A Generalisation of the Mean-Variance Analysis
Valeri Zakamouline and Steen Koekebakker
University of Agder, Faculty of Economics, Service Box 422, 4604 Kristiansand, Norway E-mails: Valeri.Zakamoulineuia.no; Steen.Koekebakkeruia.no
Abstract
In this paper we consider a decision maker whose utility function has a kink at the reference point with different functions below and above this reference point. We
also suppose that the decision maker generally distorts the objective probabilities. First we show that the expected utility function of this decision maker can be
approximated by a function of mean and partial moments of distribution. This ‘mean-partial moments’ utility generalises not only mean-variance utility of Tobin
and Markowitz, but also mean-semivariance utility of Markowitz. Then, in the spirit of Arrow and Pratt, we derive an expression for a risk premium when risk
is small. Our analysis shows that a decision maker in this framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and
aversion to uncertainty in losses. Finally we present a solution to the optimal capital allocation problem and derive an expression for a portfolio performance
measure which generalises the Sharpe and Sortino ratios. We demonstrate that in this framework the decision maker’s skewness preferences have first-order impact
on risk measurement even when the risk is small.
Keywords:
mean-variance utility, quadratic utility, mean-semivariance utility,
risk aversion, loss aversion, risk measure, probability distortion, partial moments of distribution, risk premium, optimal capital allocation, portfolio performance
evaluation, Sharpe ratio
JEL classification:
D81, G11
1. Introduction
This paper presents a uniform framework that provides general insights into a broad class of models of choice under uncertainty where the utility function has a reference
The former title of the paper was ‘Analysis of financial decision making with loss aversion’. The authors would like to thank Fred Espen Benth, Thorsten Hens, David Nawrocki, Peter
Wakker, Malevergne Yannick and the anonymous referees for their comments. The article has also benefited from comments by seminar participants at the Norwegian University of
Science and Technology, the Norwegian School of Economics and Business Administration, the Swiss Banking Institute, and the European Financial Management Association Annual
2008 Meeting. Correspondence: Valeri Zakamouline
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point andor the objective probabilities are distorted. We start with the justification of mean-partial moments utility which lays down a basis for this framework and generalises
mean-variance utility. Then we proceed to the generalisation of some central results of the mean-variance analysis, namely, the Arrow-Pratt risk aversion and risk premium,
the Arrow’s optimal capital allocation, and the Sharpe portfolio performance measure. The analysis presented in this paper also provides important new insights into the risk
and reward measurement.
Expected Utility Theory EUT of von Neumann and Morgenstern 1944 has long been the main workhorse of modern financial theory. A von Neumann and Morgenstern’s
utility function is defined over the decision maker’s wealth. The properties of a von Neumann and Morgenstern’s utility function have been studied in every detail. The
concept of ‘risk aversion’ was analysed by Friedman and Savage 1948 and Markowitz 1952. They showed that the realistic assumption of diminishing marginal utility of
wealth explains why people are risk averse. Measurement of risk aversion was developed by Pratt 1964 and Arrow 1971. These authors analysed the risk premium for small
risks and introduced a measure which is now widely known as the ‘Arrow-Pratt measure of risk aversion’. The celebrated modern portfolio theory of Markowitz and the use of
a mean-variance utility function can be justified by approximating a von Neumann and Morgenstern’s utility function by a function of mean and variance see, for example,
Samuelson 1970, Tsiang 1972 and Levy and Markowitz 1979. In this sense, the use of the Sharpe ratio Sharpe, 1966 as a measure of performance evaluation of risky
portfolios is also well justified.
However, not very long after EUT was formulated by von Neumann and Morgenstern, questions were raised about its value as a descriptive model of choice under uncertainty.
Allais 1953 and Ellsberg 1961 were among the first to challenge EUT. Influential experimental studies have shown the inability of EUT to explain many observed
phenomena and reinforced the need to rethink much of the theory. An enormous amount of theoretical effort has been devoted towards developing alternatives
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to EUT. In many of the alternative models of choice under uncertainty the decision maker’s utility has a
reference point with different functions below and above the reference point possibly with a kink at the reference point andor the decision maker distorts the objective
probability distribution. One of the first examples of such type is the utility function of Markowitz 1952 with a concave segment below the reference point and a convex
segment above the reference point. In the mean-lower partial moment model of Fishburn 1977 and Bawa 1978 the utility function is linear above the reference point and
concave below the reference point.
The most influential among all alternative models of choice under uncertainty is Prospect TheoryCumulative Prospect Theory PTCPT of Kahneman and Tversky
1979 and Tversky and Kahneman 1992. PTCPT can correctly predict individual choices even in cases in which EUT is violated.
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In PTCPT the utility function is defined over gains and losses relative to some reference point, as opposed to wealth
in EUT. The utility function has a kink at the reference point, with the slope of the loss function steeper than the gain function. This is called ‘loss aversion’ which is an
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For an excellent review of alternative theories, the interested reader can consult Starmer 2000. Besides, all financial phenomena based on nonrational behaviour among investors
constitute now the main subject of Behavioural Finance. For a review and synthesis of Behavioural Finance, the interested reader can consult Subrahmanyam 2007.
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For a brief description see, for example, Camerer 2000.
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important element of PTCPT. The marginal value of both gains and losses decreases with their size. All these properties give rise to an asymmetric S-shaped utility function,
concave for gains and convex for losses. Moreover, in PTCPT the decision maker distorts the objective probability distribution by overweighting small probabilities.
Another prominent example of an alternative descriptive model of choice under uncertainty is Disappointment Theory DT of Bell 1985 and Loomes and Sugden
1986. In DT a decision maker is assumed to be ‘disappointment averse’. In particular, this model assumes that if the outcome of a decision is worse than expected, the sense of
disappointment will be generated. On the other hand, an outcome better than expected will stimulate ‘elation’. The utility function in DT consists of two parts: the first
part is a ‘basic’ utility similar to a utility function in EUT, whereas the second part accounts for the feelings of disappointment and elation. This second utility function is
concave for outcomes worse than expected and convex for outcomes better than expected. Consequently, the total utility function is obviously concave below the reference point.
However, it is difficult to say something definite about the shape of the total utility function above the reference point. If the sense of elation is very strong, the total utility
function can be convex for outcomes better than expected.
Many researches have observed a tendency for individuals to mispercept objective probabilities, in particular, to subjectively weight objective probabilities. This effect
can be captured by a model of choice under uncertainty that incorporates ‘decision weights’ instead of objective probabilities. Theories of this type were first discussed
by Edwards 1955 and Edwards 1962. One of the best-known models of this type is Anticipated Utility TheoryRank Dependent Expected Utility AUTRDEU of Quiggin
1982. According to this theory, the decision maker’s utility function is defined in the same manner as in EUT, that is, over wealth. The main difference is the assumption
that a decision maker distorts objective probabilities using some rule of distortion. Some examples of distortion are: overweightingunderweighting small probabilities;
overweightingunderweighting probabilities of unfavourable outcomes; etc. A special case of the AUTRDEU is the Dual Theory of Yaari 1987.
Even though the models where the utility function has a reference point andor the objective probabilities are distorted have been known for quite a while, still there is only
a few studies of some effects of loss aversion and probability distortion. Moreover, practically all of these studies consider the decision making in PTCPT only. The
study of the risk premium in a behavioural framework starts with the paper by Levy and Levy 2002 who considered a decision maker equipped with a von Neumann-
Morgenstern utility function and a small gamble with two possible outcomes. They derived an expression for a risk premium which accounts for PTCPT type of probability
distortion and showed that this type of probability distortion systematically increases the risk premium. Davies and Satchell 2007 extended the model of Levy and Levy
2002 by introducing loss aversion in the decision maker’s utility.
The study of the effects of a kink in the decision maker’s utility function starts with the paper by Segal and Spivak 1990. The authors studied these effects in the
context of insurance, but also mentioned briefly that a kink in a utility function may cause the avoidance of the risky asset in the investor’s capital allocation. Further, using
a simple one-period binomial model without probability distortion and a piecewise- power utility function motivated by PTCPT, Gomes 2005 showed that loss aversion
causes avoidance of the risky asset when the risk premium is small. Berkelaar et al. 2004 considered a continuous-time model and an investor who is also equipped with a
piecewise-power utility function. They derived a closed-form solution to the investor’s
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optimal capital allocation problem and found that as the investment horizon decreases, the investor reduces the proportion of the risky asset in the complete portfolio and, thus,
invests more in the risk-free asset. Loss aversion and probability distortion may help explain the equity premium puzzle
of Mehra and Prescott 1985. Bernartzi and Thaler 1995 and subsequently Barberis et al. 2001 found that loss aversion can explain why stock returns are too high relative
to bond returns. Cecchetti et al. 2000 and Abel 2002 pointed out that pessimistic probability distortion may also cause the observed equity premium puzzle.
Despite some studies focusing on certain effects of loss aversion and probability distortion, still little is known about general implications from these alternative theories.
The goal of this paper is to present a uniform framework that is able to provide general insights into this broad class of models of choice under uncertainty. In this paper we
consider a decision maker with a generalised behavioural utility function that has a reference point. We assume that the decision maker regards the outcomes below the
reference points as losses, and the outcomes above the reference point as gains. We suppose that the behavioural utility generally has a kink at the reference point and
different functions below and above the reference point. We require only that the behavioural utility function is continuous and increasing in wealth and has at least
the first and the second one-sided derivatives at the reference point. Moreover, we also suppose that the decision maker generally distorts the objective probability distribution.
Note that our generalised framework encompasses EUT as well.
The first contribution of this paper is to provide an approximation analysis of the expected generalised behavioural utility function. Our analysis shows that the expected
generalised behavioural utility function can be approximated by a function of mean and partial moments of distribution. This ‘mean-partial moments’ utility generalises not
only mean-variance utility of Tobin and Markowitz, but also mean-semivariance utility mentioned by Markowitz 1959 and discussed further by many others. Mean-partial
moments utility appears to have reasonable computational possibilities in, for example, the optimal portfolio choice problem as well as a great degree of flexibility in modelling
different risk preferences of a decision maker.
The second contribution of this paper is, in the spirit of Pratt 1964 and Arrow 1971, to derive an expression for a risk premium when risk is small. We show that mean-
partial moments utility allows for a much richer and detailed characterisation of a risk premium. It is well known that for a decision maker with mean-variance utility the only
source of risk is variance. Moreover, the risk attitude of this decision maker is completely described by a single measure widely known as the Arrow-Pratt measure of risk aversion
which is, in fact, the aversion to uncertainty.
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In contrast, our analysis shows that a decision maker with a generalized behavioral utility distinguishes between three sources
of risk: expected loss, uncertainty in losses, and uncertainty in gains. Consequently, a decision maker in our framework exhibits three types of aversions: aversion to loss,
aversion to uncertainty in gains, and aversion to uncertainty in losses. Loss aversion leads to different weights of losses and gains in the expression for the risk premium.
As compared to the analysis presented by Davies and Satchell 2007, our results are much more concise and explicit. We also provide comparative static analysis of the
expression for the risk premium. Besides, our results are not limited to gambles with only two possible outcomes, but encompass continuous probability distributions as well.
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Here by uncertainty we actually mean a deviation measure of uncertainty which assesses the dispersion of distribution.
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The third contribution of this paper is to generalise Arrow’s famous solution to the optimal capital allocation problem for an investor. In this setting the investor’s objective
is to optimally allocate his wealth between a risk-free and a risky asset. It is widely known that a mean-variance utility maximiser will always want to allocate some wealth to the
risky asset if the risk premium is non-zero. In contrast, we show that an investor with the generalised behavioural utility function will want to allocate some wealth to the risky
asset only when the perceived risk premium is sufficiently high how high depends on the level of loss aversion and the degree of probability distortion. Otherwise, if
the perceived risk premium is small, the investor avoids the risky asset altogether and invests all wealth in the risk-free asset. This result may help explain why many investors
do not invest in equities.
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This result also illustrates that the equity premium puzzle discovered by Mehra and Prescott 1985 can be explained by either loss aversion or
pessimistic probability distortion or a combination of these effects. As compared to the analysis presented by Gomes 2005, we derive not implicit, but explicit solutions to
the optimal capital allocation problem and provide comparative static analyses of the solutions. Besides, in our setting we consider a general behaviour utility function, not
only the piecewise-power utility motivated by PTCPT. Moreover, our results are not limited to risky asset returns with only two possible outcomes, but encompass continuous
probability distributions as well. Finally, in our analysis we generally consider the case where the investor distorts the probability distribution of the risky asset returns, in the
analysis by Gomes the probability distribution is objective.
Our fourth contribution is to derive an expression for the portfolio performance measure of an investor with the generalised behavioural utility. This measure generalises
the Sharpe and Sortino
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ratios see Sortino and Price 1994. As compared to either the Sharpe or Sortino ratio where the investor’s risk preferences seemingly disappear, the
computation of the generalised performance measure usually requires knowledge of the investor’s risk preferences. Hence, this performance measure is not unique for all
investors, but rather it is an individual performance measure. The explanation for this is that in our generalised framework an investor distinguishes between several sources
of risk. Since each investor may exhibit different preferences to each source of risk, investors with different preferences might rank differently the same set of risky assets.
The fifth contribution of this paper is to provide some new insights on risk mea- surement. In modern financial theory most often one uses variance as a risk measure.
This is because in the EUT framework variance has the first-order impact on risk measurement, at least when the risk is small. We demonstrate that in many alternative
theories variance is generally not a proper risk measure even if the risk is small. In these theories it is the decision maker’s skewness preferences that have first-order impact on
risk measurement. This is the consequence of the presence of loss aversion and the fact that the decision maker’s degrees of risk aversions below and above the reference
point might be substantially different. In the latter case if probability distributions are not symmetrical, then, depending on the signs and the values of skewness preferences, either
the downside or the upside part of variance is a more adequate risk measure than variance.
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See, for example, Agnew et al. 2003 who report that about 48 of participants of retirement accounts do not invest in equities. This behavior clearly contradicts EUT.
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The Sortino ratio is a reward-to-semivariability type ratio that has been used by many researchers. Some examples are Klemkosky 1973, Ang and Chua 1979, and Ziemba
2005. The interested reader can consult Nawrocki 1999 for a brief history of downside risk measures.
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The rest of the paper is organised as follows. In Section 2 we briefly review the justification of the mean-variance analysis and present the results we want to
generalise. In Section 3 we present assumptions, definitions, and notation that we will use in our generalised framework. In Section 4 we perform the approximation analysis
and generalise mean-variance utility. In Section 5 we generalise the Arrow-Pratt risk premium. In Section 6 we discuss briefly the impact of the decision maker’s skewness
preferences on risk measurement in our generalised framework. In Section 7 we analyse the optimal capital allocation problem. In Section 8 we derive the expression for a
portfolio performance measure. Section 9 concludes the paper.
2. Expected Utility Theory and the Mean-Variance Analysis
