Statistical Analysis of Financial Data...A. Ruckstuhl -- WBL 2017, Lecture 4 of SAoFD -- Page 6 7.3...

A. Ruckstuhl -- WBL 2017, Lecture 4 of SAoFD --

ETH Zürich, WBL Angewandte Statistik 2017

Blockkurs

Statistical Analysis Statistical Analysis of Financial Dataof Financial Data

Prof. Dr. Andreas RuckstuhlProf. Dr. Andreas RuckstuhlDozent für Statistische DatenanalyseDozent für Statistische Datenanalyse

Institut für Datenanalyse und Prozess Design IDPInstitut für Datenanalyse und Prozess Design IDPZürcher Hochschule für Angewandte Wissenschaften ZHAWZürcher Hochschule für Angewandte Wissenschaften ZHAW

[email protected]@zhaw.ch30. Januar 201730. Januar 2017


Outline of Lecture 4

• Lecture 1: Financial Data and Their Properties

• Lecture 2: Model for Conditional Heteroskedasticity and Risk Measures

• Lecture 3: Statistical Issues When Applying Portfolio Theory

• Lecture 4: (Financial) Factor Models– The market model– Macroeconomic Factor Models– Fundamental Factor Model & Example with R– Cross-Sectional Fundamental Factor Model– Principal Components and Factor Analysis– * Factor Models in Return-Based Style Analysis– * Our 17 Factor Model

• Lecture 5: Copulas


7 (Financial) Factor Models7 (Financial) Factor Models

Financial factor analysis explains returns with a small number of fundamental variables called factors or risk factorrisk factor.A factor model for equity returns (or excess equity returns Rj,t - μf,t) is

Rj,t = o,j + 1,j F1,t + … + p,j Fp,t + j,t

where • Rj,t is either the return or the excess return on the jth asset at time t,• μf,t is the risk free rate at time t• F1,t, ..., Fp,t are variables, called factors or risk factorsfactors or risk factors, that represent the “state of

the financial markets and world economy” at time t.• 1,t, …, n,t are uncorrelated, mean-zero r.v. called the unique risksunique risks of the individual

stocks• “Uncorrelated” means that all cross-correlation between the returns is due to the

factors!• The factors are common to all returns j=1, …,N• k,j is called factor loading and specifies the sensitivity of the jth asset return to the

kth factor

Introduction of some types of factor models:


7.1 The market model: The root of factor models7.1 The market model: The root of factor models

The market model Rj,t – μf,t = j + j (Rm,t – μf,t ) + j,t ,

where a simple market index Rm,t is a proxy for the entire (non-observable) market portfolio.

• This model is based on the Capital Asset Pricing Model (CAPMCAPM) (*), where j would be 0.

– CAPM is an economic equilibrium model (ex ante, i.e., predictive) whereas the market model is ex post (i.e., explanatory).

– In CAPM, the market risk factor is the only source of risk besides the unique risk of each asset. Hence, it is the sole source of correlation between any two asset returns

– Test whether CAPM is valid: Test the hypothesis “j=0” in an ex post (!) setting(If is nonzero then the security (=Wertpapier) is mispriced, at least according to CAPM)

(*) CAPM was introduced by Jack Treynor, William Sharpe, John Lintner and Jan Mossin independently in the sixties, building on the earlier work of Harry Markowitz. Sharpe, Markowitz and Merton Miller jointly received the Nobel Memorial Prize in Economics for this contribution to the field of financial economics.

• The market model is a single factor model and there is fairly strong empirical evidence that one factor is not sufficient.


7.2 Macroeconomic Factor Models7.2 Macroeconomic Factor Models

Macroeconomic factor models use macroeconomic variables such as • Return on the market portfolio • Growth rate of the GDP• Interest rate on short term Treasury bills or changes in this rate• Inflation rate or changes in this rate• Interest rate spreads; e.g., difference between long-term Treasury bonds and

long-term corporate bonds

Base: The efficient market hypothesis (another piece of theory in finance) implies that stock prices change because of new information; i.e., stock returns will be influenced by unpredictable changes in macroeconomic variables.

• The factors in a macroeconomic model are not the macroeconomic variables themselves, but rather the residuals when changes in the macroeconomic variables are predicted by a times series model, such as, multivariate AR models

• The R2-values of such fits are often very low • hence such types of factor models are not used widely in practise.


7.3 Fundamental Factor Model7.3 Fundamental Factor Model

Fundamental factor models use observable asset characteristics such as • Return on portfolio of stocks; e.g., all CH stocks, SMI, KMU, ...• The difference between returns on two portfolios

Example: Fama & French Three-Factor Model (†)

The three factors are• Excess return of the market portfolio (from the market model / CAPM)• Small minus big (SMB): Difference in returns on a portfolio of small stocks

and a portfolio of large stocks (size refer to the size of the market value)• Hight minus low (HML): Difference in returns of a portfolio of high

book(‡)-to-market value (BE/ME) stocks and a portfolio of low BE/ME stocks

(†) Fama & French (1993, 1995, 1996), Univ. of Chicago 2013, Eugene Fama shared the Nobel Memorial Prize in Economics jointly with Robert Shiller and Lars Peter Hansen(‡) Book value is the net worth of a firm according to its accounting balance sheet


Example: Fama & French Three-Factor Model applied to monthly returns on three equities

GE, IBM and Mobil (Jan 1969 to Dec 1998)

Calculate excess returns; e.g.,GEf = GE – RF

> pairs(FFS1)

Left: Scatterplot matrix of the variables used in this example

Focusing on GEf,

• GEf is highly correlated with the excess market returns

• GEf is negatively related with the factor HML – GEf behaves as a value stock??!!


Fitting by Time Series Regression> (FFS.fit <- lm(cbind(GEf,IBMf,Mobilf) ~ Mkt.RF + SMB + HML, data=FFS1))Coefficients: GEf IBMf Mobilf (Intercept) 0.344260 0.146039 0.163485Mkt.RF 1.140710 0.811446 0.986718SMB -0.371926 -0.312504 -0.375296HML 0.009503 -0.298302 0.372520

• Note that GEf now has a positive relationship with HML

• All three equity returns have negative relationship to SMB;i.e., they behave like large stock as they are suppose to do


Are the residuals uncorrelated as we assume in the factor model?

> cor(FFS1.fit$residuals) GEf IBMf MobilfGEf 1.000000 0.070824 -0.25401IBMf 0.070824 1.000000 -0.10153Mobilf -0.254012 -0.101532 1.00000

> pairs(FFS1.fit$residuals, las=1)

• The correlation between GEf and Mobilf is rather far from zero> cor.test(FFS1.fit$residuals[,1], FFS1.fit$residuals[,3])t=-4.9691, df=358, p-value=1.044e-06

• There are a few outliers robust estimation? (covRobust()from R package rrcov)


Use a time-series factor model to estimate the covariance matrix R:

Use factor model Rt = β0 + βT Ft + t, where β0, β and t are unkown

The covariance matrix of Rt is R = βT F β + , (*) where is a diagonal matrix.

• F is estimated by the sample covariance of the factors> (h1 <- cov(FFS1[,c("Mkt.RF","SMB","HML")])) Mkt.RF SMB HMLMkt.RF 21.150745 4.232563 -5.104485SMB 4.232563 8.181134 -1.076021HML -5.104485 -1.076021 7.179737

• The estimate of β is the matrix of regression coefficients (without intercepts)> (h2 <- coef(FFS1.fit)[-1,]) GEf IBMf MobilfMkt.RF 1.140709644 0.8114457 0.9867185SMB -0.371925771 -0.3125040 -0.3752955HML 0.009503016 -0.2983021 0.3725198


• The estimate of is the diagonal matrix of residuals variances> n <- nrow(FFS1) >(h3 <- diag(diag((n-1)/(n-4)* cov(resid(FFS1.fit))))) [,1] [,2] [,3][1,] 16.07687 0.00000 0.00000[2,] 0.00000 31.26346 0.00000[3,] 0.00000 0.00000 27.43171

• Hence, the estimation of βT F β + is > t(h2) %*% h1 %*% h2 + h3 GEf IBMf MobilfGEf 41.03648 19.30337 19.54448IBMf 19.30337 46.75183 14.46659Mobilf 19.54448 14.46659 43.58659

• For comparison, the sample covariance matrix R is> cov(FFS1[,c("GEf","IBMf","Mobilf")]) GEf IBMf MobilfGEf 40.90213 20.87791 14.25472IBMf 20.87791 46.49057 11.51806Mobilf 14.25472 11.51806 43.35736

Largest difference is is in the covariance between GEf and Mobilf.Reason: factor model assumes a zero residual correlation, but the data show a correlation of -0.25

Note:Devide by (n-4) because we estimated 4 parameters in the factor model


7.4 Cross-Sectional Fundamental Factor Model7.4 Cross-Sectional Fundamental Factor Model

Time series factor model do not make use of variables such as dividend yields, book-to-market value, or other variables specific to the kth firm.

An alternative is a cross-sectional factor model, which is a regression model using data from many assets but from only a single holding period.

Example:Suppose that Rk, (B/M) k and Dk are the return, book-to-market value and dividend yield for the kthe asset at some fixed time t

→ cross-sectional factor model is Rk =0 + 1 (B/M) k + 2 Dk + εk

The parameter 0, 1 and 2 are unknown values (at each time point t)

Hence, in contrast to time series factor models

• 1 and 2 , which are unknown, depend on time – they play the role of factors

• The variables (B/M) k and Dk are directly measured – they play the role of loadings


7.5 Statistical Factor Analysis7.5 Statistical Factor Analysis

Let Rt be a vector of returns from d assets. Let’s assume that p latent factors Ft, called risk factorsrisk factors, explain the returns up to additive constants and some errors:

where is a p x d matrix of unknown loadings is the d dimensional vector of additive constants is the d dimensional vector of unexplained returns (=“error“) . (*)


Example: Allocate capital to the equity markets of ten different countriesIn R:> (R.fa <- factanal(R, factors=3, rotation="none"))…

Uniquenesses:HongKong Singapore Brazil Argentina UK Germany Canada France Japan USA 0.321 0.223 0.878 0.972 0.505 0.297 0.005 0.126 0.865 0.666 Loadings: Factor1 Factor2 Factor3HongKong 0.806 0.167 Singapore 0.862 0.178 Brazil 0.129 0.271 0.179 Argentina 0.111 0.124 UK 0.513 0.480 Germany 0.441 -0.167 0.694 Canada 0.997 France 0.491 -0.137 0.784 Japan 0.336 0.132 USA 0.405 0.330 0.249

Factor1 Factor2 Factor3SS loadings 2.0 1.642 1.497Proportion Var 0.2 0.164 0.150Cumulative Var 0.2 0.365 0.514

Test of the hypothesis that 3 factors are sufficient.The chi square statistic is 13.9 on 18 degrees of freedom.The p-value is 0.736

Estimates of the diagonal of

Estimates of βT

Test whether «approximation is okay»


Remarks:

• NOTE!Since factanal standardizes the variables, βT β + approximates the correlation matrix of Rt

– Let D be the diagonal matrix with the standard deviation of each asset on its diagonal. Then D (βT β + ε ) D approximates the covariance matrix R of Rt

• The latent factors Ft are used to model the correlation between the assets

• factanal would allow to rotate the β which, however, would not change F = βT β. Hence, we leave it.

• Estimation of R 65 parameters are estimated (=10 (location) + 10 (variances)+10*9/2 (covariances))

• Estimation by approximation: 53 (minus constraints) parameters are estimated (= 10 (add. Constant) + 3 (loadings) + 3*10 (Factors) + 10 (diagonal of ))


Example: Allocate capital to the equity markets of ten different countries

Bootstrapping the estimated Sharpe’s ratio of the tangency portfolio using the “factor analysis approximation” (B=250)In R:> EM.TPboot <- BootTP(Dat=100*R, mufree=1/24, nBoot=250, NoFAC=3)> boxplot(EM.TPboot[,1:2], las=1, ylab="Sharpe's ratio")> abline(h=EM.sol$tpSharpe, lty=2, lwd=2, col="gray")

- «3 factor approx» is as good as sample cov- «1 factor approx» is slightly worse, and would not pass the test on slide 6 as well


Comment on Principal Component Analysis (PCA):

● PCA aims to find low-dimensional subspaces containing most of the variation in the data● That is, the remaining components will not contain a lot of variation ...● … but may still contain correlation between the assets

● Hence, PCA cannot be used as a factor model in finance● But still a an exploratory tool to gain insights into the return data as does a

scatterplot matrix

In R:> pairs(R) # see next slide


Scatterplot matrix


7.6* Factor Models in Return-Based Style Analysis 7.6* Factor Models in Return-Based Style Analysis

In return-based style analysis, the factors are returns on representative asset classes, which are usually assimilated to market indices.

Sharpe (1988, 1992)

Example:Fund of hedge-funds (FoHF) - A less transparent financial instrument

– Many FoHF will only disclose partial information on their underlying portfolio. – The style of FoHF and their underlying investment strategy, which is the

crucial characterization of FoHF, is self-declared

Return-based style analysis is an instrument to provide a unprejudiced insight into FoHF.

Lhabitant (2001) introduced the idea: Use a multifactor model, where the factors are indices of hedge-fund sub-styles

This is a very different use of a factor model than in the previous Sections: Not the risk is of its focus but rather the composition (=factors)


Example from the block “Robust Regression”• Hedge-Fund Sub-Indices of Different Providers (CSFB/T, HFR, EDHEC)

• CSFB/T: Credit Suisse First Boston / Tremont, nowadays Credit Suisse Hedge Fund Index (www.hedgeindex.com/)

• HFR : Hedge Fund Research (https://www.hedgefundresearch.com/)• EDEC : EDEC-Risk Institute (http://www.edhec-risk.com/indexes)

http://www.hedgeindex.com/


• Residuals using robust fitting methods (→ lmrob in robustbase):

FoHF A FoHF B

homogeneous shift in the investment strategy?


Constraints on the β‘s

• From an economic perspective, it may seem natural and logical to add constraints on the β‘s:

– The β‘s should not be negative since • the short selling strategy is captured by a sub-index • FoHF only hold long positions of single hedge funds

→ hence we do not allow short sales on an sub-index – The β‘s should add up to 1 (100%) if the FoHF is invested in hedge

funds solely – and in reality?

• If the constraints are enforced in the fitting process → a “standard” residual analysis may not be sensible → standard regression inference results cannot be applied


Constraints on the β‘s – example using the FoHF dataset> round(coef(lm(FoHF ~ . -1, data=FoHF, subset=-(1:28))),2) RV CA FIA EMN ED EDD EDRA LSE GM EM SS 0.10 0.22 -0.12 0.22 -0.16 0.13 -0.17 0.42 0.24 0.05 0.06

> require(quadprog) > X2 <- data.matrix(FoHF[-(1:28),-1])> Y2 <- FoHF$FoHF[-(1:28)]> Dmat2 <- crossprod(X2)> dvec2 <- as.vector(t(as.matrix(Y2)) %*% X2)> bvec <- c(1, rep(0, p))> Amat <- rbind(c(rep(1, p)), diag(p))> sol2 <- solve.QP(Dmat=Dmat2, dvec=dvec2, Amat=t(Amat), bvec=bvec, meq=1)> names(sol2$solution) <- colnames(X2)> round(sol2$solution,2) RV CA FIA EMN ED EDD EDRA LSE GM EM SS 0.00 0.19 0.00 0.23 0.00 0.03 0.00 0.20 0.27 0.06 0.03> sum(sol2$solution) [1] 1


But a HoHF may not be invested in HF completely (i.e, sum(β‘s) < 1) or may be leveraged (i.e, sum(β‘s) > 1).

• LASSO (least absolute shrinkage and selection operator) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the statistical model it produces.

• Example: creInvest AG, May 2000 until December 2005 (b=sum(β‘s))

The result are different because the index providers use different data bases and different weighting schemes to calculate the sub-indices.


7.7* “Our” 17 Factor Model 7.7* “Our” 17 Factor Model

• Our extension of Fama & French Three-Factor Model resulted in a 17-Factor Model which is based on

– 9 factors which should cover the classical investment market. They are actual market indices

and– 8 factors which should cover the alternative investment market.

They are derived from actual market indices

Manz, Meier, Ruckstuhl, Stoz, and Weibel, The New 17-Factor Model, Research Report ZHAW, January, 2012


• Does an actively managed fund (like Fund of Hedge Fund - FoHF) pay off its higher fee? → Separate the contribution of the market (=loadings) from that of a single Fund of Hedge Fund (FoHF) manager (=intercept) to the returns

– since a single FoHF does not need to be invested in every sector,– the FoHF may be adequately described just through some of the market factors.– Since we are calibrating the factor model for the last 48 months, sometimes more than half

of the factors are not statistically significant .

– This calls for variable selection (i.e., so-called "hard thresholding")– Our focus in calibrating the factor model lies in the parameters (especially in the intercept)

and its behaviour over time.– The least-squares fit and the selection of the factors did not yield stable results due to too

many parameters and too short adjustment period, respectively.

• Way out: A gradual selection of factors based on so-called "'soft thresholding"' methods - Our choice fell on L2-Boosting


• L2Boosting in our context means:– Instead of directly solving the least-squares problem an iterative

gradient descent procedure is used– Instead of fully iterating, the procedure is stopped earlier– A single iteration step consists of fitting the best single regression

model (with one of the factors) to the residuals of the updated boosted estimator of the previous step

– cf., i.e., Bühlmann and Hothorn (2007). Boosting algorithms: Regularization, prediction and model fitting (with discussion). Statistical Science 22, 477-505

• Example: RMF Equity Hedged Strategies (a FoHF)

Intercept (=alpha) over time (Jan 2004 to Dec 2008) of a FoHF. The boxplots shows the variability of the alpha estimation over the FoHF universe on hedgegate.

Philipp, Ruckstuhl, Manz, Dettling, Dürr and Meier (2009). Performance Rating of Funds of Hedge Funds, Research Report ZHAW


Take Home Message Lecture 4Take Home Message Lecture 4

Financial factor analysis explains returns with a small number of fundamental variables called factors or risk factor

• Factor models are used to – construct estimators for the covariance matrix (→ portfolio optimization)– Identify “return and risk sources”, i.e.,

• Perform style analysis of a portfolio• Separating the contribution of the market (=loadings) from that of an

active manager • If the factors are investable, then you can mimic a less transparent

financial instrument by a portfolio of better known instruments

• Factor models are fitted by – Time series regression methods

(LS, robust, constrained LS, LASSO, gradient boosting, …)

– Cross-sectional regression methods (LS, robust, …)

– Statistical factor analysis


ﾧ

• Lecture 4 in the books:

Chapter 18

• Sections marked by * are not examined

Subsec 4.2.1, 4.2.1

Chapter 18

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