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Niklas Siebenmorgen, McKinsey & Company,

and Martin Weber ([email protected]),

Universitt Mannheim, Lehrstuhl fr ABWL und Finanzwirtschaft,

insbesondere Bankbetriebslehre, L5, 2, D - 68131 Mannheim.

Weber is also with CEPR London. This research was supported

by the Sonderforschungsbereich 504 and the Graduiertenkolleg

Allokation auf Finanz- und Gtermrkten at the

University of Mannheim. We thank Gunter Lffler, Torsten Winkler

and all members of the Behavioral Finance Group

(http://www.behavioral-finance.de) for useful comments.

We also thank the referees, Prof. Thorsten Hens and

Prof. Heinz Zimmermann, for valuable comments.

Introduction

Asset allocation is a popular way for investment

banks, insurance companies and financial advisors

to determine the optimal proportions of several

asset classes in the portfolio of an investor. In this

context asset classes are usualy short-term interest

paying assets such as cash accounts and short-

term bonds or long-term interest-paying assets

such as bonds with longer durations. Other possi-

ble assets are different types of foreign and do-

mestic stocks or stocks of blue chip companies

and small cap companies. Even non-traded assets

such as real estate or antiques are possible candi-

dates.

Asset allocation advice should depend, among

others, on the investors risk attitude and inves-tors time horizon. We will concentrate on the

influence of risk attitude as this is central to most

popular advice. Additionally, an increasing num-

ber of countries require by law that investment

advisors educate their clients about risk and also

assess their clients risk attitude, see, e.g. in Ger-

many No. 31(2) of the Wertpapierhandelsgesetz

(WpHG).[1] With respect to risk attitude, fi-

nancial advisors usually give qualitatively similar

advice: They recommend that the more risk toler-ant investors are, the more they should invest into

more risky assets. As we will see in more detail

below this implies that more risk tolerant investors

have a higher stock to bond ratio as well as that

the ratio of risky stock to less risky stock is

higher.

CANNER, MANKIW and WEIL (1997), CMW

hereafter, discuss this advice in the light of tradi-

tional mean-variance analysis and find that they

can not explain the advice in the light of mean-

variance theory. A lot of recent work has suc-

cessfully addressed the puzzle of the stock bond

ratio depending on the risk attitude using dynamic

asset allocation models (MERTON 1971,

1973).[2] This new research, however, does not

investigate the ratio of risky stocks to less risky

stocks and it is only applicable to long investment

horizons.[3]

Swiss Society for Financial Market Research (pp. 1542)

FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1 15

NIKLAS SIEBENMORGEN AND MARTIN WEBER

A BEHAVIORAL MODELFOR ASSET ALLOCATIONIN THE CASE OF CONSTANT

AND STOCHASTIC VOLATILITIES

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In the following we take a different angle to un-

derstand why advisors do not follow the re-

commendation of portfolio theory. We present a

new behavioral portfolio theory, which seeks to

describe how individual investors intuitively per-

form asset allocation. We do this, because werefuse to believe that investment advisors and

especially individual investors manage stochastic

problems in the way the literature describes it.

Like SHEFRIN and STATMAN (2000) we con-

sider behavioral arguments that play a role for

portfolio decisions. There is no reason not to be-

lieve that elements from behavioral research

should be utilized when intuitive decision-making

is to be described.

We argue that investors take three aspects intoaccount when creating an optimal portfolio: ex-

pected returns, pure risk and naive diversification.

It will be shown that recommendations by invest-

ment advisors gathered from literature and by our

own empirical study, are much closer to the re-

sults of behavioral portfolio theory than to the

results of traditional portfolio theory. Thus in-

vestment advisor follow a strategy which might be

quite clever: they do something wrong with

respect to traditional portfolio theory, which,

however, is quite appealing to the way their cli-

ents think intuitively, and the loss in efficiency

due to following this behavioral theory is not so

large.

The remainder of this paper is organized as fol-

lows. In section 2 we develop a behavioral ap-

proach to portfolio choice. Section 3 describes the

results of our own study, which is based on in-

vestment advisors recommendations and exam-

ines efficiency losses of these recommendations.

Subsequently, we compare the predictions of thebehavioral model with the data provided in CMW.

Section 4 concludes this paper.

1. A Behavioral Approach to Investors

Asset Allocation

MARKOWITZ (1952) examines prescriptively,

how to invest in several assets given the ex-

pected returns, standard deviations and the corre-lations among the assets. He assumes the investors

to be risk averse mean-variance-optimizers (--

principle). Investors weigh according to their

risk attitude the advantages of more expected

return of their portfolio (mean return) and the dis-

advantages of more portfolio risk, measured as the

variance or standard deviation of the portfolio

return. These assumptions are consistent with ex-

pected utility theory if investors utility functions

are quadratic or if returns are (log-) normal. Theefficient mean-variance-frontier can be derived by

either maximizing expected portfolio return for

each level of risk given (model opt1) or by mini-

mizing expected portfolio risk for each level of

return required (model opt2). To introduce nota-

tion, we present model opt1 below:

Model opt1:

=

n

1iiiMax

i

(1)

s.t. Risk=

= =

n

1i

n

1jijjiji = r

1n

1ii =

=and n1,..,i10 i =

with n being the number of assets to choose from,

i the proportion of asset i in the portfolio, ithe standard deviation (volatility) of asset i, i theexpected return of asset i and ij the correlationbetween the returns of asset i and asset j. We in-

troduce a short sale constraint, as we do not be-

lieve that ordinary investors are willing to accept

negative portfolio proportions.[4]

Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation

16 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1

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who ask subjects to judge assets volatility and

risk. LIPE (1998) finds that individual investors

consider return variance but they do not assess the

covariance of returns with the market return. On

the basis of his experiments, OEHLER (1995)[6]

finds similar results. His participants did not usethe explicit information about the correlations of

investment alternatives to optimize their portfo-

lios. In a questionnaire after his experiment the

participants themselves ranked the correlations as

the least important information for their deci-

sions.[7] [8] The assumption gets further support

by applying the idea of mental accounting

(THALER, 1985) to portfolio choice (see also

SHEFRIN and STATMAN, 2000). When regard-

ing each type of asset as a separate mental ac-count, it is intuitive that correlations will not be

considered.

It should be noted that this way of considering risk

can only be a first step. Several extensions are

possible. First, different measures of risk can be

used. In this paper we apply the measure which

is also used in traditional portfolio optimiza-

tion.[9]. However, other measures have been pro-

posed to describe peoples risk perception[10],

some of which are even compatible with ex-pected utility.[11] As a second extension, one can

think of taking correlations to some degree into

account, i.e. downgrading them by some specific

factor.

Naive Diversification

Even if subjects do not take correlations into ac-

count, they nevertheless emphasize the idea of

diversification. Investors tend towards naive di-

versification, i.e. investors want to split their

wealth evenly among several investment alterna-

tives perhaps because they have learned about

the advantages of diversification or perhaps be-

cause they intuitively behave this way. Naive

diversification is operationalized here by the

standard deviation of the asset proportions i .

Diversification:[12]

( ) ( )

=

=n

1i

2

in1n

1,...,Std min (4)

BENARTZI and THALER (2000) examine the

behavior of naive diversification. This means

that investors tend to distribute their capital evenly

among the available investment alternatives.[13]

Asking employees of the University of California

for their allocation of retirement contributions,

BENARTZI and THALER show that their asset

selection and therefore the risk they accept in their

portfolio strongly depends on the type of assets

that are offered to them. If being offered a fund of

stocks and a mixed fund of stocks and bonds, peo-

ple tend to invest significantly more in stocks

compared to the situation in which they are of-

fered a mixed fund and a fund just containing

bonds.[14] In the same paper an empirical study of

several contribution plans in the United States

confirms this behavior, as the average allocation

to equities strongly depends on the relative num-

ber of equity-type investment options.

FISHER and STATMAN (1997a and 1997b) also

find evidence for naive diversification in theirstudy. People tend to split their wealth by in-

vesting into all available assets or funds without

thinking about the optimal diversification strategy.

The authors also show that the allocations of mu-

tual funds and the guidelines for fiduciaries

(called ERISA) are closer to naive diversification

than to the optimal diversification described by

MARKOWITZ.

Behavioral Portfolio ModelW

We assume that our agents try to optimize their

portfolio strategy by searching for asset allo-

cations that are on the efficient frontier of these

three target variables. To be able to compare

the traditional approach with this approach we



Behavioral Portfolio ModelW

We assume that our agents try to optimize their

portfolio strategy by searching for asset allo-

cations that are on the efficient frontier of these

three target variables. To be able to compare

the traditional approach with this approach we

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propose alternative models. In these models we

define as in the MARKOWITZ model one

target function and one restriction using these

three target variables. To do this we combine

(linearly) two of the three target variables.

Consequently, there are six possiblities (see Table1 to build a behavioral model using these three

target variables:

Out of these six potential models, it does not make

sense to consider M1b, M2b and M3 for a detailed

further analysis. There are two related reasons for

that:

i) The three different risk attitudes characteriz-

ing the students were the basis for the three

portfolio recommendations of each financial

advisors. Thus, by definition, each advisorrecommended one portfolio which was the

least risky, one which was the middle risky

and one which was the most risky. A behav-

ioral portfolio model which does not replicate

these orderings should not be considered.

However, the application of models M1b,

M2b and M3 to the recommendations for three

different risk attitudes (see section 2) shows

that the (pure) risk of the three generated

benchmark portfolios often changes its order,

e.g. the benchmark portfolio of the riskiest

recommendation is less risky than the bench-

mark portfolios of the other recommendations.

ii) In deriving a portfolio recommendation, risk

attitude or expected returns need to be consid-

ered by the advisor and the investor. In order

to generate reasonable benchmark portfolios

we choose those models that are able to fix

these characteristics of a given portfolio

recommendation. Consequently, as the re-

stricitions of models M1, and M2 and M3b

contain these variables, those models will be

used to build a benchmark portfolio for the

recommended asset allocations.

M2 will be our primary model, as it ensures that

the behavioral benchmark portfolio will have the

same (pure) risk as the recommended portfolio. In

model M1 investors can restrict their portfolio

choice by a certain amount of expected return.

Model M3b is also appropriate, as its restriction

contains a linear combination of pure risk and

expected return. Model M3b produces similar

results as models M1 and M2, but will not be pre-sented in the following sections. Let us now turn

to the models M1 and M2, which will form the

basis of the further analysis.

Model M1 determines for each value of

expected return e a portfolio that is optimal

regarding diversification and pure risk. We do this

by defining a target function that combines the

two variables Diversification and Pure Risk using

a linear combination:

Model M1(): [0;1] (5)

i

Min

Diversification + (1 )Pure Risk

s.t. Expected Return = e

1n

1ii =

=and n1,..,i10 i =



Table 1: Potential Behavioral Models

Model restriction: target function:

M1 Expected Return linear combination of Diversification and Pure RiskM1b linear combination of Diversification and Pure Risk Expected ReturnM2 Pure Risk linear combination of Expected Return and Diversification

M2b linear combination of Expected Return and Diversification Pure RiskM3 Diversification linear combination of Expected Return and Pure RiskM3b linear combination of Expected Return and Pure Risk Diversification

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2.1 A Study among Investment Advisors

To compare the normative and the behavioral

portfolio selection models more thoroughly we

gathered data from banks investment advisors in

the southwest of Germany (Frankfurt, Stuttgart,

Mannheim, Heidelberg, Speyer, Karlsruhe, Offen-

burg and Freiburg). Most private investors in

Germany ask their bank to give advice on their

investment decisions. They rarely use brokerage

firms that do not offer any investment recommen-dations, to get access to the financial markets.

Therefore well-educated and well-informed bank

employees in Germany have a decisive influence

on private investors portfolio decisions. For that

we visited consultants of the departments Private

Banking who are used to managing portfolios of

at least DM 100,000. This way we reached the

highly professional advisors who have millions

of Deutschmarks of funds under management

and asked them for their recommended assetallocations for different risk attitudes. Further-

more, we asked them for their market expec-

tations. By linking the experts market expecta-

tions with their preferred asset allocations we

hope to learn more about the advisors diversifica-

tion behavior.[15]

Method

Our questionnaire consists of four pages (see ap-

pendix A). It begins with a short introduction sim-

ply telling that we intend to study recommended

portfolios of bank employees and investment con-

sultants. Then we present three fictive new clients.

Each of them is 26 years old, not married and

without any savings. They have just finished their

master in business administration at the university

will use the model M2(0) to examine the role of

the expected returns.



s.t. Pure Risk= r

1n

1ii =

=and n1,..,i10 i =

We will test our models using recommendation

data by financial advisors. We will do this by

restricting the models to the actual characteristics

of the given recommendations. For model M1

(M2), e.g., we will calculate the expected returns

(pure risk) of the given recommendations and

will use these values in the restriction of model

M1 (M2). Keeping these values fixed we have to

maximize or minimize the target function. This

procedure generates benchmark portfolios, whichwe compare with the actual recommendations.

2. The Behavioral Approach and Investment

Advisors Recommendation

We will now test if our models can explain real

world recommendations of financial advisors. In

section 2.1 the model is tested on the basis of rec-

ommendations by German investment advisors.

Besides cash and bonds, these recommendations

have different classes of risky assets. In section

2.2 we also compare the model to the data pre-

sented in CMW which relies on three assets (cash,

bond and stocks). As we do not find major

differences regarding the parameters we will use

= 0.5 (and = 0.5) in the remainder of this paper

to test the validity of our models. Furthermore, we

Model M2 assumes that investors restrict their

portfolio decisions to a certain amount of pure

risk r . Given this constant amount of pure

risk they maximize a linear combination of the

target variables Expected Return and Diversi-

fication:

Model M2(): [0;1] (6)

i

Max

Expected Return (1 )Diversification

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of Mannheim and will inherit DM 500,000

(approx. $ 250,000) within the next days. None of

them know when they will need parts of their new

wealth but they have different risk attitudes: They

prefer conservative: C, moderate: M and aggres-

sive: A, investment strategies respectively. Weasked the participants of our study to recommend

an asset allocation for each of the three fictive

clients for a time horizon of 12 months.[16] For the

allocation, the advisors are only allowed to choose

from the following asset types:

short-term cash, money market funds or

short-term bonds (TTM[17]

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=

=

=

=

=5n

1i

5n

1jk,ijk,jk,i

C

k,j

C

k,i

C

k

(similarly for Mk andA

k ) (7)

Table 2 shows, that for all portfolio types the as-

sessed volatilities Ck ,Mk and

Ak are signifi-

cantly larger than the implicit volatilities calcu-

lated by using the given correlations (Wilcoxon-

test: low risk: p = 0.002**, moderate risk: p =

0.003**, high risk: p = 0.000**). We also calcu-

late the pure risks Ck~ , Mk

~ and Ak~ that are

generated by assuming all correlations to be

100%:

=

=

=5n

1i

k,iC

k,iCk

~

(similarly for Mk andAk ) (8)

We find that the assessed volatilities are equal

(portfolios C and M) or above (portfolio A) the

values of pure risk (Wilcoxon-test: p = 0.855,

p = 0.693, p = 0.009**), see Table 2. The Results

clearly indicate, that implicit volatilities (using

correlations) cannot explain the assessed volatil-ities. The assumption that people use pure risk,

however, is compatible with the data.[25]

We now want to test whether the traditional theory

or the behavioral approach is better able to explain

the portfolio recommendations we collected. We

will compare the behavioral models M1 and M2

with the MARKOWITZ models opt1 and opt2. To

implement the models opt2 and M1 we set the

expected returns of the recommended portfolios at

e and restrict the models to this expected return

e . Similarly, we fix the observed volatility/pure

risk of the recommended portfolios at r to im-

plement the models opt1 and M2. The followingdistance measure will be used to compare the two

types of models.[26] This quadratic distance meas-

ure simply determines the distance between a

model and the recommended portfolio. Note, that

all these calculations are based on the individually

assessed return distributions.

( )

=

=5

1i

2Ck,i

Ck,i

Ck

DM

(similarly for MkDM andA

kDM ) (9)

Ck,i ,

Mk,i and

Ak,i denote the predicted portfolio

proportions of the models.

Comparing the distance measures for several

models we find significant differences. Using

the MARKOWITZ models opt1 or opt2 we

measure a higher distance than using the

behavioral models M1 and M2 with the chosenparameters = 0.5, = 0 and = 0.5. Table 3

shows the mean and median results for the three

risk classes C, M and A for each of the five

different models we compare respectivly. The last

column shows the average distance measures over

all risk classes.[27]



Table 2: Mean and Median Values of Assessed and Implicit Volatilities and Pure Risk

Mean (median) low risk (C) moderate risk (M) high risk (A)

assessed volatili ties Ck

3.40%(2.72%)

M

k6.08%

(5.32%)A

k11.08%(9.50%)

implicit volatilities Ck

2.40%(1.92%)

M

k4.47%

(3.95%)A

k7.09%

(5.94%)pure risk C

k~

3.49%(2.68%)

M

k~

5.90%(4.77%)

A

k~

8.68%(7.00%)

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findings might be that investment consultants use

the proposed asset allocations of their banks, al-

though we asked them to give their own individual

opinions and not the opinion of their banks. Con-

sequently, there might be a discrepancy between

their own market expectations and the banks rec-

ommended asset allocations that drives the devia-

tions from the rational MARKOWITZ model. Byusing historical data[32] (see appendix B) instead

of the individual market expectations we con-

trolled for this effect in an additional analysis.

With these data, the differences between the

models are smaller, but we still find significantly

better results with the behavioral models (see

Table 5). It is only the low-risk-portfolio that

cannot be explained any better by the behavioral

diversification model. For the average distance

measures of all portfolios we still find signifi-cantly better results with models M1 and M2.

Another objection might be the technique, by

which the optimal MARKOWITZ portfolio on the

efficient line is determined. So far, we have cho-

sen two ways of determining this portfolio: One

by keeping the expected returns constant (model

opt2) and one by keeping the volatility of the port-

folio constant (model opt1), and we received simi-

lar results. Alternatively, we now use the follow-

ing optimization:

Nearest MARKOWITZ solution:

( )k,ik,ik

,DMMinki,

(10)

s.t. There is a r , for which ( k,i ) is the solu-

tion of the following problem:

=

n

1i

k,ii~

~Max

i

s.t.

Risk=

= =

n

1i

n

1j

k,ijk,jk,iji~~ r

1~n

1i

i ==

and n1,..,i1~0 i =

Thus, we allow for the best (i.e., nearest to k,i )

non-dominated (notice the in the restric-

tion) solution on the efficient frontier (see Fig-

ure 1).

Analogously, we define the nearest non-

dominated solutions of models M1 and M2 (which

coincide with the nearest solutions of models

M1b and M2b, respectively). Hence, we deter-

mine the model portfolios on the efficient line that

have the lowest distance to the recommended

portfolio.

For the MARKOWITZ model we find some im-

provements compared to opt1 and opt2. We do

not find any improvements of the behavioral mod-

els using this method, probably because of the re-

striction to efficient portfolios. Nevertheless, the

differences are still significant as Table 6

shows.[33]

However, we do not think that this is an appropri-

ate way of modeling portfolio recommendations

of investment experts, because very often (12 rec-

ommendations) we derive MARKOWITZ port-

folios that are substantially different from what

has been recommended: For many of these portfo-

lios, risk and also the proportion of stocks differed

extremely from those of the recommended portfo-

Table 5: Wilcoxon-Test on the Model Differences Using Historical Market Data

M1(0.5) M2(0) M2(0.5)

opt1 P = 0.005 ** P = 0.004 ** P = 0.004 **average distancesover all portfolios opt2 P = 0.004 ** P = 0.004 ** P = 0.002 **

Nearest MARKOWITZ solution:

( )k,ik,ik

,DMMinki,

(10)

s.t. There is a r , for which ( k,i ) is the solu-

tion of the following problem:

=

n

1i

k,ii~

~Max

i

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lios. It costs further doubt on the descriptive valid-

ity of the MARKOWITZ model when the nearest

portfolio on the efficient line tends to be a portfo-

lio with totally different characteristics. In the case

of 7 questionnaires the best fitting MARKOWITZ

models (using this method) changed their risk

order, e.g. the aggressive benchmark portfoliohad less risk than the moderate benchmark port-

folio. Obviously, this again indicates that the

consultants did not want to recommend these

nearest MARKOWITZ solutions. In compari-

son, the solutions of the behavioral models hardly

changed.

Finally, we consider an investor with several bank

accounts who is consulted by more than one ex-pert on his investment decisions. Alternatively, he

Figure 1: Alternative Construction of the Benchmark Portfolios

Risk

Return

efficient frontier

recommended portfolio

portfolio of model opt1

portfolio of model opt2

Table 6: Wilcoxon-Test on the Model Differences Using the Nearest Solutions

nearest M1(0.5)

solution

nearest M2(0)

solution

nearest M2(0.5)

solution

average distancesover all portfolios

nearest MARKOWITZsolution

P = 0.026 * P = 0.011 * P = 0.008 **

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reads some financial journals with asset allocation

recommendations as they are presented in CMW.

He might decide about his money by averaging

several portfolio recommendations while his mar-

ket expectations will be comparable to the mean

market expectations of many bank consultantsand financial institutions. For this particular case,

the behavioral models (average distance measure

for M1(0.5) = 13.0%, M2(0) = 13.0%, M2(0.5) =

12.7%) seem to be 2.5 times better than the

MARKOWITZ model (average distance measure

opt1 = 31.9%, opt2 = 33.9%). Figure 2 shows the

mean recommendations for the three clients with

different risk attitudes and the results of models

opt1 and M2(0.5). Looking at aggregated market

expectations and comparing it to the mean portfo-lio recommendation even strengthens the case of

the behavioral model.[34]

Figure 2 also shows that for the behavioral

model the ratio of risky stocks to less risky stocks

increases with an increase in risk tolerance. Con-

sidering expected volatilities (Appendix B) blue

chips are less volatile (risky) than small caps and

foreign stocks which have identical expectedvolatility. According to traditional mean variance,

the ratio of foreign stocks and small caps over

blue chips is equal to 3.29 (low risk), 3.68

(moderate risk) and 3.81 (high risk). For the

advisors recommendation, we get .59 (low risk),

1.15 (moderate risk) and 1.97 (high risk), where-

as the model predicts on average .79 (low

risk), 1.78 (moderate risk) and 2.41 (high risk)

clearly more in line with the recommendations

than the mean-variance approach.



low risk

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

opt1 recommendation M2(0.5)

foreign stocks

small caps

bluechips

bonds

short-term

Figure 2: Average Portfolio Proportions

low risk

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Table 7: Historical Market Data from 1926 to 1992 (see CMW)

i Asset i iCorrelation

with bonds

Correlation

with stocks

1 Cash 0.6 % 4.3 % 0.63 0.092 Bonds 2.1 % 10.1 % 1.00 0.23

3 Stocks 9.0 % 20.8 % 0.23 1.00



0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

5%

7%

8%

10%

11%

13%

14%

16%

17%

19%

20%

Pure Risk

Portfolio

Proportions

.

Cash

Bonds

Stocks

2.2 Explaining the Investment

Recommendations in CMW

In this section we will show that our behavioral

model is also able to explain the recommendations

discussed in CMW. This is interesting as CMWpresent very popular recommendations which are

widely discussed in the literature and these rec-

ommendations are based on a different number of

assets, i.e. the three assets, cash, bond and stocks.

To be able to compare the recommendations of the

behavioral model with the investment advice of

the financial analysts in CMW, we take the same

(historical) data of the three asset types (n = 3)

stocks, bonds and cash as given in their

study:

For a first insight into the descriptive quality ofthe behavioral models, we solve for the optimal

values of i using model M2 with parameter = 0.5[35] for different values of pure risk r . The

resulting portfolio proportions are presented in

Figure 3.

Figure 3: Portfolio Proportions Using Model M2(0.5)

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Figure 4: Bond-to-Stock Ratios



-150%

-100%

-50%

0%

50%

100%

150%

200%

0% 20% 40% 60% 80% 100% 120%

Proportion of Stocks in Portfolio

Bond-to-StockRatio

naive diversifikation: model M2(0.5)

recommended portfolios

Markowitz: no riskless asset, short sale allowedMarkowitz: no riskless asset, no short sale allowed

In Figure 4 we plot the bond-to-stock-ratio of

model M2 against the stock proportion of the port-folio as it is done in the CMW study.

The thin line illustrates the optimal portfolios for

the case without riskless asset and without a short

sale constraint. The central question of the Asset

Allocation Puzzle is why the recommended port-

folios (big dots) show a decreasing tendency of

the bond-to-stock-ratio in the proportion of stocks

while the optimal portfolios show an increasing

tendency. CMW find that the short sale constraint

partially explains the puzzle: In the high-risk-area

the bond-to-stock-ratios of the recommended port-

folios coincide with the ratios of optimal portfo-

lios if short sales are not allowed (thick dotted

line) But in the low-risk-area the optimal (MAR-

KOWITZ) portfolios still do not fit the observed

recommended portfolios. The behavioral models

(here model M2(0.5) thick gray line) fit the

given recommendations quite well: The bond-to-

stock ratio shows a decreasing tendency for all

stock proportions.Alternatively, we use MARKOWITZ way to

illustrate the optimal portfolio proportions i tocompare his results with our approach. In his

study he also investigates the 3-asset-case (n = 3)

and plots the optimal 1 and 2 ( 3 is given by

213 1 = ) in a two-dimensional diagram.Figure 5 illustrates the MARKOWITZ-optimal

portfolios and the portfolios based on model

M2(0.5) for the data presented in CMW. The large

triangle is the set which MARKOWITZ calls the

attainable set, which includes all portfolios

without short sales. The dotted lines show the

optimal portfolios MARKOWITZ prescribes. The

solid line shows the results based on our model. In

each case the thin lines allow for short sales and

the thick lines show the portfolios restricted by the

short sale constraint. The big dots in the diagram

show the recommended portfolios of the four ana-

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Figure 5: Proportion of Bonds against Proportion of Stocks



-40%

-20%

0%

20%

40%

60%

80%

100%

-20% 0% 20% 40% 60% 80% 100% 120%

Proportion of stocks in portfolio

Proportionofbondsinportfolio

naive diversification, no shortsale constraint: model M2(0)

naive diversification, short saleconstraint: model M2(0)

optimal Markowitz portfolio,no short sale constraint

optimal Markowitz portfolio,short sale constraint


lysts, CMW present. Again the naive diversifica-

tion model describes the investment behavior (or to be precise the investment advice of financial

analysts) better. In the more risk averse domain,

the diagram confirms that investors tend to hold

relatively more bonds than MARKOWITZ would

prescribe.

2.3 Efficiency Losses

CMW and FISHER and STATMAN (1997b) find

that investors intuitive behavior produces portfo-

lios that are situated surprisingly close to the effi-

cient frontier. Figure 6 is based on the data pre-

sented by CMW. The dots show the recommended

portfolios while the thin line illustrates the effi-

cient MARKOWITZ frontier in the Standard De-

viation-Expected Return diagram. The thick line

shows the result of the behavioral model M2(0.5).

It is important to remember that the efficient fron-

tier of the MARKOWITZ model depends on thecorrelations between the asset returns. The behav-

ioral approach based on pure risk and naive diver-

sification does not. Hence, the efficiency losses

will strongly depend on the correlations: The effi-

ciency losses will increase when the investment

alternatives offer substantial hedging possibilities

(large negative correlations between two assets),

because the behavioral model does not take these

hedging possibilities into account.

Based on the market expectations of German fi-

nancial advisors (i.e. based on the whole data set),

we find considerable losses in efficiency. Table 2

shows the average losses in expected return per

year for each of the three portfolio types. Given

their own market assessments, the advisors

portfolio recommendations have expected re-

turns of about 1.5% below the optimal portfolios

they could have chosen. This confirms that even

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Figure 6: Efficiency Losses in CMW



0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

0% 5% 10% 15% 20% 25%

Standard deviation of Returns

ExpectedReturns

behavioral modelM2(0.5)

efficient frontier


(highly trained and highly paid) professional in-

vestment advisors do not recommend efficient

portfolios. Their tendency to ignore correlations

and to diversify naively would cost our fictive

investors of $ 250,000 between $ 3,700 (1.48%)

and $ 4,175 (1.67%) in the first year alone. As

Table 2 shows, these average efficiency losses are

much lower when we use the historical data to

evaluate the recommendations. Given these mar-ket parameters our fictive client would lose be-

tween $ 600 (0.24%) and $ 1.600 (0.64%) in the

first year.

Conclusion

We have examined the explanatory power of a

new behavioral approach to portfolio selection

based on the concepts of pure risk and naive di-

versification. By describing two simple invest-

ment models we have been able to explain profes-

sional investment advisors recommendations

relativly accurate. We tested the model on datapresented in literature as well as new data. We

asked German financial advisors, i.e. bank em-

ployees who are experienced in the field of in-

Table 2: Average Losses of Expected Return

low risk moderate risk high risk

individual market expectations 1.56% 1.48% 1.67%Historical market data 0.39% 0.24% 0.64%

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vestment consulting, to answer questions regard-

ing both, investment advice for three fictive cli-

ents with different risk attitudes and their market

expectations. With these sets of data we could

compare the traditional MARKOWITZ approach

with the new behavioral approach.Our first hypothesis is that even experienced in-

vestment advisors are not able to apply correla-

tions correctly although they are able to estimate

these correlations quite well. This hypothesis has

been confirmed. Professionals might memorize

the correlations without understanding their impli-

cations. Furthermore, we find that the behavioral

model based on naive diversification fits the given

portfolio recommendations significantly better

than the MARKOWITZ model does. We double-check our results with some alternative methods

such as using historical data and limited

risk/return restrictions. Finally, we examine effi-

ciency losses of the recommended portfolios and

find contrasting results. If we use the individual

market expectations, efficiency losses tend to be

quite substantial, but if we use the historical data,

efficiency losses are rather small, as CANNER,

MANKIW, and WEIL (1997) and FISHER and

STATMAN (1997a and 1997b) have already

found.

It is not clear, which market expectations are rele-

vant for the investment consultants when they

recommend portfolios. Do they use asset alloca-

tions of their bank, which are primarily based on

historical evaluations or are their recommenda-

tions driven by their own market expectations?

One way or another the mechanism that drives

the portfolio recommendations seems to differ

from normative theory.

Regarding the parameters and of models M1and M2, we find that the results are robust if the

diversification variable has enough weight in the

target function. The expected return of the portfo-

lio, however, does not seem to be as important for

our results as the low distance measures for model

M2(0) suggests. We suspect that this is due to the

fact that erceived volatilities and erceived ex-

pected returns of the five asset classes tend to be

correlated. Therefore the consideration of the ex-

pected returns in our models is not as important as

the consideration of the diversification term.

It will be interesting to learn more about our pro-

posed models in further studies. Especially therole of the parameters and could be reviewed.

Is it possible to identify a person-specific

parameter? How are and influenced by the

number of assets (n) and by the (historical or

perceived) market data? Is the model also

appropriate for investment problems with a much

larger number of investment alternatives. In such

situations investors and advisors try to pick

those stocks that seem to be very profitable to

them. Then the parameters and will probablyinfluence the trade-off between tendencies

towards diversification and tendencies towards

stock picking. Does our model capture this

situation as well? These are questions that should

be examined in further experiments or on the basis

of real portfolio data. It would be especially

interesting to vary the number of assets within-

subject and to consider assets whose expected

returns and volatilties are less correlated (e.g. in

an experiment) to be able to separate the influence

of these two target variables. Even more ambitious

would be to extend the idea of a different

treatment of risk to the context of dynamic asset

allocation models.

Finally, it will be an interesting field of future

research to investigate whether investors alloca-

tion behavior depends on correlations (which

might be the perceived or the historical ones)

when the induced efficiency losses are higher. As

very negative correlations between two assets

offer considerable hedging possibilities, which are

not captured by the behavioral models, it might be

the case that such low correlations influence port-

folio allocations.



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A s s e t A l l o c a t i o n Q u e s t i o n n a i r e

We, the Behavioral Finance Group at the University of

Mannheim (http://www.behavioral-finance.de), examinethe advice of investment consultants. In this context we areespecially interested in your recommendations for asset

allocations given the momentary market situation. Youradvice surely depends on your clients characteristics, so we

ask you to imagine the following scenario.

T h r e e n e w c l i e n t s i n t r o d u c e t h e m s e l v e s . T h e y h a v e n e a r l y t h e s a m e

c h a r a c t e r i s t i c s :

V o l k e r V o r s i c h t , M a r c e l M o d e r a t a n d N i k o N e r v e n s t a r k h a v e r e c e n t l y p a s s e d

t h e i r M B A - d i p l o m a a t t h e u n i v e r s i t y o f M a n n h e i m w i t h g o o d m a r k s . T h e y a r e 2 6

y e a r s o l d , s i n g l e a n d d o n o t o w n a n y r e a l e s t a t e s o r o t h e r w e a l t h . S h o r t l y ,

h o w e v e r , e a c h o f t h e m w i l l i n h e r i t D M 5 0 0 , 0 0 0 f r o m t h e i r d e c e a s e d

g r a n d m o t h e r . A s a l l o f t h e m h a v e a c c e p t e d t h e i r f i r s t j o b o f f e r a f e w w e e k s a g o

( n e t i n c o m e : D M 4 0 , 0 0 0 ) t h e y h a v e n o t e n o u g h t i m e t o i n v e s t t h e i r n e w

w e a l t h . A s k e d f o r t h e i r i n v e s t m e n t g o a l s a l l o f t h e m s a y t h a t t h e y d o n o t k n o w ,

w h e n t h e y n e e d t h e m o n e y . P e r h a p s p a r t o f t h e m o n e y i n o n e y e a r f o r a n e w

c a r o r i n f i v e y e a r s f o r a h o u s e . T h e i r k n o w l e d g e a b o u t d i f f e r e n t t y p e s o f

i n v e s t m e n t s i s r a t h e r g o o d , a s t h e y a t t e n d e d t h e c o u r s e F i n a n c e w h e r e t h e y

e v e n s t u d i e d d e r i v a t i v e s . T h e t h r e e g r a d u a t e s h o w e v e r h a v e d i f f e r e n t r i s k

a t t i t u d e s a s t h e f o l l o w i n g s t a t e m e n t s s h o w .

V o l k e r V o r s i c h t : I a m c a u t i o u s . A s a M B A - g r a d u a t e I k n o w t h a t r i s k y

a s s e t s s h o u l d h a v e h i g h e r r e t u r n s , b u t I c a n n o t b e a r t o

g a m b l e w i t h m y g r a n d m a s s a v i n g s . D e f i n i t e l y I a m

w i l l i n g t o i n v e s t p a r t o f t h e c a p i t a l i n s t o c k s a n d I a m

w i l l i n g t o a c c e p t a p o s s i b l e l o s s o f l e t s s a y 1 0 % i n a

y e a r . B u t a f t e r 1 0 y e a r s t h e r e s h o u l d a t l e a s t r e m a i n t h e

D M 5 0 0 , 0 0 0 a n d s o m e i n t e r e s t .

M a r c e l M o d e r a t : P l e a s e o f f e r m e a w e l l - b a l a n c e d i n v e s t m e n t s t r a t e g y ,

h o w t o i n v e s t t h e s e D M 5 0 0 , 0 0 0 . T h e s t r a t e g y s h o u l d

h a v e p o t e n t i a l s f o r g r o w t h a n d g a i n s w i t h o u t b e i n g t o o

r i s k y . A s a r e s u l t I a m c o m p l e t e l y a w a r e t h a t a p o s s i b l e

d r a w b a c k a t t h e m a r k e t s m i g h t p r o d u c e a p o r t f o l i o

p e r f o r m a n c e o f - 2 0 % i n o n e y e a r , w h i c h i s h a r d t o m a k e

u p f o r . T h a t s O K . B u t p l e a s e t a k e c a r e t h a t t h e p o r t f o l i o

r i s k i s n o t t o o b i g .



APPENDIX A: Questionnaire (page 1)

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N i k o N e r v e n s t a r k : A s I h a v e n e v e r d r e a m e d o f t h e s e D M 5 0 0 , 0 0 0 , I d o n o t

m i n d p o s s i b l e l o s s e s ! I a s k y o u t o i n v e s t t h i s m o n e y i n a

w a y , t h a t i t w i l l s e i z e v e r y g o o d o p p o r t u n i t i e s f o r

p o t e n t i a l g a i n s . O f c o u r s e I d o n o t w a n t t o g a m b l e w i t h

t h i s m o n e y , b u t I a m w i l l i n g t o a c c e p t t h e h i g h r i s k o f a n

a g g r e s s i v e a n d o p p o r t u n i t y - t a k i n g i n v e s t m e n t s t r a t e g y ,

t h a t m a k e s s e n s e m o m e n t a r i l y . S o I h o p e t o g e n e r a t e a

h i g h i n c o m e w i t h t h i s h e r i t a g e .

T h e f o l l o w i n g i n v e s t m e n t a l t e r n a t i v e s a r e a v a i l a b l e :

s h o r t - t e r m : i n t e r e s t - p a y i n g i n v e s t m e n t s w i t h s h o r t d u r a t i o n ( i n

D M o r E u r o ) : m o n e y m a r k e t f u n d s , c a s h a c c o u n t s ,

s h o r t - t e r m b o n d s ( t i m e t o m a t u r i t y u p t o 1 y e a r ) ,

e t c .

b o n d s : i n t e r e s t - p a y i n g i n v e s t m e n t s w i t h l o n g e r d u r a t i o n ( i n

D M o r E u r o ) : h i g h - q u a l i t y b o n d s ( t i m e t o m a t u r i t y 5

t o 2 0 y e a r s ) , l o n g - t e r m z e r o b o n d s o r c o r r e s p o n d i n g

b o n d - f u n d s

B l u e C h i p s G e r m a n s t o c k s , w h i c h b e l o n g t o t h e G e r m a n s t o c k

i n d e x D A X o r m u t u a l f u n d s i n v e s t i n g i n t h e s e s t o c k s

S m a l l C a p s o t h e r G e r m a n s t o c k s , t h a t d o n o t b e l o n g t o t h e D A X

o r c o r r e s p o n d i n g m u t u a l f u n d s

F o r e i g n s t o c k s a m i x t u r e o f f o r e i g n B l u e C h i p s , S m a l l C a p s a n d

m u t u a l f u n d s o f f o r e i g n s t o c k s , a s y o u p r e f e r i t

m o m e n t a r i l y

W h i c h p o r t f o l i o a l l o c a t i o n d o y o u a d v i s e t h e t h r e e g u y s f o r t h e n e x t 1 2 m o n t h s .

P l e a s e i n s e r t p e r c e n t a g e s .

V o l k e r V o r s i c h t M a r c e l M o d e r a t N i k o N e r v e n s t a r k

s h o r t - t e r m

b o n d s

B l u e C h i p s

S m a l l C a p s

F o r e i g n s t o c k s

S u m 1 0 0 % 1 0 0 % 1 0 0 %

P l e a s e m a k e s u r e , t h a t y o u r p o r t f o l i o p r o p o r t i o n s a d d u p t o 1 0 0 % .



Questionnaire (page 2)

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Y o u r M a r k e t E x p e c t a t i o n s .

H o w d o y o u a s s e s s t h e p e r f o r m a n c e s o f t h e m e n t i o n e d i n v e s t m e n t a l t e r n a t i v e s

a n d y o u r p o r t f o l i o p r o p o s i t i o n s i n t h e n e x t 1 2 m o n t h s ? P l e a s e c o n s i d e r a l l

i n t e r e s t p a y m e n t s , g a i n s / l o s s e s ( a l s o o f b o n d s b e c a u s e o f t h e i n t e r e s t r a t e

r i s k ) , d i v i d e n d p a y m e n t s a n d i f n e c e s s a r y e x c h a n g e r a t e r i s k s .

P l e a s e s t a t e t h e p e r f o r m a n c e ( i n % ) , o f w h i c h y o u t h i n k t h e

r e a l p e r f o r m a n c e i n 1 2 m o n t h s w i l l . . .

l o w e r b o u n d : . . . r a t h e r n o t ( i . e . i n o n l y 1 0 % o f a l l c a s e s ) r e m a i n u n d e r i t .

m e d i a n : . . . e q u a l l y l i k e l y e x e e d i t o r r e m a i n u n d e r i t .

u p p e r b o u n d : . . . r a t h e r n o t ( i . e . i n o n l y 1 0 % o f a l l c a s e s ) e x e e d i t .

l o w e r b o u n d m e d i a n u p p e r b o u n d

s h o r t - t e r m

b o n d s

B l u e C h i p s

S m a l l C a p s

F o r e i g n s t o c k s

V . V o r s i c h t

M . M o d e r a t

N . N e r v e n s t a r k

F i n a l l y w e a s k y o u f o r y o u r o p i n i o n t o w h a t e x t e n t t h e p e r f o r m a n c e s o f t h e f i v e

m e n t i o n e d i n v e s t m e n t a l t e r n a t i v e s c o h e r e ( s t a t i s t i c a l l y s p o k e n : c o r r e l a t e ) .

P l e a s e m a r k y o u r e x p e c t a t i o n s w i t h a c r o s s o n t h e s c a l e s .

- 1 0 0 % c o m p l e t e l y o p p o s i t e p e r f o r m a n c e p r o c e s s e s

0 % n o c o h e r e n c e

+ 1 0 0 % c o m p l e t e l y p a r a l l e l p e r f o r m a n c e p r o c e s s e s

c o h e r e n c e s h o r t - t e r m a n d b o n d s

% - 1 0 0 - 7 5 - 5 0 - 2 5 0 + 2 5 + 5 0 + 7 5 + 1 0 0

c o h e r e n c e s h o r t - t e r m a n d B l u e C h i p s

% - 1 0 0 - 7 5 - 5 0 - 2 5 0 + 2 5 + 5 0 + 7 5 + 1 0 0

c o h e r e n c e s h o r t - t e r m a n d S m a l l C a p s

% - 1 0 0 - 7 5 - 5 0 - 2 5 0 + 2 5 + 5 0 + 7 5 + 1 0 0

p

o

r

t

f

o

l

i

o

s

i

n

v

e

s

t

m

e

n

t

a

l

t

e

r

n

a

t

i

v

e

s




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c o h e r e n c e s h o r t - t e r m a n d f o r e i g n s t o c k s

% - 1 0 0 - 7 5 - 5 0 - 2 5 0 + 2 5 + 5 0 + 7 5 + 1 0 0

c o h e r e n c e b o n d s a n d B l u e C h i p s

% - 1 0 0 - 7 5 - 5 0 - 2 5 0 + 2 5 + 5 0 + 7 5 + 1 0 0

c o h e r e n c e b o n d s a n d S m a l l C a p s

% - 1 0 0 - 7 5 - 5 0 - 2 5 0 + 2 5 + 5 0 + 7 5 + 1 0 0

c o h e r e n c e b o n d s a n d f o r e i g n s t o c k s

% - 1 0 0 - 7 5 - 5 0 - 2 5 0 + 2 5 + 5 0 + 7 5 + 1 0 0

c o h e r e n c e B l u e C h i p s a n d S m a l l C a p s

% - 1 0 0 - 7 5 - 5 0 - 2 5 0 + 2 5 + 5 0 + 7 5 + 1 0 0

c o h e r e n c e B l u e C h i p s a n d f o r e i g n s t o c k s

% - 1 0 0 - 7 5 - 5 0 - 2 5 0 + 2 5 + 5 0 + 7 5 + 1 0 0

c o h e r e n c e S m a l l C a p s a n d f o r e i g n s t o c k s

% - 1 0 0 - 7 5 - 5 0 - 2 5 0 + 2 5 + 5 0 + 7 5 + 1 0 0

F i n a l l y p l e a s e t e l l u s , w h i c h f o r e i g n s t o c k s d i d y o u h a v e i n m i n d :

Thank you very much for your help. If you give us your email-

address, we certainly will inform you about our results. In any

case your answers will remain anonymous and will not belinked with your name.

email-address:

For your help you find attached Volume 0 from our series

"Research for practitioners".




Behavioral Finance Group

Lehrstuhl fr ABWL, Finanzwirtschaft

insbesondere Bankbetriebslehre

Universitt MannheimD-68131 Mannheim

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APPENDIX B:

Market Expectations and Historical DataExpected returns and Volatility:

mean expectation

historical data [36]

(19881999)expected return 3.2% 5.7%

short-termvolatility 0.6% 0.7%expected return 4.4% 7.1%

bondsvolatility 1.3% 6.8%expected return 12.0% 19.9%

blue chipsvolatility 8.9% 20.2%expected return 14.6% 14.3%

small capsvolatility 12.3% 18.0%expected return 15.9% 14.5%

foreign stocks volatility 12.3% 16.5%

Correlations:

mean expectationhistorical data

(19881999)

short-term bonds 36.7% 13.8%short-term blue chips 11.3% 14.1%short-term small caps 16.0% 8.4%short-term foreign stocks 7.0% 19.8%bonds blue chips 11.4% 2.0%bonds small caps 9.5% 8.1%bonds foreign stocks 14.6% 4.2%

blue chips small caps 50.0% 76.8%blue chips foreign stocks 54.1% 63.3%small caps foreign stocks 28.4% 49.5%



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APPENDIX C:Portfolio Recommendations

mean recommended proportion

low risk moderate risk high risk

short-term 29.2% 16.0% 10.3%

Bonds 43.3% 30.0% 11.9%blue chips 17.3% 25.1% 26.2%small caps 2.8% 8.2% 18.1%foreign stocks 7.3% 20.7% 33.6%

APPENDIX D:Non-aggregated Distance Measures

Participant Opt1 M2(0.5) Difference

1 52.25% 8.97% 43.28%

2 42.96% 25.45% 17.51%3 26.42% 17.06% 9.37%4 56.25% 38.39% 17.86%5 62.06% 34.63% 27.42%6 52.11% 26.41% 25.70%7 60.19% 25.18% 35.01%8 41.97% 15.63% 26.34%9 37.39% 25.25% 12.14%

10 73.11% 24.44% 48.67%11 51.86% 37.77% 14.09%12 64.19% 42.81% 21.38%13 27.98% 24.58% 3.40%14 27.58% 29.67% 2.09%15 42.11% 22.26% 19.85%16 49.78% 32.35% 17.43%17 51.61% 21.91% 29.70%18 38.88% 24.58% 14.30%19 27.07% 29.43% 2.35%20 45.01% 11.80% 33.21%21 14.36% 39.81% 25.45%22 39.61% 13.05% 26.56%23 27.81% 17.43% 10.38%



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FOOTNOTES

[1] An overview of the literature on the influence of

time horizon on asset allocation is given in KLOS,

LANGER and WEBER (2002).

[2] See BAJEUX-BESNAINOU, JORDAN and POR-

TAIT (2001), BRENNAN and XIA (2000), CAMP-

BELL and VICEIRA (2001).

[3] ELTON and GRUBER (2000) offer a different

explanation. They show that different historical

data can lead to completely different optimal

bond-to-stock ratios. Under reasonable assump-

tions they are able to find more recent market data

that fit the observed portfolio recommendations.

They propose, that the advisor supply the input

data on which suggested allocations are made.

[4] See ELTON and GRUBER (2000), page 29:Thus, an assumption of no short sales is the only

realistic assumption for the asset allocation deci-

sion,.

[5] We are aware that the behavioral approach is no

longer compatible with expected utility theory.

However, MARKOWITZ approach is also only

compatible under very restricted assumptions. In

addition, we want to explore intuitive decision

making on a portfolio level. On this level investors

use heuristics (as proposed here) which are notcompatible with expected utility theory.

[6] See section 3.3.3. in OEHLER (1995).

[7] See also section 6.3.3 in SCHROEDER-WILD-

BERG (1998).

[8] There are, however, few studies that do find some

effect of correlations on portfolio choice. KROLL

and LEVY (1992) find some effects driven by cor-

relations between different investment options. In

their new experimental design they offered more

attractive incentives to the students, who were

highly educated MBA students. Furthermore, they

published the results and strategies of all students

after each round. So, the participants had the pos-

sibility of learning and imitating successful strate-

gies.

[9] Alternatively, we use the linear combination of the

variance of the assets (n 2

i ii 1=

) as a measure

for pure risk, but we do not find different results.

[10] See E.U. WEBER (2000).

[11] See WEBER (2000).

[12] Notice that for the mean proportion holds:

n1 1

ii 1n n

1

=

=

= = .

[13] BENARTZI and THALER mention that the 1/n

heuristic or 1/n rule goes back to the 4th century,

when this rule had been proposed in the Babalo-

nian Talmud.

[14] This effect is not driven by the restrictions of in-

vestors investment alternatives.

[15] We obtained answers from employees of all major

German banks. For reasons of confidentiality we

will not mention their names explicitly.

[16] It should be noted that the questionnaire was notcompletely clear on the question of investment ho-

rizon. We explicitly asked for the advice for the

next twelve months (page 2 of the questionnaire)

but the investors investment horizon was not

clearly defined.

[17] time to maturity

[18] See also CLEMEN and REILLY (1999).

[19] Examining in particular these questionnaires we

do not get different results.

[20] Some consultants added some additional invest-

ment alternatives to their recommendations. As

we do not know their market expectations about

these additional investments we cannot use the

answers.

[21] In one case the variance-covariance-matrix result-

ing from the given answers is not positive semi-

definite, which led to a negative variance of one

portfolio. Therefore we could not use these an-

swers.

[22] We would like to emphasize that 23 is actually a

large number of observations. Most studies just

use the advise provided by headquarters.

[23] To estimate the expected returns and the volatil-

ities with the stated 10%-quantile, the 90%-

quantile and the median we used the three-point

estimator of Pearson and Tukey (as described in

KEEFER and Bodily, 1983).



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[24] We recalculated our results by assuming the

short-term volatility to be 0% and we did not find

other results.

[25] This confirms a result regarding diversification

effects in SIEBENMORGEN, WEBER and WE-

BER (2001). There we asked students who par-ticipated in a risk perception experiment to esti-

mate volatility and risk of diversified portfolios and

we found that volatility and risk of these portfolios

tended to be overestimated relatively to the indi-

vidual assets.

[26] KROLL, LEVY and RAPOPORT (1988b) use a

similar measure.

[27] The models opt1 and opt2 differ as they generate

different benchmark portfolios on the efficient fron-

tier.[28] Using opt2 instead of opt1 or M1(0.5) resp. M2(0)

instead of M2(0.5) does not change the results

substantially.

[29] We get the same results for a paired-samples T-

test.

[30] In a sensitivity check we controlled the influences

of the parameters and in the models M1 and

M2 and we found a very low sensitivity. However,

for the cases =0 and =1, in which the target

variable diversification disappears, the behav-ioral models are not better than the optimal mod-

els any more.

[31] If we test the three risk classes individually, we

also get significant results for all combinations.

[32] We changed the historical mean returns of the

asset types short-term and bonds by using the

short-term/long-term interest rates of December

1999 (3% for short-term investments and 5% for

long-term investments in bonds). We did that to

correct for the relatively low interest rates during

our study.

[33] This method is particularly interesting, since it can

be shown that certain parameters may lead to

dominated solutions in the behavioral models.

With this method we exclude dominated solutions.

Nevertheless, we find that the behavioral ap-

proaches are still significantly better.

[34] Even if we assume the volatility of the short-term

investment to be 0% (existence of a riskless as-

set), the MARKOWITZ approach produces 0%-

proportions of the short-term investment because

of the short-sale constraint.

[35] We get nearly the same results if we take otherparameters < 1 or if we use model M1 with > 0.

[36] Calculated from the monthly reports of the Ger-

man central bank Deutsche Bundesbank and the

indices DAX, SDAX, MSCI-world



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