Can We Measure Portfolio Performance?
by Steen Koekebakker and Valeri Zakamouline
Introduction
The risky assets available to investors are numerous: mutual funds, hedge funds, structured
products, equity-linked notes to name a few. The characteristics of each asset class can be
summarized in the different return distributions. Even within a single asset class the return
distributions of assets are not alike. We assume that the return distributions of all risky assets
are known and would like to choose the best asset to invest to, meaning that the risky assets
are mutually exclusive investment alternatives. How to do this? The standard approach in
financial theory and practice is to employ some portfolio performance measure to rank the
various risky investments. Each portfolio performance measure calculates a score for each
asset using its probability distribution of returns. The best asset to invest to is the asset with
the highest score.
The Sharpe ratio is a commonly used measure of portfolio performance. But because it is
based on mean-variance theory, this measure can only be used in some restrictive cases, for
example, when return distributions are normal. When return distributions are non-normal, the
Sharpe ration can lead to misleading conclusions and unsatisfactory paradoxes, see Bernardo
and Ledoit (2000) and Hodges (1998). There have been proposed numerous universal
performance measures that, in one way or the other, are alternatives to the Sharpe ratio and try
to take into account non-normality of return distributions. For some examples, see Sortino and
Price (1994), Dowd (2000), Stutzer (2000), Keating and Shadwick (2002), Gregoriou and
Gueyie (2003), Kaplan and Knowles (2004), and Ziemba (2005). The main drawback of many
of these alternative performance measures is that they lack a solid theoretical underpinning. In
this paper we review the latest results on portfolio performance measures based on either
expected utility theory or non-expected utility theory (the latter is opposed to the von
Neumann and Morgenstern expected utility theory). The main purpose of this paper is to show
that unless we know exactly the investor’s preferences, and unless all investors share the same
preferences, a single performance measure that is suitable for all investors cannot exist.
Performance Measures Based on Expected Utility Theory
Expected utility theory of von Neumann and Morgenstern has long been the main workhorse
of modern financial theory. A von Neumann-Morgenstern utility function is defined over the
investor's wealth as U(W), where U(.) is some function and W is the investor’s wealth. The
celebrated modern portfolio theory of Markowitz and the use of mean-variance utility
function can be justified by approximating a von Neumann-Morgenstern utility function by a
function of mean and variance, see, for example Samuelson (1970) and Levy and Markowitz
(1979). Besides, the use of mean-variance utility can also be justified when either the return
distribution is normal or the investor's utility function is quadratic. But when it comes to
quadratic utility, it is also well known that it has anomalous properties such as increased risk
aversion as wealth increases, to say nothing of ultimate satiation.
Mean-variance utility function is given by
1
E[U (W )]  E[W ]  Var[W ],
2
Where E[W] and Var[W] are the expected wealth and the variance of the wealth, and γ is the
investor’s risk aversion coefficient. If the investor’s utility is given by some function, then the
risk aversion coefficient is computed as
 
U ''
.
U'
This is also known as the Arrow-Pratt measure of absolute risk aversion. Mean-variance
utility function leads to a performance measure known as the Sharpe ratio
SR 
 r
,

where μ and σ are the expected returns and the standard deviation of returns of some risky
asset, and r is the risk-free rate of return. The Sharpe ratio is appealing, and rather popular in
practice, because the investor’s preferences magically disappear from the performance
measure. That is, irrespective of the level of risk aversion, all investors with either quadratic
or mean-variance utility will rank indentically different investment alternatives. But what will
be the ranking of alternative investments in the general case, that is, for a general utility
function? For a general utility function, the explicit (closed-form) solution for the
performance measure does not exist. Observe that in principle, in the framework of expected
utility theory, the portfolio evaluation problem is rather trivial, though quite demanding in
implementation. We need first to specify the investor’s utility function and then we choose the
risky asset/portfolio that maximizes the investor’s expected utility. Also, in principle, we do
not need an explicit formula for the performance measure. All we need is to specify the
investor’s utility which can rank the risky alternatives in terms of expected utilities they give.
However, the number of possible utility functions is, in principle, unlimited. Will the ranking
remain basically the same or will it be substantially different for different utility functions?
The analysis of Koekebakker and Zakamouline (2007a) suggests that the ranking of
alternative investment alternatives might be very different for different types of utility
functions.
In particular, Koekebakker and Zakamouline (2007a) suggested approximating a general
utility function using the first three moments of the return distribution, namely the mean,
variance, and skewness. They arrived to the measure that they denote as the Adjusted for
Skewness Sharpe Ratio (ASSR) given by
ASSR  SR 1
bS
SR ,
3
where SR is the standard Sharpe ratio, S is the skewness of return distribution, and b is a
parameter that reflects the investor’s preference to the skewness of return distribution. This
parameter is defined in terms of first three derivatives of the investor’s utility function
U '''
b  U' 2 .
U '' 


 U' 
Observe that, in contrast to the Sharpe ratio, the value of the ASSR is not unique and depends
on the investor’s preference for skewness as specified by a utility function. In using the ASSR
for practical purposes one cannot avoid the ambiguity in ranking different risky assets.
Koekebakker and Zakamouline (2007b) illustrated that the ranking of alternative investments
depends quite substantially on the value of the skewness preference parameter b, especially
when either the return distributions are highly skewed or the value of parameter b is rather
large with respect to 1.
Performance Measures Based on Non-Expected Utility Theory
Not very long ago after expected utility theory was formulated by von Neumann and
Morgenstern (1944), questions were raised about its value as a descriptive model of the choice
under uncertainty. Allais (1953) and Ellsberg (1961) were among the first to challenge
expected utility theory by showing that some of the assumptions behind this theory cannot be
justified by empirical studies. From the other side, the mean-variance analysis of Markowitz
was criticized because many investors do not associate the risk with the standard deviation of
returns, rather with the possibility of loss. Actually, even Markowitz recognized that the
standard deviation is not a suitable measure of risk. Markowitz (1959) also proposed to use
the semi-variance as an alternative measure of risk. Semi-variance is like variance, except that
it considers only returns below some target level. Technically, aggregating semi-variances
from assets to portfolios is extremely difficult. That is probably why this idea was not pursued
further. Later on the notion of a downside semi-variance was generalized by Fishburn (1977)
and Bawa (1978) who introduce the notion of a lower partial moment as a risk measure. The
definition of a lower partial moment of order n at some level tg is
tg
LPM n ( x, tg ) 
 (tg  x)
n
dF ( x),

where x is some random variable and F(x) is the cumulative probability distribution of x. In a
similar manner one can define an upper partial moment UPMn(x,tg). Fishburn (1977) and
Bawa (1978) proposed the mean - lower partial moment model for portfolio selection. These
authors show that the usage of the mean - lower partial moment objective corresponds to a
specific utility function of the investor. As defined over the asset’s random return x, the utility
function is given by
x

U ( x)  
n
 x   (tg  x)
x  tg,
x  tg,
where γ is the measure of the investor’s risk aversion, and n is the order of the lower partial
moment. If the risk-free rate of return is used as the target level, that is, tg=r, and n=2, then
the investor’s performance measure becomes
DSR 
 r
LPM 2 ( x, r )
.
This performance measure was introduced by Sortino and Price (1994) and Ziemba (2005).
The utility function of Fishburn (1977) and Bawa (1978) is rather specific and restrictive,
though it leads, as mean-variance utility, to a performance measure with no investor’s
preferences. Below we show that this utility function is a special case of a more general
behavioral utility function.
Influential experimental studies have shown the inability of expected utility theory to explain
many phenomena and reinforced the need to rethink much of the theory. Kahneman and
Tversky (1979) propose an alternative descriptive model of the choice under uncertainty that
they call prospect theory. Prospect theory can predict correctly individual choices even in the
cases in which expected utility theory is violated (for a brief description see, for example,
Camerer (2000)). In prospect theory, the utility function is defined over gains and losses
relative to some reference point, as opposed to wealth in expected utility theory. More
formally, the utility function is defined as
U (W  W0 ) W  W0 ,
U (W )   
U  (W  W0 ) W  W0 ,
where U+(.) is the utility function for gains, U-(.) is the utility function for losses, and W0 is
the reference point. The current level of the investor’s wealth (so-called the “status quo”)
serves usually as the reference point. However, as Kahneman and Tversky point out “gains
and losses can be coded relative to an expectation or aspiration level that differs from the
status quo” (see Kahneman and Tversky (1979) page 286). The behavioral utility function has
a kink at the origin, with the slope of the loss function steeper than the gain function. This is
what is called loss aversion which is an important element of prospect theory. In addition, in
prospect theory the investor transforms the objective probability distribution into a subjective
probability distribution.
Recently, Koekebakker and Zakamouline (2007c) performed the approximation analysis of
the investor’s optimal capital allocation problem and showed that the utility function in
prospect theory is equivalent to the following utility
1
1




E[U (W )]   E[(W  W0 )  ]   Var[(W  W0 )  ]     E[(W  W0 )  ]   Var[(W  W0 )  ]  ,
2
2




where (W  W0 )  and (W  W0 )  are the positive and negative parts, respectively, of the
difference W-W0, λ is the measure of the investor’s aversion to losses defined by

U '
,
U '
γ+ and γ- are the investor’s measures of aversion to uncertainties in gains and losses
respectively
  
U '
U'
,     ,
U ' '
U ' '
and where U '  ,U ' '  are the left-sided derivatives of U(.), and U '  ,U ' '  are the right-sided
derivatives of U(.) at the reference point. If the reference point is assumed to be the investor's
initial wealth scaled up by the risk-free rate, that is, W0=W(1+r), then it is possible to arrive at
the explicit solution of the optimal capital allocation problem, and to the closed-form solution
for the investor’s performance measure. If γ+>0 (which means that the gain function is
concave), then the performance measure of a risky asset becomes
PM 
UPM 1 ( x, r )  LPM 1 ( x, r )
LPM 2 ( x, r )  UPM 2 ( x, r )
,
where the parameter θ is the relation between the investor’s aversion to uncertainty in losses
and the investor’s aversion to uncertainty in gains


.

If γ+=0 (which means that the gain function is a straight line), then the performance measure
of a risky asset becomes
PM 
UPM 1 ( x, r )  LPM 1 ( x, r )
LPM 2 ( x, r )
.
If PM<0, it means that the investor should avoid the risky asset. The utility function of
Fishburn (1977) and Bawa (1978) is a particular case of the general behavioral utility where
γ+=0 and λ=1. In this case the performance measure PM reduces to the performance measure
DSR if the investor does not transform the original probability distribution. Observe that the
performance measure PM reduces to the Sharpe ratio when the investor has a von NeumannMorgenstern utility function for which λ=1 and γ+=γ-. But in general, to compute the
performance measure PM one needs to know the investor’s utility function U(.) (to determine
the values of λ and θ) and how the investor transforms the original probability distribution to
the subjective probability distribution. Besides, the drawback of the performance measure PM
is that it takes into account only the first two lower and upper partial moments of distribution.
That is, it does not take into account higher partial moments of distribution. Consequently,
this performance measure says nothing about the investor’s preferences to, for example, the
skewness of return distribution.
Conclusion
We reviewed the latest results on portfolio performance measures based on either expected
utility theory or non-expected utility theory. The main conclusion here is that in either case, to
compute a portfolio performance measure one needs to know the investor’s utility function.
Preliminary results on the use of the utility-based performance measures have shown that
different utility functions may result in substantially different ranking of risky portfolios,
especially when the return distributions are non-normal. We conclude that unless we know
exactly the investor’s preferences, and unless all investors share the same preferences, a single
performance measure that is suitable for all investors cannot exist.
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