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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]
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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
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
18 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1
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 =
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1 19
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.
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
20 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1
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]
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
22 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1
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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Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
24 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1
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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Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1 25
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.
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
26 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1
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
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
28 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1
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
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1 29
-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
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
30 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1
-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
recommended portfolios
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
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1 31
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
recommended portfolios
(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.
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
32 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1
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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 .
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1 33
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 % .
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
34 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1
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
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1 35
Questionnaire (page 3)
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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".
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
36 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1
Questionnaire (page 4)
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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REFERENCES
BAJEUX-BESNAINOU, I., J. V. JORDAN and R. POR-
TAIT (2001): An Asset Allocation Puzzle: Comment,
The American Economic Review, 91, pp. 11701179.
BENARTZI, S. (2001): Excessive extrapolation and
the allocation of 401(k) accounts to company stock.The Journal of Finance, 56, pp. 17471764.
BENARTZI, S. and R. H. THALER (2001): Naive Di-
versification Strategies in Retirement Saving Plans.
The American Economic Review, 91, pp. 7998.
BLUME, M.and I. FRIEND (1975): The Asset Structure
of Individual Portfolios and Some Implications for Utility
Theory. The Journal of Finance, 30, pp. 585603.
BRENNAN, M. J. and Y. XIA (2000): Stochastic Inter-
est Rates and the Bond-Stock-Mix. European Finance
Review, 4, pp. 197210.CAMPBELL, J. Y. and L. M. VICEIRA (2001): Who
should buy long-term Bonds?. The American Eco-
nomic Review, 91, pp. 99127.
CANNER, N., N. G. MANKIW and D. N. WEIL (1997):
An Asset Allocation Puzzle. The American Economic
Review, 87, pp. 181191.
CLEMEN, R. T., G. W. FISCHER and R. L. WINKLER
(2001): Assessing Dependences: Some Experimental
Results. Management Science, 46, pp. 11001115.
CLEMEN, R. T. and T. REILLY (1999): Correlationsand Copulas for Decision and Risk Analysis. Mana-
gement Science, 45, pp. 208224.
DEGEORGE, F., D. JENTER, A. MOEL and P. TU-
FANO (2001): Selling Company Shares to Reluctant
Employees: France Telecoms Experience. Working
Paper, HEC School of Management, Jouy en Josas,
France.
ELTON, E. J. and M. J. GRUBER (2000): The Ration-
ality of Asset Allocation Recommendations. Journal of
Financial and Quantitative Analysis, 35, pp. 2741.
FISHER, K. L. and M. STATMAN (1997a): The Mean-
Variance-Optimization Puzzle: Security Portfolios and
Food Portfolios. Financial Analysts Journal, 53, pp.
4150.
FISHER, K. L. and M. STATMAN (1997b): Investment
Advice from Mutual Fund Companies Closer to the
Talmud than to MARKOWITZ. The Journal of Portfolio
Management, 53, pp. 925.
JOOS, C. M. and M. KILKA (1999): Sind Aktienportfo-
lios privater Anleger ausreichend diversifiziert?. Die
Bank, 12/99, pp. 862866.
KEEFER, D. L. and S. E. BODILY (1983): Three-point
Approximations for Continuous Random Variables.
Management Science, 29, pp. 595609.KELLY, M. (1994): All their Eggs in One Basket: Port-
folio Diversification of US Households. Journal of Eco-
nomic Behavior and Organization, 27, pp. 8796.
KLOS, A., T. LANGER and M. WEBER (2002): Wel-
che Rolle spielt der Anlagehorizont bei der Beurteilung
von Investments?. Working Paper, Universitt Mann-
heim.
KROLL, Y. and H. LEVY (1992). Further Tests of the
Separation Theorem and the Capital Asset Pricing
Model. The American Economic Review, 82, pp. 664670.
KROLL, Y, H. LEVY and A. RAPOPORT (1988a): Ex-
perimental Tests of the Mean-Variance Model for Port-
folio Selection. Organizational Behavior and Human
Decision Processes, 42, pp. 388410.
KROLL, Y, H. LEVY and A. RAPOPORT (1988b): Ex-
perimental Tests of the Separation Theorem and the
Capital Asset Pricing Model. The American Economic
Review, 78, pp. 500519.
LIPE, M. G. (1998): Individual Investors Risk Judg-ments and Investment Decisions: The Impact of Ac-
counting and Market Data. Accounting, Organizations
and Society, 23, pp. 625640.
MARKOWITZ, H.M. (1952): Portfolio Selection. The
Journal of Finance, 7, pp. 7791.
MERTON, R. C. (1971): Optimum Consumption and
Portfolio Rules in a Continuous-time Model. Journal of
Economic Theory, 3, pp. 373413.
MERTON, R. C. (1973): An Intertemporal Capital As-
set Pricing Model. Econometrica, 41, pp. 867887.
OEHLER, A. (1995): Die Erklrung des Verhaltens
privater Anleger. Schaeffer-Poeschel, Stuttgart.
SCHRDER-WILDBERG, U. (1998): Entscheidungs-
und Lernverhalten an Wertpapiermrkten: Psychologi-
sche Aspekte von Brsenentscheidungen. Gabler/
DUV, Wiesbaden.
Niklas Siebenmorgen and Martin Weber: A Behavioral Model for Asset Allocation
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 17, 2003 / Number 1 41
8/2/2019 03-1 Sie Web Fmpm
28/28
SHEFRIN, H. and M. STATMAN (2000): Behavioral
Portfolio Theory. Journal of Financial and Quantitative
Analysis, 35, pp. 127151.
SIEBENMORGEN, N., E. U. WEBER and M. WEBER
(2001): Communicating Asset Risk: How the format of
historic volatility information affects risk perception andinvestment decisions. Working Paper 0038, Sonder-
forschungsbereich 504, Universitt Mannheim.
THALER, R. H. (1985): Mental Accounting and Con-
sumer Choice. Marketing Science, 4, pp. 199214.
WEBER, E. U. (2000): Decision and Choice: Risk,
Empirical Studies. To appear in: International Encyclo-
pedia of the Social & Behavioral Sciences, Elsevier
Science, Amsterdam et al.
WEBER, M. (2000): Decision and Choice: Risk, Theo-
ries. International Encyclopedia of the Social & Behav-ioral Sciences, Elsevier Science, Amsterdam et al., pp.
1336413368.
WEBER, M. and C. CAMERER (1998): The Disposi-
tion Effect in Securities Trading: An Experimental
Analysis. Journal of Economic Behavior & Organiza-
tion, 33, pp. 167184.
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