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

[pic]

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

[pic].

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

[pic],

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

[pic],

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

[pic].

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

[pic] 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

[pic]

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

[pic].

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

[pic]

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

[pic],

where [pic]and [pic] 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

[pic],

γ+ and γ- are the investor’s measures of aversion to uncertainties in gains and losses respectively

[pic], [pic],

and where [pic] are the left-sided derivatives of U(.), and [pic] 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

[pic],

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

[pic].

If γ+=0 (which means that the gain function is a straight line), then the performance measure of a risky asset becomes

[pic].

If PM