above modeled approach is actually to reach the target: diversification based on theconstraint, that the first kth principal component durations of portfolio remains same asthat of the pension liability. It may occurs, that the above approach can be infeasible. Inthis case, we can remodel the approach as follows so that we get the portfolio, which canimmunize the liability the best:

minω

k∑i=1

(PCDB

i · ωi − PCDli

)2(3.135)

ω1 + ω2 + · · ·+ ω8 = 1 (3.136)

0 ≤ ωi ≤ 1 (3.137)

In table 3.12 stands the optimal allocation of our example according to the immunizationapproach (3.132). The result of cash flows mismatching are shown in figure 3.13.

Table 3.12: Optimal Allocation of PCD ApproachBonds Maturity Weight

Bond 1 1 3.01%Bond 2 2 0.00%Bond 3 3 0.00%Bond 4 5 2.36%Bond 5 7 8.83%Bond 6 10 19.29%Bond 7 15 29.83%Bond 8 20 36.67%

Principal component duration model has several appealing features. It provides a sim-ple way to model changes in the yield curve: the factors will be selected based on theircontributions to the total variance of interest rate changes, so that only two or threecomponents usually suffice to explain more than 95% of the variation in the yield curve.Moreover, these components often serendipitously have intuitive interpretations: for ex-ample, the first component as level factor, the second component as slope factor and the


Figure 3.13: Cash Flows Comparison for PCD Model

third component as curvature.

However, the principal component duration model has also limitations. The most majorshortcomings during application of this model is that the static nature of the techniqueis unable to deal with the nonstationary time-series behavior of the interest rate changes.17

In terms of this shortcoming, application of PCA to term structure movements impliesthat the covariance matrix of interest rate changes is constant and hence the vector offactor loadings, which describe the shape of the principal components, are stationaryas well. This is critically important, because if the shapes of the principal componentschange frequently, then these components can not explain future volatility of interestrate changes any more. A large number of empirical studies find evidence, that changingvolatilities of zero coupon rates affects the stability of the principal components. Nawalkha(2005)[50] tests the effect of the changing volatilities of U.S. zero-coupon rates on theprincipal components obtained for 2000, 2001 and 2002. As a result, the first principalcomponent seems relatively stable, the second and the third principal components varysignificantly. Therefore, it suggests that we should estimate the model periodically andexamine alternative covariance matrices to check the stationarity and choose the moststable matrix.

17Nawalkha, 2005[50]

Chapter 4

Case Study: Dynamic and StochasticOptimization

In chapter 3 we have introduced 5 methods in terms of liability immunization, which arefor static and deterministic purposes. In practice, the future values of both liability sideand asset side, e.g. liability streams and interest rates, are stochastic. On this account,in this chapter using a case study we implement immunization strategies in the dynamicand stochastic framework. The stochastic framework implies that at the present pointof time the future values of liabilities and plan assets remain unknown. To simulate thestochastic process we need the scenario generator in order to generate the values of bothgain and loss accounts. However, in order to model the liabilities accurately, as describedin chapter 2, we must take some new factors into account:

• capital market risks: interest rate, inflation rate, salary trend, . . .

• biological risks: mortality, the probability of becoming disable for a person workingin the firm, . . .

• structure of the firm: number of people working in the company, their salary, age,gender, . . .

In addition, previously, for each optimization approach, we have implemented the opti-mization process once for the whole planning period, that is, the weights are set at thebeginning of the planning period and after that no further trades are undertaken, which iscalled Buy and Hold Strategy. However the remaining working life and market risk factorsvary from year to year. Moreover, the values of estimated benefits also varies with time.Therefore, the duration of liabilities is changing during the whole project period. It isnecessary to calculate the optimal allocation aiming at the liability duration variations,which describes the basic idea of dynamic optimization.

To begin with the immunization approaches in dynamic and stochastic sense, a briefrepeat of the relevant parameters in the framework will be brought out as follows:

• Liability streams: For our final pay pension plan with 10,000 pension members, wemodeled 1000 scenarios of the expected benefits for 80 years, which are also affected

65

CHAPTER 4. CASE STUDY: DYNAMIC AND STOCHASTIC OPTIMIZATION 66

Figure 4.1: Expected Benefits 80 Years

by the inflation rates. Distinct from the last chapter, we model this time the expectedbenefits with maturity of 80 years. The reason behind this is although we implementthe optimization approaches just for the first 30 years, there are still benefits afterit. In other words, the benefits will not be stopped after 30 years. Figure 4.1 gives usa straightforward demonstration of the expected benefits we use. What is left to bepointed out is that for the entire dynamic framework there is only one cone availableas shown in figure 4.1. Actually we realize the optimization approach for every timestep. Hence, there are two solution to realize a real dynamic optimization:

1. There are 30 cones corresponding to the 30 time steps available.

2. There are a tree structure of benefits cash flows for the 80 years, whose branchesimply all the possibilities how a liability stream is flowing in the future based onits current position. In other words, it is necessary to calculate the conditionalprobability so that we can capture the expected trend of liability streams.

However, both of the two solutions have expensive calculation costs, so that it pre-vents us to realize the optimization in terms of “real” dynamic. Anyway, we havesimulated the yield curves for every time step. On this account, it could be thoughtas plausible that we implement our approaches with this single cone of expectedbenefits to reach the “quasi-dynamic” purpose.

• Interest rates: As mentioned in chapter 2, we get the stochastic values of interestrates from the risklab’s economic scenario generator. We take 1000 scenarios and 30time steps for our immunization approaches. However, since we take the expectedbenefits for 80 years, it also demands to have interest rates with maturity 80 years,which unfortunately can not be generated. For this reason, we hold the interest rateswith maturities between 31 and 80 years fixed as the 30-year spot rate. That means,


we have for each time step 1000 yield curves with maturity 80 years, which appearhorizontal from year 30 to 80.

• Discount factor: According to the interest rates we calculate the discount factors indiscrete sense, that is d(t) = 1/(1 + s(t))t. On this account, we have also the valuesof 1000 scenarios and 30 time steps for discount factors. Figure 4.2 shows us theevolution of yield curves and discount curves of 30 time steps and the median of1000 scenarios.

Figure 4.2: Yield Curve and Discount Curve

(a) Interest Rate (b) Discount Factor

• Asset classes: Instead of 8 coupon bonds used in chapter 3, we bring this time 9zero coupon bonds into play. The according maturities are: 1-year, 2-year, 3-year,5-year, 7-year, 10-year, 15-year, 20-year and 30-year. As is known, the prices of thezero coupon bonds are decided by the interest rates. Therefore, at every time step,there are 1000 scenarios available to describe the price of each bond.

Now we have almost all the relevant parameters for the framework. How is the con-crete operation method due to the 1000 scenarios and 30 time steps? Basically, for everytime step according to each immunization approach we are seeking the specific optimalportfolio allocation for the single scenario. In other words, for one time step and oneimmunization model, we have 1000 best solutions to 1000 different liabilities cash flows.With respect to these 1000 solutions we investigate and analyze their characteristics, forinstance, the distribution of the values of corresponding objective function. In the next4 sections we will describe the operation method of each immunization approach in detail.

However, we do not implement the cash flow matching in terms of dynamic and stochasticframework. There are two reasons behind this:

• According to the basic idea of cash flow matching, the optimal economical portfoliowill be selected out to cover the future benefits cash flows. As soon as the portfolioallocation is set fix, all the future benefits should be covered by this portfolio in-flows. Since we have only one cone of benefits along the period, it is not necessary


to iterate the optimization process.

• We limit our asset universe in zero bonds, and in fact only 9 zero bonds. So it is notpossible to cover the yearly benefits only with these 9 bonds.

In section 4.5, we implement the Markowitz optimization in order to compare the resultswith those of the previous 4 immunization approaches.

4.1 Traditional Duration Model

In this section we introduce the operation method during the implement of the traditionalduration model in dynamic and stochastic framework. Basically, the new concept is toiterate the optimization process for 30 time steps and 1000 scenarios. However, duringthe implement we must face more problems than previously.

First of all, due to the long maturity of benefits cash flows, we can not find any bondportfolio with the same duration as liability. So as mentioned previously, we must reformour immunization approaches as follows:

1. Do

minω

(ConB · ω − Conl)2

DB · ω = Dl (4.1)∑i

ωi = 1

0 ≤ ωi ≤ 1

2. If (4.1) is infeasible, then do

minω

∥∥∥∥ DB · ω −Dl

ConB · ω − Conl

∥∥∥∥2

2∑i

ωi = 1 (4.2)

0 ≤ ωi ≤ 1

where

ω Weight of portfolioDl Liability duration

Conl Liability convexityDB = (DB1 , DB2 , . . . , DBJ

)T Duration vector of J bondsConB = (ConB1 , ConB2 , . . . , ConBJ

)T Convexity vector of J bonds.


Actually, it is difficult to immunize the liability with long maturity only by means ofbonds. Because there are hardly any bonds with maturity longer than 30 years traded inthe market. An alternative to (4.2) will be using derivatives (e.g., interest rates swaps,Treasury bond futures) to extend the portfolio duration.

Besides this, there remains still a numerical problem. That is, if we use the optimizationtoolbox of MATLAB or even other mathematical softwares, the initial values have someimpact on the final optimal result. Therefore, it is necessary to choose a plausible initialvalue. It can be realized as follows: according to the liability and bond duration we estimatethe possible weight of each bond, set the estimation as the initial input in order to continuethe implement. However, it costs too much if we do that for each time step and eachscenario. So for the simplicity we set the initial weight of each bond as:

ωi =1

number of asset classes(4.3)

Figure 4.3 illustrates the distribution of objective function values among 1000 scenariosat each time step. Note that over the project period the objective function values havedecayed from about 2× 105 to 1× 10−9. The reason behind this phenomenon is not thatthe objective functions have been changed. In table (4.1) are the median of liabilities andassets durations, from which we can see since the 5th. time step for the most scenariosthe constraints will already be satisfied. Actually, although we have changed the objec-tive function, the deviation between asset duration and liability duration remains small.The real reason is that at the beginning of the project period, the liabilities have greatduration and convexity, which can not be matched even if we invest 100% of the assetinto the 30-year bond. As time flows, the duration and the convexity of the liabilities havefallen down so that we can find a portfolio with similar duration and convexity. At last,the deviation of convexity converges even to zero. The evaluations of the median durationand convexity of liability and portfolio are shown in table F.1 in appendix.

The next questions are how the bond portfolio is allocated, which distribution the alloca-tion has for the 1000 scenarios and how will the allocation change with the time. Figure4.4 demonstrates the distribution of allocation along the project period. The first axisdenotes bond 1 to bond 9. The second axis denotes the weight of each bond and thethird axis denotes the number of the scenarios. Therefore, each single plot is actually a3-dimensional histogram. There are 30 subplots in figure 4.4, which imply 30 time steps.Apparently, at the first 3 steps, the bond portfolio consists 100% in 30-year zero bond inorder to match the liability duration and convexity. As time flows, the weight of 30-yearzero bond becomes smaller while 1-year zero bond holds also heavier weight, so do theother bonds. What can also be observed in figure 4.4 is that during the whole projectperiod the allocation is dependent on scenario, which means the allocation of portfolio isnot 100% stable. Except the first several time steps, the weight of a bond distributes overthe interval, and exposes quasi a normal distribution. For example, in figure 4.5 we cansee the distribution of 1-year zero bond at time step 13. The weight distributes over the


Figure 4.3: Distribution of Objective Function Values at Each Time Step


Table 4.1: Median of Liability and Bond DurationTime Liability Bond DurationStep Duration 1 2 3 4 5 6 7 8 9

1 31,452 0,970 1,934 2,896 4,817 6,736 9,614 14,408 19,200 28,7772 30,773 0,969 1,934 2,896 4,816 6,735 9,610 14,402 19,188 28,7613 29,962 0,969 1,933 2,895 4,815 6,732 9,606 14,398 19,184 28,7514 29,156 0,968 1,933 2,894 4,813 6,731 9,602 14,387 19,172 28,7345 28,555 0,968 1,932 2,894 4,813 6,728 9,599 14,379 19,162 28,7256 27,859 0,967 1,931 2,893 4,810 6,725 9,597 14,377 19,156 28,7097 27,241 0,967 1,931 2,893 4,811 6,725 9,594 14,372 19,149 28,7038 26,758 0,967 1,932 2,893 4,812 6,727 9,599 14,378 19,156 28,7139 26,153 0,967 1,931 2,893 4,812 6,730 9,603 14,383 19,166 28,72510 25,470 0,967 1,931 2,892 4,811 6,726 9,596 14,378 19,159 28,72011 24,926 0,967 1,930 2,892 4,810 6,726 9,597 14,380 19,157 28,71312 24,162 0,966 1,930 2,892 4,812 6,727 9,596 14,376 19,152 28,70513 23,621 0,967 1,931 2,892 4,811 6,725 9,598 14,378 19,154 28,70714 23,032 0,967 1,931 2,892 4,811 6,725 9,596 14,375 19,155 28,71115 22,389 0,967 1,930 2,892 4,810 6,725 9,593 14,375 19,155 28,70916 21,791 0,967 1,931 2,893 4,813 6,729 9,598 14,378 19,156 28,70717 21,250 0,967 1,931 2,894 4,814 6,728 9,597 14,371 19,147 28,70118 20,695 0,967 1,931 2,892 4,812 6,726 9,595 14,376 19,157 28,71319 20,063 0,967 1,931 2,893 4,811 6,726 9,595 14,371 19,146 28,69520 19,449 0,967 1,930 2,891 4,809 6,724 9,592 14,368 19,143 28,68821 18,953 0,967 1,931 2,893 4,812 6,727 9,595 14,372 19,152 28,70722 18,408 0,966 1,930 2,892 4,810 6,725 9,593 14,369 19,145 28,69423 17,850 0,967 1,931 2,894 4,812 6,728 9,596 14,372 19,146 28,70024 17,380 0,967 1,931 2,893 4,812 6,728 9,597 14,376 19,158 28,71425 16,859 0,968 1,932 2,894 4,814 6,727 9,596 14,375 19,153 28,70526 16,414 0,968 1,932 2,895 4,816 6,731 9,602 14,385 19,163 28,72627 15,896 0,968 1,933 2,897 4,817 6,732 9,601 14,384 19,167 28,72828 15,361 0,967 1,931 2,895 4,815 6,729 9,599 14,382 19,167 28,72629 14,859 0,967 1,931 2,894 4,813 6,728 9,597 14,375 19,154 28,71130 14,412 0,967 1,931 2,893 4,812 6,727 9,599 14,380 19,159 28,714


Figure 4.4: Distribution of Portfolio Allocation along 30 Years


interval [0.04, 0.35].

Figure 4.5: Distribution of 1-year Bond Weight at Time Step 13

Figure 4.6(a) gives us a direct demonstrate of the allocation trend over the 30 years. Herethe portfolio weight is taken as the mean value of 1000 scenarios. On this account thisfigure can not tell us the exact weight of a bond along the period, but the developmentof allocation along the 30 years. Note that the allocation has a quasi continuous trendin figure 4.6(a), which is not the case if we only take one scenario instead of taking themean value of 1000 scenarios. The allocation of this case is demonstrated in figure 4.6(b).There are some springs in this figure, e.g. at time step 21 and 26. The reason of thisdifference is that if we take one scenario, the portfolio allocation is specific for this sce-nario. For instance, we focus on the period of time step 21 to time step 26. The weightsof 1-year, 2-year and 30-year bond decrease with time while the weights of 7-year and10-year bond increase. At the time step 25 the allocation reaches its marginal value sothat at the next time step we get a totally new allocation. The weight of 1-year bondreaches suddenly the previous level again. The 15-year bond and 20-year bond have alsomuch heavier weight than the step before. In case that we take the mean value of 1000allocations, the extreme effect of one scenario will be eliminated. So the curves whichdescribe the bond weight development seem to be much more continuous. However, thegeneral trends the portfolio allocation in the two figures have the similar characteristic.This is the reason that we take the mean value for analyzing the trend of the portfolioallocation. We can find out in figure 4.6(a) that the portfolio begins with only 30-yearzero bond. In the first 15 years, it consists only in 30-year and 1-year bonds. As time flowsthe weight of other bonds become heavier so that at the time step 30 although the 30-yearbonds still holds the heavier weight compared to the other bonds, the dominance is notso outstanding as before. 1-year, 10-year, 15-year and 20-year bonds hold almost sameweights and the whole portfolio consists of all of the 9 bonds at the end the project period.


Figure 4.6: Allocation Trend for Traditional Duration Matching

(a) Mean Value of Allocation Trend (b) Allocation Trend of one Scenario

4.2 Generalized Duration Vector Model

The operation method of implementing the generalized duration vector model in thestochastic and dynamic framework remains similar as that of implementing the traditionalduration model. As described previously in section 3.3.3, the immunization approach isformed this time as:

minω1,ω2,...,ωK

K∑k=1

ω2k (4.4)

DPm =

K∑k=1

ωk ·Dkm = Dl

m m = 1, . . . ,M (4.5)∑kωk = 1 (4.6)

0 ≤ ωk ≤ 1 (4.7)

with Dkm =

∑tNt=t1

wt · (tα)m and wt =[

CFt

eR to f(s) ds

]/V0.

Here K will be taken as 9 to describe the number of bonds available and M is taken as 3referring to the length of the duration vector. As previously we set the value of α as 0.5.

This time it is even more difficult to satisfy all the constraints at the same time that attraditional duration model, since we have more constraints. Analogously, if the constraints


can not be satisfied, we must reform the immunization approach:

minω1,ω2,...,ωK

∥∥∥∥∥∥∥DP

1 −Dl1

...DP

M −DlM

∥∥∥∥∥∥∥2

2

(4.8)

∑kωk = 1 (4.9)

0 ≤ ωk ≤ 1 (4.10)

As at this model we choose the continuous discount rate with the form as: d(t) = e−t·s(t)

and the zero bonds instead of coupon bonds. So the duration vector with length 3 of 9zero bonds can be expressed as nothing but: DB

1

DB2

DB3

=

Maturity(1×α)B

Maturity(2×α)B

Maturity(3×α)B

(4.11)

where MaturtiyB denotes the Maturity of the bond. We give a brief proof of this result:

Proof:

DBm =

tN∑t=t1

wt · (tα)m

=

tN∑t=t1

CFt

eR to f(s) ds

V0

· (tα·m)

zero bond=

CFMaturityB

eRMaturityBo f(s) ds

V0

·Maturity(α·m)B

= 1 ·Maturity(α·m)B

= Maturity(α·m)B (4.12)

Table 4.2 shows us the duration vector of each bond.

As the next step we analyze analogously the distribution of the objective function values.It must be pointed out this time is that if the constraints are all satisfied, the objectivefunction value is not the squared deviation between the portfolio duration vector andthe liability duration. On this account we demonstrate this time the squared deviation infigure 4.7, not the objective function values.

The median deviation begins at about over 1500 but ends at 1 × 10−28, since at thebeginning of the project period all the constraints can not be simultaneously satisfied.


Figure 4.7: Squared Deviation of Duration Vector Model


Table 4.2: Duration Vector of Each Bond

Duration Vector 1. order 2. order 3. orderBond 1 1 1 1Bond 2 1,414 2 2,83Bond 3 1,732 3 5,20Bond 4 2,236 5 11,18Bond 5 2,646 7 18,52Bond 6 3,162 10 31,62Bond 7 3,873 15 58,09Bond 8 4,472 20 89,44Bond 9 5,477 30 164,32

From about the 20 time step the constraints are satisfied for the most scenarios. So thedeviations come closely to zero, which can also be observed in table 4.3. Note that at thefirst 6 time steps the median value of portfolio duration vector is exactly the durationvector of 30-year bond, which implies the optimal portfolio of the first several years willconsists 100% of 30-year bond. It is also consistent with the traditional duration model.From time step 21 the median values on the portfolio side are exactly as same as thoseon the liability side, which implies the constraint (4.5) is satisfied and from this momentwe use the first immunization approach.

Now let us come back to the portfolio allocation. We demonstrate the distribution of port-folio allocation again by means of 30 subplots. Basically, figure 4.8 has similar exteriorappearance to the figure 4.4, for example, they start at 100% 30-year bond and reachdiversification at last. Referring to the concrete allocation at some time step there existdifferences between the two models, which can be easily observed in figure 4.9. By meansof figure 4.9 we can find that both of models have similar allocation trends over 30 years,which is also consistent with our argument previous. The reason behind it is that the basicideas behind these two models are similar. Actually, for traditional duration model, wefocus on the duration and convexity, which come from the Taylor-Expansion 2th order ofthe yield function while for generalized duration vector model we are interested in dura-tion vector with longer length. The allocation trend of generalized duration vector modelappears like the allocation wave of tradition duration model by being pushed forward. Forinstance, in the first 15 years, the optimal portfolio1 of traditional duration model consistsin only 30-year and 1-year bonds. However, since the 8th time step the optimal portfolioof generalized duration vector model consists not only of 30-year and 1-year bonds butalso in 20-year bond in spite of its little proportion. At the time step 15 the portfolioconsists already of 6 bonds. As it is expected that bonds with long maturities have betterperformance than those with short maturities in an arbitrage-free market, the portfoliovalue of the generalized duration vector model will be greater than that of traditionalduration model. Between the time step 20 and 25 for traditional duration model 2-yearbond plays a great role in the portfolio allocation, which is not the case for the gener-alized duration vector model. Instead of it the 5-year bond holds a larger weight during

1It refers to the mean allocation here.


Table 4.3: Median of Liability and Portfolio Duration VectorTime Liability PortfolioStep 1. order 2. order 3. order 1.order 2.order 3.order

1 5,463 32,576 204,572 5,477 30,000 164,3172 5,401 31,840 197,833 5,477 30,000 164,3173 5,332 31,100 191,211 5,477 30,000 164,3174 5,262 30,267 183,778 5,477 30,000 164,3175 5,211 29,638 178,038 5,477 30,000 164,3176 5,134 28,837 171,214 5,477 30,000 164,3177 5,088 28,299 166,516 5,477 30,000 164,3178 5,028 27,675 161,269 5,385 29,403 160,9549 4,974 27,098 156,246 5,249 28,520 155,98410 4,904 26,399 150,528 5,094 27,516 150,33011 4,851 25,804 145,672 4,962 26,662 145,52112 4,769 24,992 139,432 4,792 25,563 139,33013 4,713 24,473 135,032 4,673 24,791 134,97914 4,643 23,798 129,895 4,544 23,887 129,88015 4,581 23,178 125,121 4,502 23,216 125,11616 4,517 22,537 120,144 4,462 22,559 120,14217 4,455 21,969 115,765 4,416 21,983 115,76318 4,389 21,364 111,228 4,361 21,370 111,22719 4,319 20,723 106,506 4,306 20,727 106,50520 4,248 20,096 102,035 4,245 20,097 102,03521 4,198 19,625 98,539 4,198 19,625 98,53922 4,125 18,995 94,143 4,125 18,995 94,14323 4,064 18,472 90,400 4,064 18,472 90,40024 4,000 17,934 86,614 4,000 17,934 86,61425 3,940 17,405 83,059 3,940 17,405 83,05926 3,878 16,894 79,674 3,878 16,894 79,67427 3,818 16,406 76,373 3,818 16,406 76,37328 3,744 15,812 72,561 3,744 15,812 72,56129 3,681 15,314 69,368 3,681 15,314 69,36830 3,614 14,813 66,222 3,614 14,813 66,222


Figure 4.8: Distribution of Portfolio Allocation along 30 Years


Figure 4.9: Mean of Allocation Trend

(a) Traditional Duration Model (b) Generalized Duration Vector Model

this period. At the end of the project period the allocation of generalized duration vectormodel becomes better-distributed over 9 bonds, which means there is not great differenceamong the weights of the 9 bonds.

4.3 Key Rate Duration Model

It is expected that the portfolio allocation will have great changes for key rate durationmodel compared to the previous two models. Because the basic idea of key rate durationmodel is quite different from the other two. However, it is just our estimation till now. Inorder to analyze the immunization result we begin with our implement in dynamic andstochastic framework again.

As described in the previous chapter, the immunization approach of key rate durationmodel is formulated as follows:2

minω

(EKRD)T · Σ · EKRD

1T · ω = 1 (4.13)

0 ≤ ωi ≤ 1

where

EKRD = KRDl −KRDP

= KRDl −KRDB · ω

2Σ denotes the covariance matrix of key rate changes.


Because of the characteristics of this immunization approach, there does not exist thecase that the constraints can not be satisfied. On this account it is not necessary to com-plement an additional approach.

In order to implement the approach there remain two steps to be done:

• The choice of the key rates

• The valuation of the covariance matrix Σ

For these two steps we need the historical data of spot rates. As mentioned in section 3.4we load the weekly data from Bloomberg in time horizon 2001.08.17-2006.12.01. Accordingto the correlation matrix of spot rate changes the key rate buckets will be chosen. Thistime the key rates are chosen as:

1-year, 3-year, 5-year, 8-year, 13-year, 18-year, 24-year and 29-year.

The respective correlation matrix can be found in Appendix C.

The corresponding Σ, which is defined as:

Σ =

cov(∆s1, ∆s1) cov(∆s1, ∆s2) · · · cov(∆s1, ∆s9)cov(∆s2, ∆s1) cov(∆s2, ∆s2) · · · cov(∆s2, ∆s9)

......

. . ....

cov(∆s9, ∆s1) cov(∆s9, ∆s2) · · · cov(∆s9, ∆s9)

(4.14)

is shown in table 4.4.

Table 4.4: Covariance Matrix of Key Rate ChangesMaturity Key Rate 1 2 3 4 5 6 7 8

1 0,00553 0,00721 0,00649 0,00523 0,00417 0,00341 0,00304 0,002843 0,00721 0,01166 0,01108 0,00938 0,00793 0,00681 0,00620 0,005915 0,00649 0,01108 0,01115 0,00979 0,00856 0,00758 0,00702 0,006788 0,00523 0,00938 0,00979 0,00910 0,00829 0,00763 0,00721 0,0070213 0,00417 0,00793 0,00856 0,00829 0,00804 0,00766 0,00741 0,0072818 0,00341 0,00681 0,00758 0,00763 0,00766 0,00791 0,00756 0,0074924 0,00304 0,00620 0,00702 0,00721 0,00741 0,00756 0,00765 0,0075629 0,00284 0,00591 0,00678 0,00702 0,00728 0,00749 0,00756 0,00758

As described in the obvious two sections we iterate the implement procedures for 1000scenarios and 30 time steps. Although we implement the approach in dynamic framework,the value of Σ and the choice of key rates will be set fix over these 30 years, which isexactly the shortcoming described in section 3.4. In an analogous manner, we focus atfirst on the distribution of the objective function values.

Figure 4.10 illustrates the distribution of the objective function values at each time step.The objective function values start at about 0.5 and decays monotonically to 2.5× 10−4.The reason behind this is that the squared value of the excess key rate duration (EKRD)


Figure 4.10: Distribution of Objective Function Values at Each Time Step


becomes smaller with time. The key rate duration matrix of bond portfolio is relativelystable over the whole 30 years and 1000 scenarios, which can be concluded from figure4.11. There are 8 subplots in figure 4.11, which demonstrate 8 3-dimensional histograms ofbond key rate duration according to 8 key rates. The x-axis denotes the 9 bonds and they-axis reflects the key rate duration value of the bond with respect to each key rate. Thez-axis refers the number of the scenarios. Note that this time we consider simultaneously1000 scenarios over 30 time steps, which implies that we have 30000 scenarios altogether.Apparently, the key rate duration matrixes of all the 30000 scenarios are consistent witheach other and the median value of them is shown in table 4.5.

Figure 4.11: Distribution of Key Rate Duration Corresponding to 8 Key Rates

However, the values of the key rate duration on the liability side changes with time, whichis demonstrated in table 4.6. The key rate duration values according to the key rates withshort maturities increase with time, while those according to the key rates with long ma-turities decay with time. For example, the key rate duration value respected to 1-year keyrate starts at 0.0237 and ends at 0.0705 while the key rate duration respected to 29-yearkey rate starts at 26.2072 and ends at 4.2730. What is relatively interesting is that thekey rate duration values with respect to key rate 6 and 7 start at 1.4554 and 2.1746,increases, and then decays to 2.8168 and 2.5388. From this characteristic we assume thatthe proportions of bonds with long maturities will decrease with time while those of bonds


Table 4.5: Median of Bonds Key Rate DurationKey Rate 1 2 3 4 5 6 7 8

Bond 1 0,9677 0 0 0 0 0 0 0Bond 2 0,9661 0,9661 0 0 0 0 0 0Bond 3 0 2,8945 0 0 0 0 0 0Bond 4 0 0 4,8143 0 0 0 0 0Bond 5 0 0 2,2434 4,4868 0 0 0 0Bond 6 0 0 0 5,7607 3,8404 0 0 0Bond 7 0 0 0 0 8,6299 5,7532 0 0Bond 8 0 0 0 0 0 12,7763 6,3882 0Bond 9 0 0 0 0 0 0 0 28,7254

Figure 4.12: Allocation Trend of Mean

with short maturities will increase and the proportions of bonds with maturities near to20 will increase at first and then decay. To check whether the assumption is right we usethe mean value of portfolio allocation for 1000 scenarios again.

Figure 4.12 shows us the allocation trend of key rate duration model. Apparently, theweight of 30-year bond decays with time and the weights of 1-year to 10-year bonds in-crease along the 30 years. What is probably not obvious to find out is that the weightsof 15-year and 20-year bonds have the same trend as we assumed before. However, theconcrete weights of these two bonds can be found in table D.3 of Appendix D . Duringyear 20 to 25 the weights of both bonds reach their maximums and then there exist a lightdecline. Why is the portfolio allocation dependent on the trend of the liability key rateduration? As we demonstrate in figure 4.11 the bond key rate duration remains stableduring the time horizon. The covariance matrix Σ will be fix, too. Therefore, in orderto minimize the objective function value we should adjust the proportion, which has theidentical trend as the liability key rate duration.


Table 4.6: Median Value of Liability Key Rate DurationKey Rate 1 2 3 4 5 6 7 8Maturity 1 3 5 8 13 18 24 29

1 0,0237 0,0701 0,1510 0,3894 0,7976 1,4554 2,1746 26,20722 0,0242 0,0717 0,1530 0,4070 0,8442 1,5262 2,2691 25,27153 0,0246 0,0723 0,1560 0,4230 0,8883 1,6080 2,3581 24,26964 0,0250 0,0729 0,1620 0,4416 0,9380 1,6871 2,4632 23,21785 0,0250 0,0729 0,1662 0,4600 0,9798 1,7523 2,5601 22,34516 0,0252 0,0757 0,1730 0,4852 1,0332 1,8304 2,6609 21,31917 0,0252 0,0778 0,1764 0,5079 1,0719 1,8990 2,7621 20,58318 0,0255 0,0788 0,1807 0,5253 1,1173 1,9630 2,8490 19,74679 0,0263 0,0796 0,1898 0,5455 1,1574 2,0367 2,9359 18,958710 0,0269 0,0822 0,2014 0,5719 1,2098 2,1200 3,0316 18,022111 0,0271 0,0861 0,2084 0,5945 1,2528 2,2016 3,1226 17,189312 0,0283 0,0925 0,2162 0,6276 1,3089 2,2893 3,2089 16,149713 0,0301 0,0968 0,2268 0,6579 1,3614 2,3689 3,2932 15,417614 0,0321 0,0994 0,2388 0,6808 1,4089 2,4534 3,3815 14,475815 0,0333 0,1023 0,2492 0,7072 1,4720 2,5377 3,4670 13,587916 0,0340 0,1089 0,2627 0,7380 1,5384 2,6212 3,5328 12,662117 0,0354 0,1153 0,2708 0,7705 1,6005 2,7068 3,5792 11,919118 0,0378 0,1218 0,2829 0,8023 1,6655 2,7949 3,5849 11,105819 0,0404 0,1259 0,2977 0,8387 1,7365 2,8771 3,5652 10,304020 0,0425 0,1312 0,3137 0,8799 1,8093 2,9534 3,5130 9,573321 0,0435 0,1377 0,3254 0,9189 1,8742 3,0295 3,4429 8,987322 0,0458 0,1463 0,3404 0,9676 1,9504 3,0864 3,3473 8,290423 0,0482 0,1532 0,3560 1,0122 2,0181 3,1251 3,2511 7,684124 0,0513 0,1581 0,3756 1,0547 2,0944 3,1329 3,1604 7,109125 0,0535 0,1649 0,3986 1,0999 2,1681 3,1183 3,0533 6,579426 0,0548 0,1749 0,4151 1,1464 2,2439 3,0866 2,9604 6,080127 0,0576 0,1865 0,4340 1,1991 2,3112 3,0319 2,8643 5,594928 0,0622 0,1971 0,4557 1,2481 2,3594 2,9601 2,7408 5,108029 0,0667 0,2063 0,4826 1,3034 2,3847 2,8897 2,6427 4,664530 0,0705 0,2173 0,5087 1,3639 2,3976 2,8168 2,5388 4,2730


Now let us compare the mean allocation trend of key rate duration model with other 2models. The portfolio allocation has a great change. At begin, about in the first year, theportfolio consists only of 30-year bond analogously. However, from even the second yearthe portfolio absorbs new bonds. As time flows, the portfolio consists of more and morebonds. At about time step 20, it contains already all 9 bonds. It is to be pointed out,that there is a sequence of the optimal portfolio to absorb new bonds. It begins alwayswith the long-term bonds, and then the intermediate-term bonds, at last the short-termbonds. The valuations in table 4.5 and 4.6 lead us to the reason of this characteristic.The liability key rate durations with respect to short-term key rates are relatively smallin the first several years. According to the objective function, the excess key rate duration(EKRD) will even be squared. On this account, even if we do not invest in short-termbonds at all, it will not have large impact on the objective function value. As describedpreviously, the liability key rate durations respected to the long-term bonds go down withtime, so that we don’t need to focus on the 30-year bond. In the next several years wemay invest a part of the plan asset in intermediate-term bonds in order to reduce theexcess key rate duration. With the growing liability key rate durations according to theshort-term key rates we should attach more importance to the short-term bonds. Thatis the rough reason of such allocation trend. The concrete valuation of portfolio key ratedurations can be found in table F.2 of appendix F What we can still conclude from figure4.12 is that the portfolio value of the key rate duration model will be greater than thatof the previous two models along the period, since plan asset will be invested more intointermediate-term bonds instead of short-term bonds and it is expected that bonds withlonger maturity can achieve greater performance. This assumption will be proved in thechapter 5.

4.4 Principal Component Duration Model

Now we come to our last immunization model: Principal Component Duration Model.Before we implement this model in dynamic and stochastic framework, we make a pre-diction that the allocation trend of this model is more or less similar as that of the keyrate duration model, as the principal component duration can actually be expressed as alinear combination of key rate durations. Whether the assumption is right or not, let usprove it in this section.

Analogously, we start with the immunization approach. As described in section 3.5, theimmunization model is formulated as follows:


minω

ωT · ω (4.15)PCDB1

1 PCDB21 · · · PCDBm

1

PCDB12 PCDB2

2 · · · PCDBm2

......

. . ....

PCDB1k PCDB2

k · · · PCDBmk

1 1 · · · 1

·

ω1

ω2...

ωm

=

PCDl

1

PCDl2

...PCDl

k

1

(4.16)

0 ≤ ωi ≤ 1 (4.17)

However, the immunization approach can be formulated as above just in case that theconstraints (4.16) and (4.17) are satisfied. If the approach is not infeasible, we shouldreformulate it as:

minω

k∑i=1

(PCDB

i · ωi − PCDli

)2ω1 + ω2 + · · ·+ ωm = 1 (4.18)

0 ≤ ωi ≤ 1

Here m denotes the number of bonds and k represents the number of the necessary prin-cipal components, which during our implement are set as 8 and 3 separately.

For reminding we give the expressions of the principal component duration again, whichis nothing but:

PCDj =m∑

i=1

KRDi · lij j = 1, . . . , k (4.19)

As we do not change the covariance matrix of spot rates changes, the factor loadings (lij)3

remain same along the 30 years and 1000 scenarios. On this account if we take all the 8principal components into consideration, the immunization approach (4.18) will get thesame optimal portfolio as the optimization approach (4.13) of the traditional key rateduration model formulated as follows, which is mentioned in (3.4):

minω

(KRDl −KRDP )2

1T · ω = 1 (4.20)

0 ≤ ωi ≤ 1.

3See page 59


Figure 4.13: Impact of the First 3 PCs on the TSIR

Figure 4.13 gives us a visualization of the factor loadings, which reflects also the impactof the first three principal components on the term structure of interest rates.

Because of the different definitions of the objective functions it is not worthy analyzingthe distribution of the objective function values. For this reason we plot the distribution ofsquared deviation between portfolio and liability principal component durations instead,which is shown in figure 4.14.

Analogously, the deviations decline monotonically with time. They start at around 0.05and at time step 25 has a huge fall from 10−7 to 10−30, which implies that from timestep 25 the constraints (4.16) and (4.17) of the immunization approach for almost all thescenarios can be satisfied and from this time step the optimal allocation of almost all thescenarios can be obtained by means of approach (4.15).

As described in the last section, the bond key rate durations remain relatively stableover the project period and among 1000 scenarios. The factor loadings are also set to beconstant. It is supposed that the bond principal component duration is also consistentwith each other since the PCD is actually the linear combination of key rate durations.We plot the distribution of bonds principal component duration according to the firstthree principal components in figure 4.15. Analog with key rate duration, we analyze thedistribution among all the 1000 scenarios and 30 years. From figure 4.15 we can proveour supposition that the PCD of each bond remain stable among the 1000 scenarios andduring the 30 years. The median values of bonds PCD are shown in table 4.7.

Although it is known that the first 3 liability key rate durations increase monotonicallyalong the period, we can not suppose that the liability PCD will have the same trends askey rate durations. Table 4.8 illustrates the median value of liability principal componentdurations, which have the exact characteristic as the liability key rate durations, however


Figure 4.14: Distribution of Squared Deviation of Principal Component Duration


Table 4.7: Median of Bonds PCD

Bond Maturity PCD 1 PCD 2 PCD 3

1 1 0,0540 -0,0411 0,02312 2 0,1488 -0,0831 0,02103 3 0,2842 -0,1259 -0,00634 5 0,4889 -0,1234 -0,05435 7 0,6511 -0,0752 -0,07176 10 0,8803 0,0347 -0,07897 15 1,2326 0,3018 -0,02838 20 1,5604 0,6134 0,08219 30 2,2166 1,1085 0,2323

Table 4.8: Median of Liability PCD

Time Steps PCD 1 PCD 2 PCD 3 Time Steps PCD 1 PCD 2 PCD 3

1 0,0237 0,0701 0,1510 16 0,0340 0,1089 0,26272 0,0242 0,0717 0,1530 17 0,0354 0,1153 0,27083 0,0246 0,0723 0,1560 18 0,0378 0,1218 0,28294 0,0250 0,0729 0,1620 19 0,0404 0,1259 0,29775 0,0250 0,0729 0,1662 20 0,0425 0,1312 0,31376 0,0252 0,0757 0,1730 21 0,0435 0,1377 0,32547 0,0252 0,0778 0,1764 22 0,0458 0,1463 0,34048 0,0255 0,0788 0,1807 23 0,0482 0,1532 0,35609 0,0263 0,0796 0,1898 24 0,0513 0,1581 0,375610 0,0269 0,0822 0,2014 25 0,0535 0,1649 0,398611 0,0271 0,0861 0,2084 26 0,0548 0,1749 0,415112 0,0283 0,0925 0,2162 27 0,0576 0,1865 0,434013 0,0301 0,0968 0,2268 28 0,0622 0,1971 0,455714 0,0321 0,0994 0,2388 29 0,0667 0,2063 0,482615 0,0333 0,1023 0,2492 30 0,0705 0,2173 0,5087


Figure 4.15: Distribution of Bond PCD corresponding to the 3 Components

only by accident. The median of portfolio PCDs are demonstrated in table F.3.

The next step is to compare the optimal allocation of the principal component durationmodel with that of the other three models, which is demonstrated in figure 4.16. Notethat what is in figure 4.16 illustrated is the mean value of 1000 scenarios of each model.It observes that the allocation trend of principal component duration model is relativelydifferent from the first 2 models, but a bit similar as the key rate duration model. Onereason behind this is that the calculation of PCDs relies on the valuations of KRDs. How-ever, there exists still deviations between the last two models. For example, the optimalallocation of PCD model starts with 30-year and 15-year bonds instead of 30-year and20-year bonds, which is the case of KRD model. In addition, the weight of the 10-yearbond doesn’t monotonically increase as what happens at the KRD model, but have aconvexity between year 16 and 23.


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4.5 Markowitz Optimization Using the Resampling

Method

In order to exam the results of our immunization optimization approaches more precisely,we compare the results with those according to the Markowitz optimization approaches,which implies not the traditional Markowitz optimization, but the so-called ResamlingOptimization using the resampling method. Therefore, we give here a brief description ofthis method.

4.5.1 A Description of the Problem

It was already in 1952 that Harry Markowitz laid the foundation for the so called Mean-Variance-Analysis, which creates most efficient portfolios with respect to the fact, thatfor a given level of risk the portfolio with highest expected return or for a given level ofexpected return the portfolio with least risk will be selected out. The set of all portfoliosthat are risk-return efficient are said to form the Markowitz efficient frontier. This is theso-called “traditional” portfolio optimization, which can be mathematically described asfollows:

maxω

n∑i=1

ωiµi (4.21)

ωT Ωω = σ (4.22)n∑

i=1

ωi = 1 (4.23)

or, alternatively

minω

1

2· ωT Ωω (4.24)

µT ω = µ (4.25)n∑

i=1

ωi = 1 (4.26)

where ω refers to the weights of a portfolio with n assets, µ refers to the expected returnsof the asset classes, σ to the standard deviation of the portfolio we are going to optimizedand Ω to the covariance matrix with size n× n.

However, the nobel objectives of mean variance optimization by Markowitz has also somelimitations and disadvantages, such as the optimization results are very sensitive withrespect to the input data. The model is to be implemented based on the assumption thatthe input information is 100 percent certain, which is actually unrealistic. Hence, the im-pact of estimation error on portfolio optimization could be very serious.


Scherer described this problem as portfolio optimization suffering from error maximiza-tion: “The optimizer tends to pick those assets with very attractive features (high returnand low risk and/or correlation) and tends to short or deselect those with the worst fea-tures. These are exactly the cases where estimation errors is likely to be highest, hencemaximizing the impact of estimation error on portfolio weights. The quadratic program-ming optimization algorithm takes point estimates as input and treats them as if theywere known with certainty (which they are not) will react to tiny differences in returnsthat are well within measurement error.” This is exactly the reason that mean-varianceoptimized portfolios suffer from instability and ambiguity.4

4.5.2 Michaud’s Methodology

A solution to the mean-variance optimization impasse is suggested by Michaud in 1998and patented by Richard Michaud and Robert Michaud. It is the form of resampling thedata inputs, by means of a Monte Carlo simulation procedure, to adjust for data uncer-tainty.

The procedure can be summarized as follows:Suppose we are given the true distribution parameters covariance matrix Ω0 and themean return vector µ0. Then we generate a random sample based on the same distri-bution with m observations from the original sample. Using the traditional Markowitzoptimization allows us to get the optimal portfolios for this sample, while the efficientfrontier is composed of portfolios varying from the minimum-variance to the maximum-variance portfolio. With other words, we calculate at first the portfolios with minimumand maximum variance. The next step, we divide the difference of return of these twoportfolios into nP , which refers to the number of portfolios we wish to have. By calculat-ing the expected return of these portfolios we reach our goal that we get the nP points ofthis efficient frontier. Repeating this procedure N times let us achieve N efficient frontiersrepresented by nP Portfolios with the corresponding allocation vector ω1, ω2, . . . , ωnP .In case that we use each set of allocation vectors ωi = (ω1

i , ω2i , . . . , ω

nPi ) i ∈ [1, N ] back

to the original variance-covariance matrix Ω and the mean return vector µ0, we can geta new efficient frontier which is below the original efficient frontier. It is comprehensiblebecause the new allocation vector is only optimal for Ωi, i ∈ [1, N ], not optimal for Ω0.The result of the resampling procedure is that estimation error in the inputs parametersis transformed as the uncertainty of the optimal weight vector.

The remaining operation to get the resampling allocation is to compute the average port-folio:

ωresampledj =

1

N

N∑i=1

ωij (4.27)

where ωij denotes the weight vector of the jth portfolio along the frontier for the ithresampling.

4See Jiao, 2004[37]


4.5.3 Optimal Portfolio of Reampling Optimization

Back to our case study, we have 1000 scenarios and 30 time steps. Every time we selecta sample with population 100 and search the efficient frontier for this sample. Then thisprocess will be repeated 20 times, which is represented by N in (4.27). Note that we seek20 optimal portfolios on each efficient frontier and then according to (4.27) we captureone efficient frontier composed by 20 points. Analogously, we iterate it for 30 time stepsso that for each time step we have a efficient frontier. Note that the expected return usedduring this approach is actually calculated as the deviation between the logarithms of theplan asset return and DBO return5 , expressed as:

Expected Return = log(1 + ReturnPA)− log(1 + ReturnDBO) (4.28)

where ReturnPA denotes the return of a rand portfolio6. The reason why we take thelogarithm instead of expected return itself into consideration is that the distribution of theresults will be more symmetric in this way. The covariance matrixes are taken accordingto the expected return. The entire Markowitz optimization using the resampling methodwill be implemented separately with the following instruments:

1. 9 government zero-coupon bonds we have used previously

2. new asset universe including

• Corporate Bonds

• 9 Government Bonds

• 9 Inflation Bonds

• European Stocks

• Real Estate

• Absolute Return

The next step is to compare the optimal allocation of immunization approaches with thatof the Markowitz optimization using resampling method. In sense of immunization, weseek after the optimal portfolios which can reduce the interest risk exposures best. Onthis account it is meaningful to compare the portfolio of minimum variance (MVP) withthe immunization portfolios. Figure 4.17 illustrates the MVP of Markowitz Optimizationusing resampling method and the concrete values of MVP can be found in table E.1. Thetrend of the allocation over the 30 years does not have the continuous characteristic asimmunization portfolios. However, the single outstanding commonality is the weight of30-year bond decreases with time. At a glance of the corresponding covariance matrixeswe can find out that the magnitudes of the components related to 30-year bonds increase

5We take the logarithms instead of (ReturnPA − ReturnDBO) because it can avoid negative orpositive skew of the results distribution as well as possible.

6Rand portfolios mean the portfolio containing only one instrument.


Figure 4.17: MVP of Resampling Optimization with 9 Bonds

with the period. That the reason why fewer and fewer assets will be invested in 30-yearbond.

In the same way we implement the Markowitz optimization using resampling method with22 instruments. Figure 4.18 demonstrates the mean-variance-frontiers over the 30 years.We take the MVP into consideration, that is, we take the first point of the respectivefrontier with the minimal variance and then study the characteristic of the portfolioallocation. The trend of the portfolio allocation is demonstrated in figure 4.19. This timethe whole asset will be invested mainly in corporate bonds, 30-year government bonds,European stocks and absolute return. The exact portfolio allocations will be shown in thetable E.2.


Figure 4.18: Mean-Variance-Frontiers over 30 Years

Figure 4.19: MVP of Resampling Optimization with 22 Instruments

Chapter 5

Result Interpretation

In this chapter we compare the results of different dynamic and stochastic immunizationapproaches in our previously described framework. Since we will reset the portfolio allo-cation at the beginning of each year, the comparison is based on the assumption that webuy the instruments at the beginning of the year and sell it at the end of that year. Inthe next year, the process will be iterated. For the simplicity we ignore here the transac-tion cost. In order to capture the comparison results we need some criteria. For example,funding level, surplus return, surplus volatility and so on. On this account we analyze theresults of comparison depending on the different criteria one by one.

5.1 Plan Asset, Plan Asset before Contributions

As described in chapter 2, plan asset is in fact the amount invested in the pension plan.The criterium “Plan Asset before Contributions” describes the amount left in the pensionplan before the new yearly contributions are paid to the pension plan. According to thedifferent contribution strategies the amount of the contributions will be calculated. Here,for our comparison purpose, we set the contribution strategy as:

• Lower Lever: 100%

• Upper Lever: 115%

• Withdraw Lever: 140%

• Recovery Plan: 10 year

As mentioned in chapter 2, if the “plan asset before contributions” are high enough, forexample, higher than 140% of the DBO, then no contributions will be extra paid to thepension plan. But in case that the “plan asset before contributions” is lower than 100% ofthe DBO, then the recovery plan will go into effect. Figure 5.1 shows us the comparisonresult of the four immunization approaches according to the criteria “plan asset beforecontributions” and its volatility. Note that we begin with the plan asset as 100% DBO.

98

CHAPTER 5. RESULT INTERPRETATION 99

Figure 5.1: Plan Asset before Contributions and its Volatility

Figure 5.2: DBO of 80 Years


On the left side of figure 5.1 we are shown the comparison result respected with thecriterium “Plan Asset before Contributions” and the right plot demonstrates the com-parison result according to the volatility of the previous criterium. Obviously, there doesnot exist a great deviation among the four immunization approaches respecting the firstcriterium. There exists dominance for the key rate duration model and the principal com-ponent duration model. However the dominance is not very outstanding. All of the fourimmunization models have the same trend of bPA1. The bPAs increase with time, whichis actually calculated as follows:

bPAt = PAt−1 −Benefitst + ReturnPAt; (5.1)

where ReturnPAt denotes the return of the plan asset at the time step t. The reason ofthe increasing trend is that the DBO increase in the first 30 years, which is illustrated infigure 5.2. In order to hold the funding level the plan asset must be set almost as high asthe DBO. But on the other hand, the benefit of the current year does not have the samestrong increase as the DBO. These are the main factors leading to the “bPA” increase.

What should be paid attention to is that at the beginning of the project period, the initialplan asset is set to be 100% DBO2. Then it will be generated using the iteration formula(5.2):

PAt = bPAt + Contributionst (5.2)

The right side of the figure 5.1 shows us the standard deviation of bPA over the first 30years. Analogously, the result of key rate duration model and principal component dura-tion model have the slight dominance. The magnitude of the standard deviation increasesover the time with the magnitude of bPA.

5.2 Contributions, Present Value of Contributions

Contribution is the extra money which will be invested in the pension plan in order tokeep the funding level. As mentioned in chapter 2, the sponsor is not able or willing to payany amount of the contributions. So we set the upper bound of the current contributionas 10% of the current DBO. On one hand, we pursue to get lower contributions; on theother hand we hope the volatility of contributions are also lower. Figure 5.3 shows us thecomparison results of these two criteria.

First, we focus on the left side of figure 5.3. Basically the four immunization models havethe similar characteristics respecting contributions. During about the year 12 and 25 thelast 3 immunization models, namely “General Duration Vector Model”, “Key Rate Dura-tion Model” and “Principal Component Duration Model” need lower contributions than

1We use “bPA” as the shortcut of “Plan Asset before Contributions”.2According to different pension plans and contribution strategies the initial value of the plan asset are

different.


Figure 5.3: Contributions and its Volatility

Figure 5.4: Present Value of Contributions and its Volatility


the first model. On this account we can predicate that the last two models will reach thehigher funding lever than the first two. We will check whether the predication is right ornot in the next section. The contributions do not have the similar trend as the bPA. Theyincrease in the first 17 years and reach the top point, then they decay in the last years.

With respect to the volatility, the last three models accomplish lower volatility over themain period of the whole 30 years, while the last 3 years the contribution volatility oftraditional duration model is lower than those of the other 3 models.

Besides Contribution, we use the present value of contributions as another criterium tocompare the four immunization models. It will be calculated that the rest contributionsare discounted by the current discount rates. For example, the present value of contri-butions at time step 10 is calculated that the contributions from year 11 to year 31 arediscounted by the discount rate of time step 10. In this way we can capture the presentvalues of the contributions at each time step, which is illustrated in figure 5.4.

The result is consistent with that of contributions. It is also simple to find the reason.Because we discount the contributions with the same discount rates for the four models.So the models with lower contributions will accomplish the lower present values of con-tributions. The values decrease with time because we implement the optimization onlyfor the first 30 years; in addition, as “Final Pay” the pension plan will absorb no newmembers. Then rest sum of contributions will decrease, so do the present values.

Respecting the volatility, this time the traditional duration model accomplishes the lowervolatility than the other 3 models. However, the dominance is also not very outstanding.Besides, they have the decreasing trend as the present values.

5.3 Funding Level

Funding level is always a relevant criterium to judge a pension plan. It is defined asdescribed in chapter 2:

FLt =PAt

DBOt

(5.3)

On this account, to reach the high funding level, we need the high plan asset. However itis not expected to pay high amount of the contributions. So the plan asset before contri-butions are expected to be great enough, which is already mentioned previously. Since wehave already got the results of the ’plan asset before contributions’ and ’contributions’ itis easy to predicate the comparison result of funding level. It predicates that the last twoimmunization models can reach the higher funding level than the first two by using thesame DBO. Let’s focus on the figure 5.5, which shows us the actual result.


Figure 5.5: Funding Level and its Volatility

Analogously, we focus ourselves on the funding level. In the first 17 or 18 years, there is aobvious dominance for the key rate duration model and principal component model, whilein the rest years the dominance still exists but not outstanding any more. The fundinglevels for all the four models begin at 1 and reach their highest points at circa 1.038. Thenthere comes a huge fall for all of the four models. The funding level for the traditionalduration model falls even to 0.968. There are several reasons behind this huge fall. We setthe initial plan asset as 100% initial DBO. So the starting funding level is 1. But DBO hasa great speed of increasing. However the upper bound of the contributions is set to be 10%DBO. Besides we limit our instrument universe only in government bonds. Therefore, theplan asset return can not be so great. Thanks to the recovery plan, the funding level turnsup again. What may be strange is that why the funding level can not reach the lower level100% of DBO in the next 10 years since it is already below the lower level today, which isillustrated in figure 5.5. While we plot the median value of 1000 scenarios for analyzing.So it is possible that the median value of funding level in 10 years is still below the lowerlevel, which does not mean that the funding level of the same scenario is still under thelower level. What is also be noted is that the funding level has close relation with pensionplans or contribution strategies. If we change the contribution strategies, the values offunding levels will also change.

One step forward, let’s check the result of the volatility of funding level. In the first 8


Figure 5.6: Surplus Return and Surplus Volatility

years, the key rate duration model and the principal component duration model have aslight dominance at the volatility. But since then the volatility for the traditional durationvector becomes lowest during the rest period. Compared with traditional duration model,the funding level volatility for the last two models is only maximal 0.7 to 0.8 percentileshigher, which is actually not so informative.

5.4 Surplus Return and Surplus Volatility

Now let us come to our last criteria, Surplus Return and Surplus Volatility. There areseveral different definitions of surplus return and its volatility. Here in our analyzing wedefine the surplus return as the difference of the rate of return between the plan assets andthe liabilities. The surplus volatility is the respective standard deviation of the surplusreturn. In addition, return of plan assets is iteratively calculated as:

RenditePAt =bPAt − (PAt−1 −Benefitst)

PAt−1 −Benefitst

(5.4)


where we use the notation RenditePA to express the rate of plan assets return. Analo-gously, the rate of return on the liability side is calculated as:

RenditeDBOt =DBOt − SCt − (DBOt−1 −Benefitst)

DBOt−1 −Benefitst

. (5.5)

The definition of the “RenditePA” is comprehensible while the “RenditeDBO” is notsimple to understand. Here we consider the DBO as one kind of plan assets and its re-turn is defined as the difference between actual current DBO and predicated DBO in thelast year. Then the rate of return is calculated as the relative DBO return. The compar-ison results according to the surplus return and its volatility is demonstrated in figure 5.6.

As shown on the left side of figure 5.6 it is difficult to argue which model has this timethe dominant performance. For example, in the first 15 years key rate duration modeland principal component duration model realize the best performance. However the dom-inance is relatively slight. Between the year 15 and 20 generalized duration vector modelrealizes the best surplus return, while in the next 5 years it is traditional duration modelthat has the dominance. In the last 5 years, the last 3 models have the similar perfor-mance. Therefore, from the point of view of dominance period, the KRD model and thePCD model will be preferred to. Besides, over the whole project period the KRD modeland the PCD model have lower volatility than that of the other two models.

5.5 Comparison with Results of Resampling Opti-

mization

After analyzing the results using several different criteria among the four immunizationmodels, we compare the immunization results with the portfolio from Markowitz Opti-mization using Resampling method, which we have introduced in section 4.5. As described,we implement the Markowitz Optimization twice, once using 9 government bonds and onceusing 22 instrument classes.

5.5.1 Plan Assets before Contributions

Figure 5.7 illustrates the comparison results according to the criterium ’Plan Asset beforeContributions’. Obviously, the portfolio, which consists of the 9 government zero-bondsand is generated by resampling optimization approach, has the worse performance thanthe other 4 immunization models with the same asset universe. On the contrary, the port-folio consisting of 22 instruments has a outstanding dominance over the whole projectperiod. The reason is that since we have used more instruments, and as shown in section?? it consists mainly of corporate bond, 30-year zero bond, European stocks and absolutereturn, the portfolio can realize better return.


Figure 5.7: bPA Comparison with Markowitz Optimization Results

In addition, although the portfolio with 22 instruments accomplish the higher bPA, itsvolatility is the lowest. By comparison with other four immunization models, the volatilityof Markowitz PF93 is also lower over 30 years.

5.5.2 Contributions and Present Value of Contributions

The next step, we compare the results with the criteria “Contribution” and “PresentValue of Contributions” introduced previously.

Figure 5.8 and 5.9 show us the corresponding comparison results in terms of ’Contri-butions’ and ’Present Value of Contributions’. The Markowitz PF22 needs the lowestcontributions over the 30 years, while the Markowitz PF9 has a relatively worse perfor-mance. On this account, the present value of contributions has the same characteristics.At each time step the present value of contributions for Markowitz PF22 is lower than allof the other portfolios, while the Markowitz PF9 has the highest present value of contri-butions. Besides, PF9 has even the highest volatilities not only for contributions but forpresent value of contributions as well, while the volatility of Markowitz PF22 is quite low.

3We use the ’PF9’ as the shortcut of MVP generated by resampling optimization with 9 bonds andthe ’PF22’ as the shortcut of MVP with 22 instruments.


Figure 5.8: Contributions Comparison with Markowitz Optimization Results

Figure 5.9: PV Contributions Comparison with Markowitz Optimization Results


5.5.3 Funding Level

From the last two sections we can already predicate the performance of all the modelsin terms of ’Funding Level’. The Markowitz PF22 will accomplish the highest fundinglevel, while Markowitz PF9 the lowest. Although the Markowitz PF22 has the lowestcontributions, compared with its dominance for ’plan asset before contributions’, it canstill realize the highest plan assets. This is also the actual reason why the MarkowitzPF9’s funding level will be lower. By means of figure 5.10 it proves that our predicationis correct.

5.5.4 Surplus Return and Surplus Vola

All of the models have the same RenditeDBO, so the factor, which influence the surplusreturn is RenditePA, which is calculated as:

RenditePAt =bPAt − (PAt−1 −Benefitst)

PAt−1 −Benefitst

(5.6)

Since Benefits are also same among the different models, the current bPA and the pre-vious PA play an important roll for calculating RenditePAt. As described in, the valueof bPAt − (PAt−1 − Benefitst) is actually ReturnPA. Then the Markowitz PF22 hassimultaneously the greatest denominator and numerator. Hence, it is difficult to predicatethe performance with respect to the criterium “Surplus Return”. Let us directly see theresult from figure 5.11.

Analogous as the other immunization models, the surplus returns for Markowitz PF 22and PF9 also have continuous springs over the 30 years. However the springs of theMarkowitz PF9 is much greater than those of the other models. Although at some timesteps, it accomplish greater surplus returns, at other time steps the disadvantages aremore outstanding. The springs of Markowitz PF22 are also greater in comparison withother immunization models. But the average surplus returns are still larger than those ofthem.

In terms of surplus volatility, both of the Markowitz portfolios hold lower volatility thanthe other immunization models. However, the dominance is also not so great.

5.6 Conclusion

By means of several criteria we have we have compared the results of different immu-nization approaches and those of the MVP with different instrument classes generated by


Figure 5.10: Funding Level Comparison with Markowitz Optmization Results

Figure 5.11: Surplus Return Comparison with Markowitz Optimization Results


the Markowitz optimization with resampling method. In connection with the results thefollowing comments and conclusions should be pointed out:

• The main goal of immunization approaches is to protect the holding assets yieldagainst market risks. Therefore, we are seeking after that portfolio which has themost similar characteristics in terms of market risks as those of the liability. Fromthis perspective, the criteria we used for comparison can not show us how good theimmunization effect is. However, as described the criteria used in the last sectionsare also the important factors which the investors and the sponsors are interestedin. It is necessary to implement the comparison from several different points of view.Since each immunization model have different performance directed towards differ-ent criteria, we can not simply reach the conclusion that one immunization modelis the best one while the others should be given up. Even by immunization effectdescribed in chapter 3, the more complex models have achieve better immunizationeffects, but they have also own respective shortcomings. On this account it is not tochoose the best model but the most suitable model according to the specific pensionplan and the demand of the sponsors.

• From the results of comparison we can find out that the key rate duration model andthe principal component duration model have dominance at some criteria. However,the dominance is not that outstanding. One reason is that the instruments-class islimited in 9 government bonds. The yield difference of these government bonds ismore or less slight. It is expected the difference of comparing results will be widerby expanding the instruments universe.

• The Markowitz optimization we used is not the traditional one which concentratesitself only on asset side. We chose the MVPs with respect to the according sur-plus return and the surplus volatility. In other words, the values of liability sideare also taken into consideration during the optimization. In case we take the 22instruments-classes, at almost all the comparing criteria this MVP holds relativelygreat dominance thanks to the high return achieved by expanded instruments, suchas Corporate Bonds, European Stocks and so on. However, if we only use the pre-vious 9 government bonds, the comparing results of the Markowitz MVP are notas nice as those of other immunization models at most of the criteria. It impliesthat by using the same instruments universe our immunization results, especiallythose of key rate duration model and principal component duration model have stilladvantages compared with the results of MVP by resampling optimization.

Chapter 6

Summary and Outlook

This thesis aims at introducing and comparing different Liability Driven Investment (LDI)Optimization approaches, which are demonstrated in terms of immunization. First of all,we introduced the immunization models from the theoretic perspective. In this part, allthe inputs are considered as deterministic and static. Then, with help of a case studywe implement the models in stochastic and dynamic framework which is more consistentwith the realistic market. Thanks to the Risklab Economic Scenario Generator (ESG) wesimulated 1000 scenarios of the valuations on both liabilities side and assets side. Thebasic idea of the dynamic and stochastic optimization is to consider one scenario as deter-ministic and static. Then we looked for the optimal portfolio for this scenario accordingto different immunization approaches. At last we investigated the mean portfolio of these1000 scenarios to analyze its characteristics.

Using some criteria we compared the results of different immunization models not onlyamong themselves but with that of MVP generated by Markowitz optimization using re-sampling method as well. The immunization models such as the “key rate duration model”and the “principal component duration model” have dominance at most criteria comparedwith the “traditional duration model” and the “generalized duration vector model”. Evencompared with the MVP generated by resampling optimization by using the same in-struments universe, these two models still have advantages. However, we can not for thatreason draw the conclusion that all the other models are not the proper immunizationmodels. The more complex the immunization model is, the better the immunization effectwill be. Nevertheless, every model has its own advantages and shortcomings. Therefore,to choose a proper immunization models is more dependent on the specific pension planand the sponsor‘s demand.

There exist still several points which can not be realized during our implement in rangeof this thesis:

• As mentioned in chapter 4 we use 1000 streams to simulate the expected benefitsprocess. However, if we want to implement the models in “real” dynamic frameworkwe should have 1000 liability cash flows at each time step. An alternative will bethat we begin with these 1000 streams and study all the possibilities how the currentstream will flow in the future. Both of the ways demand high computational costs.

111

CHAPTER 6. SUMMARY AND OUTLOOK 112

For that reason we implement the optimization still with the only data we have.Therefore, in a sense the dynamic framework, in which we implement our models,is simplified.

• In range of this thesis we immunize our assets portfolio only against interest rate riskexposures. However, inflation has also effects on the liability and asset sides. Hence,in terms of a complete LDI optimization it is also important during the implementa-tion to find the optimal allocation, which is captured also based on the considerationof inflation rate risk exposure. Scherer mentioned we could protect the assets againstinflation risks by using inflation swaps, which is also a good idea for the future study.

• As we implement the LDI optimization in terms of portfolio immunization, the op-timal portfolios we found out have low volatility. At the same time, they can onlyachieve low return. In order to achieve higher returns, we should expand the instru-ments universe. In this sense futures and swaps will be a good choice. It is expectedthat the asset portfolio can achieve greater return if the portfolio absorbs futuresand swaps.

As one of the latest buzzword being used by consultants and fund managers Liabilitydriven investment will win more and more focusses in the field of pension fund manage-ment. On this account, more studies are expected to be brought out not only in researchfield but also in practice sense and this thesis gives some rough ideas that hopefully willlead the readers to more better studies in the field of LDI.

Appendix A

List of Symbols

aglt total actuarial gains and losses at time taglat actuarial gains and losses due to unexpected movements in assets at

time tagllt actuarial gains and losses due to unexpected movements in liabilities

at time tBenefitst paid benefits at time tCF cash flowsConB convexity of bondsConl convexity of liabilitiescorridort corridor at time tDB duration of bondsDl duration of liabilitiesDBLt defined benefit liabilities at time tDBOt defined benefit obligation is the present value of liabilitiesEKRDt excess key rate duration at time teDBOt expected DBO at time tFLt funding level at time tICt interest cost at time t

KRCBit key rate convexity of bond i at time t

KRC lt key rate convexity of liabilities at time t

KRDBt key rate duration of bonds at time t

KRDlt key rate duration of liabilities at time t

llcorr lower level of a corridorPAt plan assets at time tPC principal components

PCCBit principal components convexity of bond i at time t

PCC lt principal components convexity of liabilities at time t

PCDBit principal components duration of bond i at time t

PCDlt principal components duration of liabilities at time t

PEt pension expense at time traglt recognized actuarial gains and losses at time t

113

APPENDIX A. LIST OF SYMBOLS 114

SCt service cost at time tuaglt unrealized actuarial gains and losses at time tulcorr upper level of a corridorω portfolio weight in asset class iwlcorr withdrawal level of a corridor

Appendix B

Summary of Immunization Models’Features

115

APPENDIX B. SUMMARY OF IMMUNIZATION MODELS’ FEATURES 116

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Correlation Matrix of Spot RatesChanges

117

APPENDIX C. CORRELATION MATRIX OF SPOT RATES CHANGES 118

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Appendix D

Mean of Optimal PortfolioAllocation

120

APPENDIX D. MEAN OF OPTIMAL PORTFOLIO ALLOCATION 121

Table D.1: Traditional Duration Matching Model

Time Bond Bond Bond Bond Bond Bond Bond Bond BondStep 1 2 3 4 5 6 7 8 9

1 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 100,00%2 0,31% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 99,69%3 0,90% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 99,10%4 1,93% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 98,07%5 3,01% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 96,99%6 4,48% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 95,52%7 6,02% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 93,98%8 7,67% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 92,33%9 9,57% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 90,43%10 11,72% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 88,28%11 14,05% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 85,95%12 16,22% 0,03% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 83,75%13 18,44% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 81,56%14 20,57% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 79,43%15 22,56% 0,17% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 77,27%16 23,96% 0,91% 0,00% 0,00% 0,05% 0,01% 0,00% 0,00% 75,07%17 23,86% 2,95% 0,00% 0,00% 0,28% 0,06% 0,00% 0,00% 72,84%18 20,79% 7,79% 0,00% 0,00% 0,96% 0,20% 0,01% 0,01% 70,23%19 13,61% 15,38% 0,00% 0,00% 3,03% 0,63% 0,01% 0,01% 67,34%20 5,98% 21,34% 0,00% 0,00% 7,08% 1,71% 0,01% 0,01% 63,87%21 1,60% 22,54% 0,00% 0,00% 11,92% 3,49% 0,02% 0,02% 60,39%22 0,21% 20,32% 0,00% 0,00% 16,37% 6,27% 0,02% 0,03% 56,77%23 0,19% 17,39% 0,02% 0,05% 19,01% 9,92% 0,12% 0,15% 53,14%24 1,28% 14,66% 0,14% 0,22% 18,93% 13,79% 0,72% 0,93% 49,32%25 2,84% 12,41% 0,34% 0,52% 17,10% 17,32% 1,73% 2,17% 45,58%26 6,62% 9,19% 0,86% 1,43% 13,46% 17,69% 4,20% 5,46% 41,08%27 10,69% 6,16% 1,64% 2,61% 9,77% 15,82% 7,53% 9,40% 36,37%28 13,77% 3,79% 2,65% 4,06% 7,68% 12,39% 10,85% 13,08% 31,72%29 14,00% 3,24% 3,61% 5,31% 7,41% 10,65% 12,66% 14,91% 28,22%30 12,90% 3,63% 4,52% 6,23% 7,90% 10,28% 13,44% 15,71% 25,39%

Table D.2: Generalized Duration Vector ModelTime Bond Bond Bond Bond Bond Bond Bond Bond BondStep 1 2 3 4 5 6 7 8 9

1 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 100,00%2 0,03% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 99,97%3 0,20% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 99,80%4 0,69% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 99,31%5 1,37% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 98,63%6 2,19% 0,01% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 97,80%7 3,38% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 96,62%8 4,61% 0,14% 0,09% 0,00% 0,00% 0,00% 0,00% 0,00% 95,16%9 6,29% 0,26% 0,10% 0,03% 0,00% 0,00% 0,00% 0,00% 93,31%10 8,41% 0,28% 0,34% 0,10% 0,00% 0,00% 0,00% 0,00% 90,87%11 9,98% 0,83% 0,91% 0,42% 0,01% 0,00% 0,00% 0,00% 87,85%12 11,49% 1,68% 1,45% 0,45% 0,05% 0,01% 0,00% 0,00% 84,86%13 11,02% 2,86% 2,65% 1,63% 0,09% 0,00% 0,00% 0,00% 81,76%14 9,36% 3,45% 4,78% 3,44% 0,38% 0,01% 0,00% 0,00% 78,57%15 6,77% 3,64% 6,89% 6,34% 1,02% 0,02% 0,00% 0,00% 75,30%16 3,95% 3,06% 7,16% 11,30% 2,57% 0,11% 0,01% 0,01% 71,81%17 1,50% 2,00% 7,02% 15,13% 5,49% 0,38% 0,07% 0,08% 68,33%18 0,43% 0,66% 4,67% 18,47% 9,90% 0,91% 0,18% 0,22% 64,55%19 0,09% 0,31% 2,43% 17,64% 15,59% 2,07% 0,43% 0,53% 60,92%20 0,00% 0,07% 1,52% 15,13% 20,23% 3,81% 0,97% 1,21% 57,05%21 0,00% 0,06% 1,49% 12,54% 22,52% 6,02% 1,82% 2,23% 53,31%22 0,00% 0,12% 2,13% 12,01% 20,73% 8,82% 3,11% 3,80% 49,28%23 0,00% 0,23% 3,28% 12,42% 17,53% 10,68% 4,84% 5,98% 45,03%24 0,00% 0,54% 4,51% 12,54% 15,10% 11,45% 6,67% 8,25% 40,94%25 0,01% 0,93% 5,60% 12,28% 13,66% 11,60% 8,37% 10,33% 37,23%26 0,03% 1,50% 6,33% 11,80% 12,92% 11,81% 9,88% 11,97% 33,76%27 0,05% 2,24% 6,85% 11,34% 12,45% 12,03% 11,19% 13,30% 30,56%28 0,14% 3,26% 7,19% 10,87% 12,07% 12,31% 12,42% 14,44% 27,29%29 0,35% 4,08% 7,35% 10,59% 11,93% 12,67% 13,39% 15,19% 24,46%30 0,66% 4,66% 7,45% 10,47% 11,97% 13,08% 14,18% 15,67% 21,86%

APPENDIX D. MEAN OF OPTIMAL PORTFOLIO ALLOCATION 122

Table D.3: Key Rate Duration Model


1 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,02% 99,98%2 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,49% 1,51% 98,00%3 0,00% 0,00% 0,00% 0,00% 0,00% 0,02% 1,44% 2,81% 95,73%4 0,00% 0,00% 0,00% 0,00% 0,00% 0,09% 3,04% 4,25% 92,62%5 0,00% 0,00% 0,00% 0,00% 0,00% 0,32% 4,37% 5,58% 89,73%6 0,00% 0,00% 0,00% 0,00% 0,03% 0,43% 6,43% 6,98% 86,13%7 0,00% 0,00% 0,00% 0,00% 0,02% 0,80% 8,19% 8,44% 82,55%8 0,00% 0,00% 0,00% 0,01% 0,08% 1,17% 10,15% 9,38% 79,22%9 0,00% 0,00% 0,00% 0,02% 0,12% 1,85% 12,05% 10,41% 75,54%10 0,00% 0,00% 0,00% 0,04% 0,16% 2,80% 14,06% 11,21% 71,73%11 0,00% 0,00% 0,01% 0,09% 0,44% 4,15% 15,47% 12,13% 67,72%12 0,00% 0,00% 0,01% 0,10% 0,60% 6,10% 15,88% 13,26% 64,05%13 0,00% 0,00% 0,01% 0,24% 1,15% 7,79% 15,80% 14,39% 60,61%14 0,00% 0,01% 0,03% 0,46% 1,90% 9,26% 15,55% 15,50% 57,28%15 0,01% 0,02% 0,06% 0,77% 2,83% 10,34% 15,55% 16,37% 54,06%16 0,02% 0,04% 0,11% 1,26% 3,94% 10,92% 15,67% 17,21% 50,83%17 0,06% 0,09% 0,21% 1,81% 5,04% 11,25% 15,82% 17,92% 47,81%18 0,13% 0,18% 0,41% 2,49% 6,14% 11,36% 15,99% 18,58% 44,71%19 0,24% 0,31% 0,69% 3,19% 6,78% 11,55% 16,17% 19,11% 41,96%20 0,42% 0,49% 1,04% 3,81% 7,32% 11,75% 16,40% 19,57% 39,21%21 0,58% 0,61% 1,43% 4,34% 7,72% 11,98% 16,65% 19,96% 36,73%22 0,78% 0,77% 1,91% 4,84% 8,08% 12,22% 16,89% 20,24% 34,28%23 0,99% 0,96% 2,45% 5,31% 8,42% 12,47% 17,10% 20,41% 31,88%24 1,14% 1,20% 3,00% 5,77% 8,79% 12,78% 17,27% 20,39% 29,66%25 1,22% 1,45% 3,51% 6,19% 9,16% 13,13% 17,39% 20,24% 27,68%26 1,27% 1,76% 3,98% 6,63% 9,59% 13,56% 17,48% 19,96% 25,76%27 1,31% 2,16% 4,43% 7,08% 10,01% 13,96% 17,49% 19,61% 23,95%28 1,45% 2,69% 4,91% 7,51% 10,44% 14,38% 17,44% 19,16% 22,02%29 1,71% 3,12% 5,29% 7,87% 10,83% 14,84% 17,41% 18,66% 20,26%30 2,00% 3,51% 5,65% 8,23% 11,25% 15,31% 17,39% 18,08% 18,59%

Table D.4: Principal Component Duration Model


1 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,01% 0,00% 99,99%2 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 1,29% 0,00% 98,71%3 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 3,02% 0,00% 96,98%4 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 5,56% 0,00% 94,44%5 0,00% 0,00% 0,00% 0,00% 0,00% 0,06% 7,94% 0,00% 92,00%6 0,00% 0,00% 0,00% 0,00% 0,00% 0,18% 10,90% 0,00% 88,92%7 0,00% 0,00% 0,00% 0,00% 0,00% 0,33% 13,90% 0,00% 85,77%8 0,00% 0,00% 0,00% 0,00% 0,03% 0,64% 16,62% 0,00% 82,70%9 0,00% 0,00% 0,00% 0,00% 0,05% 1,10% 19,59% 0,00% 79,26%10 0,00% 0,00% 0,00% 0,00% 0,12% 1,61% 22,77% 0,00% 75,50%11 0,00% 0,00% 0,00% 0,01% 0,27% 3,34% 24,58% 0,01% 71,80%12 0,00% 0,00% 0,00% 0,02% 0,27% 5,46% 25,77% 0,03% 68,45%13 0,00% 0,00% 0,00% 0,03% 0,75% 9,35% 23,84% 0,03% 66,00%14 0,00% 0,00% 0,00% 0,08% 1,45% 13,85% 20,62% 0,09% 63,91%15 0,00% 0,00% 0,01% 0,18% 2,59% 18,59% 16,30% 0,20% 62,13%16 0,00% 0,00% 0,01% 0,37% 4,42% 22,87% 11,39% 0,45% 60,49%17 0,00% 0,00% 0,05% 0,72% 6,62% 25,91% 6,92% 0,97% 58,81%18 0,00% 0,01% 0,12% 1,48% 9,30% 26,93% 3,55% 2,10% 56,50%19 0,00% 0,01% 0,27% 2,57% 12,00% 24,60% 3,18% 3,90% 53,47%20 0,00% 0,02% 0,59% 4,07% 13,41% 21,77% 3,94% 6,58% 49,62%21 0,00% 0,06% 0,95% 5,42% 14,25% 18,98% 5,36% 9,10% 45,88%22 0,00% 0,12% 1,51% 7,09% 13,98% 16,13% 7,38% 12,15% 41,63%23 0,01% 0,25% 2,43% 8,36% 13,12% 13,90% 9,59% 14,99% 37,37%24 0,06% 0,53% 3,43% 8,88% 12,22% 12,93% 11,49% 16,70% 33,75%25 0,10% 0,90% 4,31% 9,06% 11,49% 12,59% 13,09% 17,75% 30,71%26 0,23% 1,34% 4,93% 9,05% 11,20% 12,77% 14,29% 18,09% 28,10%27 0,43% 1,86% 5,36% 9,06% 11,10% 13,06% 15,21% 18,15% 25,77%28 0,82% 2,49% 5,59% 9,09% 11,19% 13,51% 15,95% 17,90% 23,46%29 1,32% 2,91% 5,70% 9,28% 11,47% 13,99% 16,37% 17,45% 21,51%30 1,80% 3,24% 5,84% 9,58% 11,85% 14,46% 16,61% 16,90% 19,71%

Appendix E

MVP of Resampling Optimization

123

APPENDIX E. MVP OF RESAMPLING OPTIMIZATION 124

Table E.1: MVP of Resampling Optimization with 9 BondsTime Bond Bond Bond Bond Bond Bond Bond Bond BondStep 1 2 3 4 5 6 7 8 9

1 10,47% 0,00% 0,40% 1,44% 0,00% 2,27% 1,90% 2,90% 80,62%2 10,28% 0,00% 0,00% 0,00% 0,00% 3,61% 0,00% 7,51% 78,60%3 6,74% 0,11% 1,93% 0,00% 6,49% 0,00% 0,00% 6,58% 78,14%4 2,13% 0,33% 1,80% 1,87% 0,57% 2,80% 0,00% 26,33% 64,17%5 3,91% 6,72% 1,40% 1,01% 0,00% 0,00% 0,00% 16,82% 70,13%6 9,14% 0,00% 0,85% 1,03% 1,17% 2,17% 6,73% 7,96% 70,96%7 8,44% 3,94% 0,00% 0,00% 2,01% 2,43% 0,00% 12,36% 70,81%8 12,49% 2,11% 0,00% 2,72% 1,45% 0,00% 0,00% 12,44% 68,78%9 6,17% 2,13% 0,00% 4,52% 4,10% 3,22% 8,22% 5,12% 66,52%10 7,88% 2,64% 5,08% 3,83% 2,81% 0,00% 2,98% 6,32% 68,45%11 10,83% 8,79% 1,47% 2,12% 1,11% 0,00% 0,00% 9,92% 65,77%12 10,89% 0,35% 5,66% 2,18% 0,00% 3,23% 0,00% 16,59% 61,09%13 1,63% 5,54% 0,99% 0,00% 2,92% 3,99% 4,04% 34,02% 46,88%14 13,07% 1,66% 0,13% 3,13% 0,00% 0,00% 5,49% 30,40% 46,12%15 10,45% 2,56% 6,90% 6,22% 2,39% 0,00% 2,06% 9,57% 59,86%16 4,05% 5,39% 3,30% 0,71% 0,39% 7,23% 5,75% 32,24% 40,94%17 9,93% 1,42% 0,00% 0,72% 2,55% 7,30% 3,36% 41,02% 33,69%18 8,93% 8,75% 4,06% 0,00% 4,30% 0,78% 2,09% 29,91% 41,17%19 7,37% 1,34% 0,60% 4,50% 0,00% 1,56% 18,34% 36,98% 29,30%20 14,24% 6,23% 0,40% 5,91% 3,32% 0,25% 3,28% 27,66% 38,71%21 12,24% 0,00% 1,17% 0,00% 3,14% 14,54% 6,41% 31,38% 31,12%22 7,76% 3,28% 9,67% 8,22% 2,29% 7,66% 1,50% 29,08% 30,53%23 6,36% 4,04% 7,11% 4,50% 1,19% 8,62% 5,56% 37,73% 24,90%24 9,06% 4,41% 18,55% 0,00% 10,73% 0,00% 0,00% 17,21% 40,03%25 6,65% 5,26% 0,00% 4,88% 4,32% 0,00% 20,90% 46,03% 11,96%26 15,22% 3,03% 1,20% 1,52% 11,08% 5,05% 9,14% 29,49% 24,25%27 8,66% 11,32% 3,96% 5,32% 0,00% 1,83% 13,83% 43,19% 11,89%28 11,12% 7,25% 8,64% 5,86% 0,00% 4,99% 4,32% 40,04% 17,78%29 12,72% 3,05% 12,36% 3,32% 12,36% 0,00% 6,51% 28,90% 20,77%30 14,50% 4,15% 1,19% 5,00% 10,28% 5,20% 7,25% 39,17% 13,26%

APPENDIX E. MVP OF RESAMPLING OPTIMIZATION 125

Table E.2: MVP of Resampling Optimization with 22 Instruments

(a) MVP 1Time Corporate Government Bonds EuropeanStep Bonds 1 2 3 4 5 6 7 8 9 Stocks

1 2,17% 0% 0% 0% 0% 0% 0% 0% 0% 82,66% 1,62%2 7,79% 0% 0% 0% 0% 0% 0% 0% 0% 81,38% 3,34%3 9,31% 0% 0% 0% 0% 0% 0% 0% 0% 80,55% 2,65%4 14,82% 0% 0% 0% 0% 0% 0% 0% 0% 78,03% 2,82%5 14,70% 0% 0% 0% 0% 0% 0% 0% 0% 77,83% 1,06%6 16,44% 0% 0% 0% 0% 0% 0% 0% 0% 76,92% 1,13%7 10,22% 0% 0% 0% 0% 0% 0% 0% 0% 77,01% 1,03%8 19,44% 0% 0% 0% 0% 0% 0% 0% 0% 73,78% 2,44%9 22,63% 0% 0% 0% 0% 0% 0% 0% 0% 71,70% 2,44%10 19,88% 0% 0% 0% 0% 0% 0% 0% 0% 70,65% 1,25%11 19,49% 0% 0% 0% 0% 0% 0% 0% 0% 69,54% 1,83%12 23,31% 0% 0% 0% 0% 0% 0% 0% 0% 69,16% 2,97%13 22,65% 0% 0% 0% 0% 0% 0% 0% 0% 67,63% 0,66%14 29,40% 0% 0% 0% 0% 0% 0% 0% 0% 63,54% 2,27%15 28,03% 0% 0% 0% 0% 0% 0% 0% 0% 63,77% 1,80%16 31,85% 0% 0% 0% 0% 0% 0% 0% 0% 61,39% 2,01%17 35,60% 0% 0% 0% 0% 0% 0% 0% 0% 58,48% 1,88%18 34,31% 0% 0% 0% 0% 0% 0% 0% 0% 57,53% 1,51%19 37,75% 0% 0% 0% 0% 0% 0% 0% 0% 56,60% 2,18%20 38,61% 0% 0% 0% 0% 0% 0% 0% 0% 54,07% 1,38%21 39,60% 0% 0% 0% 0% 0% 0% 0% 0% 52,97% 1,52%22 44,48% 0% 0% 0% 0% 0% 0% 0% 0% 49,47% 1,78%23 44,50% 0% 0% 0% 0% 0% 0% 0% 0% 48,72% 1,31%24 48,16% 0% 0% 0% 0% 0% 0% 0% 0% 47,74% 1,19%25 48,24% 0% 0% 0% 0% 0% 0% 0% 0% 45,32% 1,54%26 49,38% 0% 0% 0% 0% 0% 0% 0% 0% 44,55% 2,29%27 52,65% 0% 0% 0% 0% 0% 0% 0% 0% 40,56% 0,89%28 53,11% 0% 0% 0% 0% 0% 0% 0% 0% 41,02% 1,69%29 58,83% 0% 0% 0% 0% 0% 0% 0% 0% 37,84% 1,93%30 56,06% 0% 0% 0% 0% 0% 0% 0% 0% 37,83% 1,32%

(b) MVP 2Time Inflation Bonds Real AbsoluteStep 1 2 3 4 5 6 7 8 9 Estate Return

1 0% 0% 0% 0% 0% 0% 0% 0% 0,16% 0% 13,39%2 0% 0% 0% 0% 0% 0% 0% 0% 0,54% 0% 6,95%3 0% 0% 0% 0% 0% 0% 0% 0% 0,14% 0% 7,36%4 0% 0% 0% 0% 0% 0% 0% 0% 0,32% 0% 4,02%5 0% 0% 0% 0% 0% 0% 0% 0% 0% 0,22% 6,18%6 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 5,52%7 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 11,73%8 0% 0% 0% 0% 0% 0% 0% 0% 0,11% 0% 4,24%9 0% 0% 0% 0% 0% 0% 0% 0% 0,00% 0% 3,23%10 0% 0% 0% 0% 0% 0% 0% 0% 0,39% 0% 7,83%11 0% 0% 0% 0% 0% 0% 0% 0% 0,03% 0% 9,11%12 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 4,57%13 0% 0% 0% 0% 0% 0% 0% 0% 0,08% 0% 8,98%14 0% 0% 0% 0% 0% 0% 0% 0% 0,20% 0% 4,59%15 0% 0% 0% 0% 0% 0% 0% 0% 0,16% 0% 6,24%16 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 4,75%17 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 4,04%18 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 6,65%19 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 3,47%20 0% 0% 0% 0% 0% 0% 0% 0% 0% 0,30% 5,63%21 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 5,91%22 0% 0% 0% 0% 0% 0% 0% 0% 0,24% 0% 4,03%23 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 5,47%24 0% 0% 0% 0% 0% 0% 0% 0% 0% 0,32% 2,59%25 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 4,89%26 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 3,78%27 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 5,90%28 0% 0% 0% 0% 0% 0% 0% 0% 0% 0,08% 4,11%29 0% 0% 0% 0% 0% 0% 0% 0% 0,06% 0,03% 1,31%30 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 4,79%

Appendix F

Evaluations of Chapter 4

126

APPENDIX F. EVALUATIONS OF CHAPTER 4 127

Table F.1: Median Durations and Convexities of Liabilities and AssetsTime Duration Convexity Time Duration ConvexityStep Liability Portfolio Liability Portfolio Step Liability Portfolio Liability Portfolio

1 31,45 28,78 1322,59 855,74 16 21,79 21,79 674,71 639,352 30,77 28,76 1269,45 854,79 17 21,25 21,25 643,88 621,683 29,96 28,75 1210,46 854,09 18 20,69 20,69 615,43 606,804 29,16 28,71 1151,20 852,33 19 20,06 20,06 582,22 582,075 28,56 28,55 1108,08 845,68 20 19,45 19,45 550,43 550,436 27,86 27,86 1055,69 825,35 21 18,95 18,95 525,76 525,767 27,24 27,24 1013,65 805,63 22 18,41 18,41 498,72 498,728 26,76 26,76 978,11 792,00 23 17,85 17,85 474,39 474,399 26,15 26,15 938,33 773,88 24 17,38 17,38 450,85 450,8510 25,47 25,47 894,52 753,02 25 16,86 16,86 427,08 427,0811 24,93 24,93 858,49 735,69 26 16,41 16,41 406,89 406,8912 24,16 24,16 813,15 712,96 27 15,90 15,90 385,08 385,0813 23,62 23,62 780,38 696,09 28 15,36 15,36 362,63 362,6314 23,03 23,03 745,35 677,44 29 14,86 14,86 343,03 343,0315 22,39 22,39 708,21 658,24 30 14,41 14,41 324,55 324,55

Table F.2: Median of Portfolio Key Rate DurationKey Rate 1 2 3 4 5 6 7 8Maturity 1 3 5 8 13 18 24 29

1 0,000 0,000 0,000 0,000 0,000 0,000 0,000 28,7772 0,000 0,000 0,000 0,000 0,000 0,000 0,000 28,7593 0,000 0,000 0,000 0,000 0,000 0,000 0,000 28,7284 0,000 0,000 0,000 0,000 0,000 0,336 0,168 27,9845 0,000 0,000 0,000 0,000 0,000 1,036 0,439 26,4116 0,000 0,000 0,000 0,000 0,240 1,544 0,533 24,6727 0,000 0,000 0,000 0,000 0,655 1,719 0,595 23,8478 0,000 0,000 0,000 0,000 0,861 1,987 0,630 22,5819 0,000 0,000 0,000 0,000 1,243 2,157 0,671 21,62510 0,000 0,000 0,000 0,000 1,479 2,330 0,715 20,50711 0,000 0,000 0,000 0,000 1,646 2,505 0,797 19,46312 0,000 0,000 0,000 0,436 1,680 2,675 0,902 18,13613 0,000 0,000 0,000 0,552 1,717 2,789 0,954 17,36614 0,000 0,000 0,000 0,692 1,733 2,883 1,007 16,32615 0,000 0,000 0,000 0,804 1,759 2,985 1,054 15,41716 0,000 0,000 0,118 0,861 1,782 3,113 1,111 14,43917 0,000 0,000 0,193 0,904 1,801 3,212 1,153 13,66518 0,000 0,000 0,272 0,948 1,818 3,299 1,190 12,85619 0,000 0,003 0,329 0,988 1,838 3,370 1,220 12,02120 0,010 0,018 0,368 1,017 1,865 3,440 1,249 11,24721 0,012 0,034 0,392 1,043 1,897 3,506 1,274 10,57422 0,015 0,062 0,420 1,067 1,925 3,553 1,291 9,80823 0,017 0,081 0,447 1,092 1,953 3,588 1,302 9,14024 0,020 0,099 0,474 1,125 1,981 3,595 1,302 8,50825 0,025 0,116 0,503 1,163 2,005 3,588 1,293 7,92726 0,029 0,131 0,531 1,208 2,029 3,561 1,277 7,34827 0,033 0,153 0,566 1,251 2,045 3,518 1,255 6,83628 0,039 0,172 0,601 1,295 2,055 3,454 1,227 6,27929 0,045 0,185 0,632 1,338 2,066 3,391 1,199 5,79330 0,052 0,198 0,648 1,384 2,077 3,323 1,165 5,342

Table F.3: Median of Portfolio PCDTime Steps PCD 1 PCD 2 PCD 3 Time Steps PCD 1 PCD 2 PCD 3

1 2,221 -1,111 -0,233 16 1,720 -0,709 -0,1162 2,219 -1,110 -0,233 17 1,682 -0,681 -0,1103 2,217 -1,109 -0,232 18 1,641 -0,652 -0,1044 2,209 -1,104 -0,231 19 1,594 -0,622 -0,0985 2,172 -1,073 -0,221 20 1,550 -0,593 -0,0916 2,123 -1,032 -0,207 21 1,518 -0,571 -0,0857 2,087 -1,003 -0,198 22 1,475 -0,541 -0,0788 2,050 -0,971 -0,188 23 1,436 -0,516 -0,0719 2,014 -0,942 -0,179 24 1,399 -0,491 -0,06510 1,970 -0,907 -0,167 25 1,364 -0,467 -0,05911 1,934 -0,876 -0,157 26 1,327 -0,443 -0,05312 1,882 -0,835 -0,144 27 1,292 -0,420 -0,04713 1,847 -0,806 -0,135 28 1,249 -0,392 -0,04114 1,803 -0,772 -0,127 29 1,215 -0,369 -0,03615 1,763 -0,742 -0,122 30 1,180 -0,346 -0,030

Appendix G

Evaluations of Chapter 5

Here are the lists of the evaluations in chapter 5. What is to be noticed is that theevaluations of time step 0 are the initial values of the results, so they are not listed in thetables.

128

APPENDIX G. EVALUATIONS OF CHAPTER 5 129

Table G.1: Plan Assets before ContributionsTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duration 9 22

1 9,12E+08 9,12E+08 9,12E+08 9,12E+08 9,13E+08 9,18E+082 9,75E+08 9,75E+08 9,76E+08 9,76E+08 9,78E+08 9,87E+083 1,03E+09 1,03E+09 1,03E+09 1,03E+09 1,03E+09 1,04E+094 1,10E+09 1,10E+09 1,10E+09 1,10E+09 1,11E+09 1,12E+095 1,16E+09 1,16E+09 1,17E+09 1,17E+09 1,17E+09 1,18E+096 1,23E+09 1,23E+09 1,23E+09 1,23E+09 1,24E+09 1,26E+097 1,33E+09 1,33E+09 1,34E+09 1,34E+09 1,33E+09 1,36E+098 1,43E+09 1,43E+09 1,44E+09 1,45E+09 1,43E+09 1,47E+099 1,51E+09 1,51E+09 1,52E+09 1,53E+09 1,52E+09 1,57E+0910 1,60E+09 1,60E+09 1,62E+09 1,62E+09 1,60E+09 1,66E+0911 1,68E+09 1,69E+09 1,72E+09 1,72E+09 1,69E+09 1,77E+0912 1,78E+09 1,78E+09 1,82E+09 1,82E+09 1,79E+09 1,86E+0913 1,89E+09 1,90E+09 1,92E+09 1,92E+09 1,91E+09 1,97E+0914 1,98E+09 1,99E+09 2,03E+09 2,03E+09 2,01E+09 2,09E+0915 2,07E+09 2,10E+09 2,13E+09 2,13E+09 2,10E+09 2,21E+0916 2,17E+09 2,22E+09 2,25E+09 2,25E+09 2,23E+09 2,31E+0917 2,29E+09 2,35E+09 2,37E+09 2,37E+09 2,35E+09 2,45E+0918 2,38E+09 2,46E+09 2,48E+09 2,48E+09 2,42E+09 2,56E+0919 2,52E+09 2,58E+09 2,59E+09 2,58E+09 2,58E+09 2,67E+0920 2,66E+09 2,71E+09 2,71E+09 2,71E+09 2,66E+09 2,82E+0921 2,77E+09 2,82E+09 2,83E+09 2,83E+09 2,80E+09 2,92E+0922 2,92E+09 2,95E+09 2,94E+09 2,94E+09 2,90E+09 3,05E+0923 3,07E+09 3,08E+09 3,08E+09 3,08E+09 3,07E+09 3,20E+0924 3,19E+09 3,18E+09 3,19E+09 3,18E+09 3,11E+09 3,29E+0925 3,35E+09 3,33E+09 3,32E+09 3,33E+09 3,35E+09 3,46E+0926 3,42E+09 3,43E+09 3,43E+09 3,42E+09 3,38E+09 3,62E+0927 3,48E+09 3,50E+09 3,51E+09 3,50E+09 3,45E+09 3,66E+0928 3,57E+09 3,59E+09 3,60E+09 3,60E+09 3,53E+09 3,81E+0929 3,62E+09 3,65E+09 3,65E+09 3,64E+09 3,58E+09 3,84E+0930 3,69E+09 3,73E+09 3,71E+09 3,71E+09 3,67E+09 3,96E+09

Table G.2: bPA VolatilityTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22



Table G.3: Plan Assets before ContributionsTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22


Table G.4: bPA VolatilityTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22



Table G.5: ContributionsTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22


Table G.6: Volatility of ContributionsTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22



Table G.7: Present Value of ContributionsTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22


Table G.8: Volatility of Present ValueTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22



Table G.9: Funding LevelTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22

1 1,03 1,03 1,03 1,03 1,04 1,042 1,04 1,04 1,04 1,04 1,04 1,053 1,03 1,03 1,03 1,03 1,04 1,054 1,03 1,03 1,03 1,03 1,03 1,055 1,02 1,02 1,02 1,02 1,03 1,046 1,01 1,01 1,02 1,02 1,02 1,047 1,01 1,01 1,02 1,02 1,02 1,048 1,01 1,01 1,02 1,02 1,01 1,039 1,00 1,00 1,01 1,01 1,01 1,0310 1,00 1,00 1,02 1,02 1,00 1,0311 0,99 1,00 1,01 1,01 0,99 1,0312 0,99 0,99 1,01 1,01 0,99 1,0313 0,98 0,99 1,00 1,00 0,99 1,0214 0,98 0,99 1,00 1,00 0,98 1,0215 0,98 0,99 1,00 1,00 0,98 1,0316 0,97 0,99 1,00 1,00 0,99 1,0217 0,97 1,00 1,00 1,00 0,99 1,0218 0,97 1,00 1,00 1,00 0,98 1,0319 0,97 1,00 1,00 1,00 0,99 1,0220 0,98 1,00 1,00 0,99 0,98 1,0321 0,99 1,00 1,00 1,00 0,99 1,0322 0,99 1,00 0,99 0,99 0,99 1,0323 1,00 1,00 1,00 0,99 0,99 1,0324 1,00 1,00 1,00 1,00 0,98 1,0225 1,00 1,00 1,00 1,00 1,00 1,0326 0,99 0,99 1,00 0,99 0,98 1,0427 0,99 0,99 0,99 0,99 0,98 1,0328 0,98 0,99 0,99 0,99 0,97 1,0429 0,98 0,99 0,99 0,99 0,97 1,0330 0,98 0,99 0,99 0,99 0,98 1,03

Table G.10: Volatility of Funding LevelTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22

1 0,05 0,05 0,05 0,05 0,04 0,042 0,05 0,06 0,05 0,05 0,04 0,043 0,06 0,06 0,06 0,06 0,05 0,054 0,06 0,06 0,06 0,06 0,05 0,055 0,06 0,06 0,06 0,06 0,05 0,056 0,06 0,06 0,06 0,06 0,05 0,057 0,06 0,06 0,06 0,06 0,05 0,068 0,06 0,07 0,06 0,06 0,06 0,069 0,07 0,07 0,07 0,07 0,06 0,0610 0,06 0,07 0,07 0,07 0,06 0,0711 0,07 0,07 0,07 0,07 0,06 0,0712 0,07 0,07 0,07 0,07 0,06 0,0713 0,07 0,07 0,07 0,07 0,07 0,0714 0,07 0,07 0,07 0,07 0,06 0,0715 0,07 0,07 0,07 0,07 0,06 0,0716 0,06 0,07 0,07 0,07 0,07 0,0717 0,06 0,07 0,07 0,07 0,06 0,0718 0,07 0,07 0,07 0,07 0,06 0,0819 0,07 0,07 0,07 0,07 0,07 0,0720 0,07 0,07 0,07 0,07 0,06 0,0721 0,07 0,07 0,07 0,07 0,07 0,0822 0,07 0,07 0,07 0,07 0,06 0,0823 0,07 0,07 0,07 0,07 0,06 0,0724 0,07 0,07 0,07 0,07 0,06 0,0725 0,07 0,07 0,07 0,07 0,07 0,0826 0,07 0,07 0,07 0,07 0,06 0,0827 0,07 0,07 0,07 0,07 0,06 0,0728 0,07 0,07 0,07 0,07 0,06 0,0829 0,06 0,07 0,07 0,07 0,06 0,0830 0,06 0,07 0,06 0,07 0,06 0,08


Table G.11: Surplus ReturnTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22

1 0,036 0,036 0,036 0,036 0,040 0,0452 -0,001 -0,002 0,000 0,000 0,000 0,0043 -0,006 -0,007 -0,004 -0,004 -0,003 0,0014 -0,006 -0,007 -0,003 -0,003 -0,004 -0,0025 -0,004 -0,005 -0,004 -0,004 -0,008 -0,0046 -0,004 -0,005 -0,004 -0,004 -0,007 -0,0057 -0,003 -0,001 -0,001 0,000 -0,007 0,0078 -0,003 -0,001 -0,001 -0,001 -0,007 -0,0039 -0,009 -0,007 -0,007 -0,006 -0,002 -0,00510 -0,004 -0,004 -0,002 -0,001 -0,006 0,00211 -0,009 -0,008 -0,007 -0,007 -0,013 0,00112 -0,004 -0,002 -0,002 -0,002 -0,001 -0,00113 -0,010 -0,007 -0,006 -0,005 0,000 -0,00914 -0,009 -0,004 -0,006 -0,006 -0,013 0,00215 -0,007 0,000 -0,003 -0,002 -0,006 0,00016 -0,013 -0,004 -0,007 -0,007 -0,001 -0,00617 -0,006 0,000 -0,004 -0,004 -0,006 -0,00418 -0,002 0,002 -0,001 -0,002 -0,013 0,00319 -0,006 -0,004 -0,007 -0,007 0,004 -0,00620 0,001 -0,003 -0,005 -0,004 -0,023 -0,00221 0,003 -0,003 -0,004 -0,004 0,009 0,00422 0,000 -0,005 -0,005 -0,005 -0,009 -0,00223 0,000 -0,005 -0,005 -0,005 -0,001 -0,00524 -0,001 -0,002 -0,003 -0,003 -0,022 -0,01325 -0,008 -0,008 -0,008 -0,008 0,017 0,01226 -0,009 -0,005 -0,004 -0,004 -0,023 0,00827 -0,011 -0,008 -0,008 -0,008 -0,006 -0,01528 -0,012 -0,012 -0,012 -0,012 -0,016 0,00229 -0,003 -0,003 -0,003 -0,004 -0,005 -0,00730 -0,008 -0,009 -0,009 -0,009 -0,002 0,000

Table G.12: Surplus VolatilityTime Traditional Generalized Key Rate Principal Compo- MVP MVPStep Duration Duration Vector Duration end Duation 9 22

1 0,054 0,054 0,054 0,054 0,044 0,0412 0,068 0,069 0,068 0,068 0,062 0,0593 0,070 0,070 0,069 0,069 0,062 0,0604 0,070 0,072 0,070 0,070 0,063 0,0605 0,070 0,073 0,070 0,070 0,063 0,0616 0,066 0,070 0,066 0,066 0,060 0,0587 0,068 0,072 0,068 0,068 0,063 0,0628 0,067 0,071 0,067 0,067 0,061 0,0609 0,066 0,069 0,065 0,065 0,062 0,05910 0,065 0,069 0,065 0,066 0,060 0,05911 0,062 0,067 0,062 0,063 0,057 0,05512 0,062 0,065 0,061 0,062 0,058 0,05613 0,059 0,062 0,058 0,058 0,058 0,05814 0,059 0,062 0,058 0,058 0,054 0,05515 0,056 0,058 0,054 0,055 0,053 0,04916 0,056 0,058 0,054 0,054 0,055 0,05017 0,055 0,054 0,052 0,052 0,050 0,04718 0,057 0,054 0,052 0,052 0,051 0,04819 0,055 0,052 0,051 0,051 0,053 0,04820 0,054 0,050 0,049 0,049 0,052 0,04721 0,051 0,048 0,048 0,048 0,051 0,04422 0,050 0,047 0,047 0,047 0,048 0,04323 0,047 0,046 0,046 0,046 0,046 0,04224 0,045 0,044 0,043 0,044 0,046 0,04225 0,044 0,043 0,043 0,043 0,055 0,04326 0,045 0,043 0,043 0,043 0,043 0,04227 0,045 0,041 0,041 0,041 0,041 0,04528 0,044 0,041 0,042 0,041 0,041 0,04329 0,040 0,038 0,038 0,038 0,038 0,03830 0,037 0,036 0,036 0,036 0,038 0,039

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