Summary A Generalisation of the Mean Variance An

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

In this paper we considered a decision maker who generally distorts the objective probabilities and whose utility function has a kink at the reference point with different functions for losses and gains. The approximation analysis presented in the paper generalised mean-variance utility of Tobin and Markowitz and mean-semivariance utility of Markowitz, the Arrow-Pratt measure of risk, the Arrow’s solution to the optimal capital allocation problem, and the Sharpe and Sortino portfolio performance measures. All in all, this paper presented a simple uniform framework that provides general insights into a broad class of the models of choice under uncertainty. Our analysis showed that a decision maker in this generalised framework distinguishes between three sources of risk: expected loss, uncertainty in losses, and uncertainty in gains. Consequently, a decision maker exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. We demonstrated that an investor in our setting will want to allocate some wealth to the risky asset only when the perceived risk premium is sufficiently high. Otherwise, if the risk premium is small, the investor avoids the risky asset and invests only in the risk-free asset. As compared with the Sharpe and Sortino ratios where the investor’s risk preferences apparently disappear, we showed that the performance measure of an investor in our C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd framework is generally not unique, but rather an individual performance measure. We demonstrated that in our generalised framework variance is generally not a proper risk measure even if the risk is small. Our analysis showed that it is the decision maker’s skewness preferences that have first-order impact on risk measurement. Unfortunately due to the length limitation, the role of probability distortion remains largely hidden in our analysis. In the paper we managed to consider only the role of the optimistic and pessimistic probability distortions in the investor’s optimal capital allocation problem. It is rather straightforward to show that the pessimistic probability distortion systematically increases the risk premium, whereas the optimistic proba- bility distortion systematically decreases the risk premium. However, there might be many other types of probability distortion. To get more insights into the impact of probability distortion on the financial decision making, the interested reader is advised to consult the papers by Levy and Levy 2002, Davies and Satchell 2007, Cecchetti et al. 2000 and Abel 2002. Appendix A: Computation of Moments of Distribution with and without Distortion of Probabilities Preamble We suppose that a gamble X can be represented by an objective probability distribution p 1 , . . . , p n over a fixed set of outcomes x 1 , . . . , x n where p i is the probability of x i . That is, any gamble can be represented by a vector of outcomes and probabilities X = x 1 , p 1 ; . . . ; x n , p n . We suppose that the outcomes of the gamble are ordered such that x 1 is worst and x n is best. Moreover, we suppose that x 1 · · · x k 0 x k+1 · · · x n . The cumulative distribution function of X is given by F X x m = m i =1 p i , 1 ≤ m ≤ n. We will illustrate the computation of moments of distribution using a fair in the objec- tive world gamble X = −1, 2 3 ; 2, 1 3 and a complementary gamble −X = −2, 1 3 ; 1, 2 3 . Without distortion of probabilities the first central moment of X is computed as E[X ] = n i =1 x i p i , 43 and, for example, the lower and upper partial moments of order 1 around 0 are computed as LMP 1 X , 0 = − k i =1 x i p i , UMP 1 X , 0 = n i =k+1 x i p i . 44 Using these rules, we arrive at the following values for the first central and partial moments of X and −X E[X ] = −E[−X] = 0, LMP 1 X , 0 = UMP 1 X , 0 = LMP 1 −X, 0 = UMP 1 −X, 0 = 2 3 . C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd With distortion of probabilities, the objective probability p i of the outcome x i is replaced with a distorted probability q p i where q. is computed using some weighting function w . such that w 0 = 0 and w1 = 1. Otherwise, the rules of computation of central and partial moments are the same as those given by 43 and 44. The cumulative distorted distribution function of X is given by Q X x m = m i =1 q p i , 1 ≤ m ≤ n. Probability distortion in prospect theory In PT the objective probability p of an outcome is replaced with a distorted probability q p = w p. Tversky and Kahneman 1992 proposed the following functional form for the weighting function 16 w p = p δ p δ + 1 − p δ 1 δ , 45 where 0 δ 1. Using δ = 0.65, a decision maker in PT perceives the gambles as X = −1, 0.6618;2, 0.4218 and −X = −2, 0.4218;1, 0.6618. The values of the first central and partial moments of X and −X E [X ] = −E[−X] = 0.1817, LMP 1 X , 0 = UMP 1 −X, 0 = 0.6618 LMP 1 −X, 0 = UMP 1 X , 0 = 0.8436. Observe that for both gambles X and −X n i =1 q p i = 1.0836 = 1, that is, the decision weights do not sum to 1. Probability distortion in anticipated utility theory In AUTRDEU the objective probability p of an outcome is replaced with a distorted probability qp such that q p 1 = w p 1 , q p i = w i j =1 p j − w i −1 j =1 p j ∀ i 1. Note that n i =1 q p i = w1 = 1, so that in AUTRDEU the decision weights sum to 1. 16 Ingersoll 2008 shows that this probability weighting function is not increasing for all parameter values and, therefore, can assign negative decision weights to some outcomes. Consequently, this weighting function should be used with caution. C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd A pessimistic and optimistic probability distortion can be modelled using the weight- ing function w p = p δ where 0 δ 1 for a pessimistic distortion and δ 1 for an optimistic distortion. A pessimistic decision maker with δ = 0.5 perceives the gambles as X = −1, 2 3 ; 2, 1 − 2 3 and −X = −2, 1 √ 2 ; 1, 1 − 1 √ 2 . Computation of the first central and partial moments of X and −X for this pessimistic decision maker gives E [X ] = − √ 6 − 2 0, E [−X] = − 3 √ 2 − 1 0, LP M 1 X , 0 = 2 3 LP M 1 −X, 0 = √ 2, U P M 1 X , 0 = 2 1 − 2 3 U P M 1 −X, 0 = 1 − 1 √ 2 . An optimistic decision maker with δ = 2 perceives the gambles as X = −1, 4 9 ; 2, 5 9 and −X = −2, 1 9 ; 1, 8 9 . Computation of the first central and partial moments of X and −X for this optimistic decision maker gives E [X ] = 2 3 0, E [−X] = 2 3 0, LP M 1 X , 0 = 4 9 , LP M 1 −X, 0 = 2 9 , U P M 1 X , 0 = 10 9 , U P M 1 −X, 0 = 8 9 . Probability distortion in cumulative prospect theory In CPT the objective probability p of an outcome is replaced with a distorted probability qp such that q p 1 = w − p 1 , q p i = w − i j =1 p j − w − i −1 j =1 p j ∀ 1 i ≤ k, q p n = w + p n , q p i = w + n j =i p j − w + n j =i+1 p j ∀ k + 1 ≤ i n. Tversky and Kahneman 1992 proposed to use the same functional form 45 for both weighting functions w − · and w + ·, but with different coefficient δ for gains and losses. They estimated that for losses δ − = 0.69 and for gains δ + = 0.61. A decision maker in CPT perceives the gambles as X = −1, 0.6573;2, 0.4375 and −X = −2, 0.4075;1, 0.6677. Observe that for gamble X n i =1 q p i = 1.0949 = 1, whereas for gamble −X n i =1 q p i = 1.0752 = 1, C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd that is, the decision weights do not sum to 1. The values of the first central and partial moments of X and −X E [X ] = 0.2177, E [−X] = −0.1472, LP M 1 X , 0 = 0.6573, LP M 1 −X, 0 = 0.8149, U P M 1 X , 0 = 0.8750, U P M 1 −X, 0 = 0.6677. Appendix B: Solution to the Optimal Capital Allocation Problem with Probability Distortion In this appendix we provide an example which demonstrates how a probability distortion might cause either avoidance of the risky asset or the existence of two local maxima in the optimal capital allocation problem. For simplicity, we assume that the investor is equipped with Quadratic utility see Section 4 and the probability distortion is as in AUTRDEU. Also, for the sake of simplicity of exposition, we assume a simple one- period binomial model for the return on the risky asset x. In particular, in the objective world the return on the risky asset is ether μ − d or μ + d with equal probabilities. Note that μ is the expected return on the risky asset in the objective world. The ordered vector of the outcomes and probabilities of x − r is μ − d − r, 1 2 ; μ + d − r, 1 2 . Observe that in the objective world E[x − r] = −E[r − x] = μ − r. In the subjective world of the investor, the ordered vector of the outcomes and probabilities of x − r is μ − d − r, w0.5; μ + d − r, 1 − w0.5, whereas the ordered vector of the outcomes and probabilities of r − x is r − μ − d, w0.5; r − μ + d, 1 − w0.5, where w· is some weighting function. This means that under the distortion of probabilities E [x − r] = μ − r + d1 − 2w0.5, E [r − x] = r − μ + d1 − 2w0.5. Recall that a pessimistic and optimistic probability distortion can be modelled using the weighting function w p = p δ where 0 δ 1 for a pessimistic distortion and δ 1 for an optimistic distortion. With this type of probability distortion E [x − r] = μ − r + d 1 − 1 2 δ −1 , E [r − x] = r − μ + d 1 − 1 2 δ −1 . Observe that for an optimistic investor E [x − r] μ − r, E [r − x] r − μ. If, for example, μ = r, then for an optimistic investor E [x − r] 0, E [r − x] 0, which means that for an optimistic investor the expected risk premia of both the buy- and-hold and short selling strategies are positive when the expected risk premium is zero in the objective world. In contrast, for a pessimistic investor E [x − r] μ − r, E [r − x] r − μ. C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd If, for example, μ = r, then for a pessimistic investor E [x − r] 0, E [r − x] 0. Note that in order a pessimistic investor finds it optimal to buy-and-hold the risky asset the condition is E[x − r] 0, the expected return on the risky asset in the objective world should exceed μ r − d 1 − 1 2 δ −1 . Suppose that μ = 10, d = 15, and r = 5. This means that the expected risk premium in the objective world is 5 for the buy-and-hold strategy and −5 for the short selling strategy. Consider a pessimistic investor with δ = 0.5. In this case the expected risk premia of the buy-and-hold and short selling strategies E [x − r] = 10 − 5 + 15 × 1 − √ 2 ≈ −1.21, E [r − x] = 5 − 10 + 15 × 1 − √ 2 ≈ −11.21. That is, for this pessimistic investor the expected risk premia of both the strategies are negative. This means that for this pessimistic investor neither the buy-and-hold nor short selling strategy is optimal. This investor will, therefore, invests only in the risk-free asset. This pessimistic investor will buy-and-hold some amount of the risky asset only if μ 5 − 15 × 1 − √ 2 ≈ 11.21, that is, if in the objective world the expected return exceeds 11.21. Consider now an optimistic investor with δ = 2. For this investor the expected risk premia of the buy-and-hold and short selling strategies E [x − r] = 10 − 5 + 15 × 1 − 1 2 = 12.5, E [r − x] = 5 − 10 + 15 × 1 − 1 2 = 2.5. That is, for this optimistic investor the expected risk premia of both the strategies are positive. This means that for this investor there are two local maxima in the optimal capital allocation problem: one local maximum for the short selling strategy, and the other local maximum for the buy-and-hold strategy. Suppose that the absolute risk aversion of the investor is γ = 0.01. Then the optimal amount of the risky asset that should be sold short is given by a SS = E [r − x] γ E [r − x 2 ] ≈ 142.86, and the optimal amount that should be bought-and-held is given by a BH = E [x − r] γ E [x − r 2 ] ≈ 384.62. The optimal strategy is the strategy that gives the highest expected utility. The compu- tations of expected utilities of both these strategies gives E SS [U ∗ W ] = 1 2 E[r − x] 2 γ E [r − x 2 ] ≈ 1.79, C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd E BH [U ∗ W ] = 1 2 E[x − r] 2 γ E [x − r 2 ] ≈ 24.04. Clearly, the investor should pursue the buy-and-hold strategy since it gives the highest expected utility. An interesting case to consider is the optimal capital allocation decision of an optimistic investor when μ = r = 5. In this case using the same data as above a SS = a BH ≈ 333.33 and E SS [U ∗ W ] = E BH [U ∗ W ] = 12.5. In this case the optimistic investor is indifferent between the short selling and buy-and-hold strategies. Both the strategies give the same positive expected utility References Abel, A. B., ‘An exploration of the effects of pessimism and doubt on asset returns’, Journal of Economic Dynamics and Control, Vol. 267–8, 2002, pp. 1075–92. Agnew, J., Balduzzi, P. and Annika, S., ‘Portfolio choice and trading in a large 401k plan’, American Economic Review, Vol. 931, 2003, pp. 193–215. Allais, M., ‘Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’Ecole Americaine’, Econometrica, Vol. 21, 1953, pp. 503–46. Ang, J. S. and Chua, J. H., ‘Composite measures for the evaluation of investment performance’, Journal of Financial and Quantitative Analysis, Vol. 142, 1979, pp. 361–84. Arrow, K. J., Essays in the Theory of Risk-Bearing North-Holland, 1971. Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., ‘Coherent measures of risk’, Mathematical Finance, Vol. 93, 1999, pp. 203–28. Barberis, N., Huang, M. and Santos, T., ‘Prospect theory and asset prices’, Quarterly Journal of Economics, Vol. 1161, 2001, pp. 1–53. Bawa, V. S., ‘Safety-first, stochastic dominance, and optimal portfolio choice’, Journal of Financial and Quantitative Analysis, Vol. 132, 1978, pp. 255–71. Bell, D. E., ‘Disappointment in decision making under uncertainty’, Operations Research, Vol. 331, 1985, pp. 1–27. Berkelaar, A., Kouwenberg, R. and Post, T., ‘Optimal portfolio choice under loss aversion’, Review of Ecnomics and Statistics, Vol. 864, 2004, pp. 973–87. Bernartzi, S. and Thaler, R., ‘Myopic loss aversion and the equity premium puzzle’, Quarterly Journal of Economics, Vol. 1101, 1995, pp. 73–92. Borch, K., ‘A note on uncertainty and indifference curves’, Review of Economic Studies, Vol. 361, 1969, pp. 1–4. Camerer, C. F., ‘Prospect theory in the wild: evidence from the field’, in D. Kahneman and A. Tversky, eds. Choices, Values and Frames Cambridge University Press, New York, 2000, pp. 288–300. Cecchetti, S. G., Pok-sang, L. and Nelson, C. M., ‘The behavioural components of risk aversion’, American Economic Review, Vol. 904, 2000, pp. 787–805. Davies, G. B. and Satchell, S. E., ‘The behavioural components of risk aversion’, Journal of Mathematical Psychology, Vol. 511, 2007, pp. 1–13. Edwards, W., ‘The prediction of decisions among bets’, Journal of Experimental Psychology, Vol. 503, 1955, pp. 201–14. Edwards, W., ‘Subjective probabilities inferred from decisions’, Psychological Review, Vol. 69, 1962, pp. 109–35. Ellsberg, D., ‘Risk, ambiguity, and the savage axioms’, Quarterly Journal of Economics, Vol. 754, 1961, pp. 643–69. Feldstein, M. S., ‘Mean-variance analysis in the theory of liquidity preference and portfolio selection’, Review of Economic Studies, Vol. 361, 1969, pp. 5–12. Fishburn, P. C., ‘Mean-risk analysis with risk associated with below-target returns’, American Economic Review, Vol. 672, 1977, pp. 116–26. C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd Friedman, M. and Savage, L. J., ‘The utility analysis of choices involving risk’, Journal of Political Economy, Vol. 564, 1948, pp. 279–304. Gomes, F. J., ‘Portfolio choice and trading volume with loss-averse investors’, Journal of Business, Vol. 782, 2005, pp. 675–706. Ingersoll, J., ‘Non-monotonicity of the Tversky-Kahneman probability-weighting function: a caution- ary note’, European Financial Management, Vol. 143, 2008, pp. 385–90. Kahneman, D. and Tversky, A., ‘Prospect theory: an analysis of decision under risk’, Econometrica, Vol. 4712, 1979, pp. 263–91. Klemkosky, R. C., ‘The bias in composite performance measures’, Journal of Financial and Quantitative Analysis, Vol. 83, 1973, pp. 505–14. K¨obberling, V. and Wakker, P. P., ‘An index of loss aversion’, Journal of Economic Theory, Vol. 122, 2005, pp. 119–31. Levy, H. and Markowitz, H., ‘Approximating expected utility by a function of mean and variance’, American Economic Review, Vol. 693, 1979, pp. 308–17. Levy, M. and Levy, H., ‘Arrow-Pratt risk aversion, risk premium and decision weights’, Journal of Risk and Uncertainty, Vol. 253, 2002, pp. 265–90. Loomes, G. and Sugden, R., ‘Disappointment and dynamic consistency in choice under uncertainty’, Review of Economic Studies, Vol. 532, 1986, pp. 271–82. Markowitz, H., Portfolio Selection. Efficient Diversification of Investments John Wiley, New York, 1959. Markowitz, H., ‘The utility of wealth’, Journal of Political Economy, Vol. 602, 1952, pp. 151–58. Mehra, R. and Prescott, E. C., ‘The equity premium: a puzzle’, Journal of Monetary Economics, Vol. 15, 1985, pp. 145–61. Nawrocki, D. N., ‘A brief history of downside risk measures’, Journal of Investing, Vol. 3, 1999, pp. 9–25. Pratt, J. W., ‘Risk aversion in the small and in the large’, Econometrica, Vol. 321–2, 1964, pp. 122–36. Quiggin, J., ‘A theory of anticipated utility’, Journal of Economic Behavior Organization, Vol. 34, 1982, pp. 323–43. Samuelson, P. A., ‘The fundamental approximation theorem of portfolio analysis in terms of means, variances, and higher moments’, Review of Economic Studies, Vol. 374, 1970, pp. 537–42. Segal, U. and Spivak, A., ‘First order versus second order risk aversion’, Journal of Economic Theory, Vol. 511, 1990, pp. 111–25. Sharpe, W. F., ‘Mutual fund performance’, Journal of Business, Vol. 311, 1966, pp. 119–38. Sharpe, W. F., ‘Morningstar’s risk-adjusted ratings’, Financial Analysts Journal, Vol. 544, 1998, pp. 21–33. Sortino, F. A. and Price, L. N., ‘Performance measurement in a downside risk framework’, Journal of Investing, Vol. 33, 1994, pp. 59–65. Starmer, C., ‘Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk’, Journal of Economic Literature, Vol. 382, 2000, pp. 332–82. Subrahmanyam, A., ‘Behavioural finance: a review and synthesis’, European Financial Management, Vol. 141, 2007, pp. 12–29. Tobin, J., ‘Comment on borch and feldstein’, Review of Economic Studies, Vol. 361, 1969, pp. 13–14. Tsiang, S. C., ‘The rationale of the mean-standard deviation analysis, skewness preference, and the demand for money’, American Economic Review, Vol. 623, 1972, pp. 354–71. Tversky, A. and Kahneman, D., ‘Advances in prospect theory: cumulative representation of uncer- tainty’, Journal of Risk and Uncertainty, Vol. 54, 1992, pp. 297–323. von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior Princeton University Press, 1944. Yaari, M. E., ‘The dual theory of choice under risk’, Econometrica, Vol. 551, 1987, pp. 95–115. Ziemba, W. T., ‘The symmetric downside-risk sharpe ratio’, Journal of Portfolio Management, Vol. 321, 2005, pp. 108–22. C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd