Performance Measurement in the Generalized Framework

If it is optimal for the investor to sell short the risky asset, then the first-order derivative of a with respect to γ + is ∂ a ∂γ + = − E[r − x] − λ − 1LPM 1 r − x, 0 UPM 2 r − x, 0 λγ − LP M 2 r − x, 0 + γ + U P M 2 r − x, 0 2 0. Similarly, if it is optimal for the investor to buy-and-hold the risky asset, then the first- order derivative of a with respect to γ − is ∂ a ∂γ − = − λ E[x − r] − λ − 1LPM 1 x − r, 0 LPM 2 r − x, 0 λγ − LP M 2 r − x, 0 + γ + U P M 2 r − x, 0 2 0. If it is optimal for the investor to sell short the risky asset, then the first-order derivative of a with respect to γ − is ∂ a ∂γ − = − λ E[r − x] − λ − 1LPM 1 r − x, 0 LPM 2 r − x, 0 λγ − LP M 2 r − x, 0 + γ + U P M 2 r − x, 0 2 0. Next consider the dependence of the optimal amount that should be invested in the risky asset or sold short on the loss aversion parameter λ. The computation of the first-order derivative of a, given by 29, with respect to λ gives ∂ a ∂λ = − A × LPM 1 x − r, 0 + γ − B × LPM 2 r − x, 0 λγ − LP M 2 r − x, 0 + γ + U P M 2 r − x, 0 2 , where A = λγ − LP M 2 r − x, 0 + γ + U P M 2 r − x, 0, B = E[x − r] − λ − 1LPM 1 x − r, 0. Since A 0 due to 28 and B 0 due to 27, the sign of ∂ a ∂λ is obviously negative if γ − ≥ 0. However, if γ − 0, then the sign of ∂ a ∂λ might be positive. The latter means that when the investor appreciates uncertainty in losses, then an increase in loss aversion might increase the optimal amount invested in the risky asset. This is obviously counter-intuitive. We believe that the explanation for this paradox lies in the fact that the piecewise-quadratic utility function 18 is not always increasing for all values of W − W . This might result in a number of paradoxes similar to those for the standard quadratic utility function. Note that if γ − 0, then by condition 28 we must have γ + 0. In this case the piecewise-quadratic utility is increasing only in the range 1 γ − W − W 1 γ + . For example, if W − W decreases below 1 γ − , then the investor’s utility begins to increase which is not sensible. Generally, when the piecewise-quadratic utility function is not increasing for all W − W , we can stumble upon some paradoxes. Some of these paradoxes will be due to the violation of the first-order stochastic dominance principle. For the standard quadratic utility function this type of paradoxes were first noted by Borch 1969 and Feldstein 1969.

8. Performance Measurement in the Generalized Framework

In this section we discuss performance measurement in our generalised framework. By a performance measure in finance one means a score attached to each risky asset. This score is usually used for ranking of risky asset. That is, the higher the performance measure of an asset, the higher the rank of this asset. The goal of any investor who uses a particular performance measure is to select the asset with the highest score. Note C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd that for us a performance measure is not just some arbitrary reward-to-risk ratio. In our definition a performance measure is related to the level of expected utility provided by the asset. That is, the higher the performance measure of an asset, the higher level of expected utility the asset provides. We believe that this is the most natural definition of a performance measure within any utility theory. First of all, note the following property of a performance measure: Proposition 1. A positive linear transformation of a performance measure produces an equivalent performance measure. Proof. Suppose that PM A is the performance measure of asset A and PM B is the performance measure of asset B. Define performance measures PM ′ A = c PM A + d and PM ′ B = c PM B + d for any real c and d such that c 0. Then PM ′ A and PM ′ B are equivalent to PM A and PM B in the sense that they produce equal ranking of the assets. That is, if PM A PM B , then also PM ′ A PM ′ B . Theorem 11. If conditions 28 and 32 in Theorem 8 are satisfied, then the performance measure of an asset might be given by PM = max[PM B H , PM S S , 0], 39 where PM B H = E [x − r] − λ − 1LPM 1 x − r, 0 λγ − LP M 2 x − r, 0 + γ + U P M 2 x − r, 0 is the performance measure associated with the buy-and-hold strategy, and PM S S = E [r − x] − λ − 1LPM 1 r − x, 0 λγ − LP M 2 r − x, 0 + γ + U P M 2 r − x, 0 is the performance measure associated with the short selling strategy. Proof. According to Definition 7, a performance measure of an asset is related to the level of the investor’s expected utility associated with the investment in this asset. According to Theorem 8, if it is optimal to buy-and-hold the risky asset, then the investor’s maximum expected utility is E B H [U ∗ W ] = 1 2 PM 2 B H such that PM BH 0 observe that numerator in PM BH is positive due to 27. In this case PM BH PM SS because: if there exists a local maximum for the short selling strategy as well, then E B H [U ∗ W ] E S S [U ∗ W ] = 1 2 PM 2 S S since we assume that the buy-and-hold strategy gives higher expected utility and PM SS 0; if there is no local maximum for the short selling strategy, then PM SS ≤ 0. Similarly, if it is optimal to sell short the risky asset, then the investor’s maximum expected utility is E S S [U ∗ W ] = 1 2 PM 2 S S such that PM SS 0. In this case PM SS PM BH because: if there exists a local maximum for the buy-and-hold strategy, then E S S [U ∗ W ] E B H [U ∗ W ] = 1 2 PM 2 B H and PM BH 0; if there exists no local maxi- mum for the buy-and-hold strategy, then PM BH ≤ 0. Finally, if it is optimal to avoid the risky asset, then the investor’s maximum expected utility is zero. In this case PM BH ≤ 0 and PM SS ≤ 0. Corollary 12. In the EUT framework, the performance measure 39 reduces to a performance measure that is equivalent to the Sharpe ratio 12. C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd Proof. For a von Neumann-Morgenstern utility function λ = 1 and γ − = γ + = γ . The conditions 28 and 32 imply that γ 0. This gives P M = 1 √ γ E[x − r] E[x − r 2 ] . Since a positive linear transformation of a performance measure produces an equivalent performance measure and γ 0, the performance measure PM is equivalent to the Sharpe ratio 12. Observe that the computation of the Sharpe ratio does not require the knowledge of the investor’s coefficient of absolute risk aversion γ . By contrast, to compute the performance measure PM one generally needs to define the values of λ, γ − , and γ + and, possibly, the rule of distortion of the objective probability distribution. This means that the performance measure PM is not unique for all investors, but rather an individual performance measure. That is, investors with different preferences different degrees of loss aversion and risk aversions in the domains of losses and gains might rank differently the same set of risky assets. Corollary 13. To compute a performance measure that is equivalent to PM given by 39, one needs to define maximum two parameters that describe the investor’s preferences. Proof. We consider only the case when it is optimal to buy-and-hold the risky asset. In this case the performance measure is given by PM B H = E [x − r] − λ − 1LPM 1 x − r, 0 λγ − LP M 2 x − r, 0 + γ + U P M 2 x − r, 0 . Observe now that if γ + 0, then an equivalent performance measure is given by PM ′ B H = E [x − r] − λ − 1LPM 1 x − r, 0 λθ LPM 2 x − r, 0 + UPM 2 x − r, 0 , 40 where θ = γ − γ + . Note that to compute PM ′ BH we need to define only two parameters that describe the investor’s preferences: λ and θ , where the latter is the relation between the investor’s risk aversions in the domain of losses and the domain of gains. Similarly, if γ − 0, then an equivalent performance measure is given by PM ′ B H = E [x − r] − λ − 1LPM 1 x − r, 0 λLPM 2 x − r, 0 + θUPM 2 x − r, 0 , 41 where θ = γ + γ − . If γ − = 0 in this case γ + 0 due to condition 28 then an equivalent performance measure is given by PM ′ B H = E [x − r] − λ − 1LPM 1 x − r, 0 U P M 2 x − r, 0 . If γ + = 0 in this case γ − 0 due to condition 28, then an equivalent performance measure is given by PM ′ B H = E [x − r] − λ − 1LPM 1 x − r, 0 LP M 2 x − r, 0 . 42 In the last two cases we need to define only one parameter, λ. C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd Table 4 Return on the risk-free asset and probability distributions of returns of four risky assets: A, B, C, and D Each risky asset provides an uncertain return which depends on the future state of the world. The probabilities of the states are equal. State 1 2 3 4 Probability 0.25 0.25 0.25 0.25 Return asset A −10 20 30 Return asset B −15 4 20 30 Return asset C −6 −3 13 35 Return asset D −10 4 10 35 Risk-free return 4 4 4 4 Remark 6. Observe that all performance measures given by equations 39, 40, 41, and 42 can still be interpreted as reward-to-risk ratios. Note that when the investor exhibits loss aversion, the reward measure should be adjusted for the investor’s degree of loss aversion and the expected loss. The risk measure depends heavily on the investor’s risk preferences. It is widely known that the Sharpe ratio is a meaningful measure of portfolio performance when the risk can be adequately measured by variance, for example, when returns are normally distributed. The literature on performance evaluation that tries to take into account non-normality of return distributions is a vast one. There have been proposed dozens of alternative performance measures. Unfortunately, most of these alternative performance measures lack a solid theoretical underpinning. However, the following performance measure can be justified by our analysis: Corollary 14. If λ = 1, γ + = 0, and the investor does not distort the objective probability distribution, then the performance measure PM given by 39 is equivalent to the following Sortino ratio So R = E[x − r] LPM 2 x, r . Proof. Start with the equivalent performance measure 42, remove the probability distortion and set λ = 1. We end this section with an example that demonstrates the impact of the investor’s skewness preferences and loss aversion on performance measurement. In particular, in this example we will compute performance measures of some risky assets. The data for the example is provided in Table 4 which presents the probability distribution of four risky assets. Observe that the probabilities of the states of the world are alike so that the presented results do not depend on the probability distortion where a decision weight is computed using only the objective probability of the state. Table 5 presents the descriptive parameters of the return distributions of the risky assets. In the example we use the piecewise-quadratic utility function 18 which can result in different shapes and preferences depending on the set of parameters λ, γ − , γ + . Recall that the solution for the optimal capital allocation problem generally does not exists for the bilinear utility Behavioural IV , and, therefore, it is omitted. C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd Table 5 Descriptive parameters of the return distributions of four risky assets presented in Table 4 The table reports the expected returns, the standard deviations of returns, the first and the second lower and upper partial moments of the distributions of returns, and the skewnesses of returns. Observe that asset A has greater expected return and lower variance than assets B, C, and D. Note that the probability distribution of asset A is symmetrical, whereas the probability distribution of asset B exhibits negative skewness and the probability distributions of assets C and D exhibit positive skewnesses. Parameter Asset A Asset B Asset C Asset D Expected return, E[x] 10.00 9.75 9.75 9.75 Std. deviation, √ LPM 2 + U PM 2 16.91 17.98 17.26 17.27 UPM 1 10.50 10.50 10.00 9.25 LPM 1 4.50 4.75 4.25 3.5 √ U PM 2 15.26 15.26 16.14 15.79 √ LPM 2 7.28 9.50 6.10 7.00 Skewness −0.3019 0.5894 0.4750 First, we consider the investor’s choice between assets A, B, and C. Observe that asset A has greater expected return and lower variance than assets B and C. Standard intuition says that the investor should prefer asset A to assets B and C. However, here we demonstrate that this intuition may lead us astray if the investor is equipped with the generalised behavioural utility function. Note that the probability distribution of asset A is symmetrical, whereas the probability distribution of asset B exhibits negative skewness and the probability distribution of asset C exhibit positive skewness. We compute the performance measures of five different investors, each having a distinctly shaped utility function. In the computation of the performance measures we make sure that in the optimal allocation the investor’s marginal utility is positive in all states. This is essential, because otherwise it is very easy to arrive at spurious results. The results of the computations are presented in Table 6. Observe that the investor with Quadratic utility considers asset A to be more attractive than assets B and C, whereas the investor with convex loss function Behavioural II utility considers asset B to be more attractive than assets A and C. For the rest of investors asset C is more attractive than assets A and B. Next we demonstrate that the ranking of assets generally depends on the level of loss aversion. In particular, we consider the investor’s choice between assets A and D. As compared to asset D, asset A has greater expected return and lower variance. The main difference between these two assets is that asset D has notably lower expected loss than asset A. First, we assume that the investor has the same values of γ − = γ + 0, but he may have different values of loss aversion. In this case the piecewise-quadratic utility can be interpreted as Quadratic utility with added loss aversion. The computation of the performance measures as given by 41 for assets A and D yields PM A = 0.3548 PM D = 0.3329 when λ = 1, PM A = 0.0815 PM D = 0.1207 when λ = 2. Second, we assume that the investor has γ − 0, γ + = 0, and he may have different values of λ. In this case the piecewise-quadratic utility can be interpreted as the C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd Table 6 The performance measures of three risky assets for investors with different shapes of the piecewise-quadratic utility function For Quadratic utility we compute the performance measure using 12, for Behavioural I utility using 42, for Behavioural II using 40, and for Behavioural III and Behavioural V utilities using 41. For each shape of the utility function the table reports the corresponding values of λ, γ − , and γ + . Note that for Behavioural I, Behavioural III, and Behavioural V utilities the absolute values of γ − and γ + are distinctly different. The highest performance measure for every investor is underlined. That is, the underlined performance measure marks out the asset which is considered to be the most attractive for a distinct investor. Performance Measure Utility λ γ − γ + Asset A Asset B Asset C Quadratic 1 0.04 0.04 0.3548 0.3198 0.3332 Behavioural I 1 0.04 0.00 0.8242 0.6053 0.9421 Behavioural II 2 −0.04 0.04 0.1331 0.1380 0.1100 Behavioural III 2 0.10 0.02 0.1214 0.0664 0.1333 Behavioural V 2 0.10 −0.02 0.1946 0.0864 0.3169 Behavioural I utility with added loss aversion. The computation of the performance measures as given by 42 for assets A and D yields PM A = 0.8242 PM D = 0.8214 when λ = 1, PM A = 0.2060 PM D = 0.3214 when λ = 2. Observe that when there is no loss aversion, the investor prefers asset A to asset D. However, the investor with loss aversion prefers asset D to asset A.

9. Summary