Vulnerability, Uncertainty, and Risk ©ASCE 2014
Asset Allocation with Disappointment Aversion
Downloaded from ascelibrary.org by Columbia University on 06/04/15. Copyright ASCE. For personal use only; all rights reserved.
Yuxin Xie1, Athanasios A Pantelous2 1
Institute for Financial and Actuarial Mathematics (IFAM), Department of Mathematical Sciences, University of Liverpool, United Kingdom.
2
Institute for Financial and Actuarial Mathematics (IFAM), Department of Mathematical Sciences, and Institute for Risk and Uncertainty, University of Liverpool, UK. Email:
[email protected],
[email protected]. ABSTRACT
This paper investigates disappointment aversion (DA) in financial markets within a typical asset allocation framework. Drawing upon the seminal study of Ang et al. (2005), we incorporate disappointment aversion (that is, aversion to outcomes that are worse than prior expectations) within a simple theoretical portfolio-choice model. Based on the results of this model, we then empirically address the portfolio allocation problem of an investor who chooses between a risky and a risk-free asset using international data from 19 countries, see Xie et al. (2013). Our findings strongly support the view that disappointment aversion leads investors to reduce their exposure to the stock market (i.e., disappointment aversion significantly depresses the weights to equities in all cases considered). Overall, it appears that disappointment aversion plays an important role in explaining the equity premium puzzle around the world.
1. Introduction In a sense of classical economics, it is a truism that the utility of money exhibits a diminishing character. A concave-convex pattern in the gain-loss domain is widespread (Rabin, 2000a, 2000b). However, back to the pioneering work of Allais (1953) and Edwards (1955, 1963), solid evidence were reported that individuals systematically violate the basic assumptions of expected utility theory (EU). People are more encouraged to go for a certain prospect than to risk a higher reward. For example, one might be very likely to turn down a 50-50 gamble between $11 and -$10. This tendency is known as the Allais Paradox, which is particularly useful in understanding the equity premium puzzle and many other financial anomalies like disposition effect (Odean, 1998) etc. Since the late of 20th century, behavioral finance has emerged to provide a supplement to the stand paradigm on decision-making theories. One of a most popular
Vulnerability, Uncertainty, and Risk
1180
Downloaded from ascelibrary.org by Columbia University on 06/04/15. Copyright ASCE. For personal use only; all rights reserved.
Vulnerability, Uncertainty, and Risk ©ASCE 2014
approach is the prospect theory introduced by Tversky and Kahneman (1979, 1992). Fundamental to the prospect theory is the concept of loss aversion (LA), which suggests that decisions are formed by the potential losses and gains rather than the final wealth. Afterwards, agents evaluate these outcomes based on their own reference points. An individual is a loss averse if one unit of loss issues more negative utility than one unit of gains can please him. Namely, the utility function is concave over gains, convex over losses, and more sensitive in the loss-region. To summarize, the prospect theory can be shown as the following form:
xa , x‡0 v( x) = , b -g (- x) , x < 0 where γ is the coefficient controls the degrees of loss aversion, α and β are the curvature parameter, the median value of α and β are reported by Kahneman and Tversky at both 0.88 while the median γ is 2.25. Gul (1991) proposed disappointment aversion (DA) based on an idea that agents would form an endogenous expected certainty equivalent by comparing upcoming prospects. Outcomes that outperform the equivalent will be marked as elations; on the other hand, outcomes will disappoint agents when they are worse than the equivalent point. Formally, the DA preference can be defined as:
A ( xt + h - E t xt + h ) , xt + h ‡ Et xt + h vDA = xt + h + ( xt + h - E t xt + h ) , xt + h < Et xt + h The above function is a compact way to show the disappointment aversion. Specifically, with only one parameter extension of the conventional expected utility, E refers to the expected value of certainty equivalent, A (0 mw } = 0. E ( ( (3) ¶W ¶W
where 1 is an indicator function and E refers to the expected value of certainty equivalent. According to Eq. (3) above, the DA utility function only concentrates on the differentiation between terminal wealth levels m w , previous losses and gains will not be taken into account directly. Let a represent the proportion of equity investment. The ending period wealth (denoted by W) is then defined as follows
W = aW0 ( exp( y) - exp(r) ) + W0 exp ( r ) .
(4)
In this framework, the investor chooses between the risky asset y (i.e., equity) and the risk-free asset “r” (i.e., treasury bills). The term a refers to the proportion of wealth invested in the risky asset while a * is the optimal weight. If m w is known, the
a * can be calculated by solving Eq. (3). The tricky part is that m w is also a function of a , which means that a system of simultaneous equations has to be solved (Eq. (2) and (3)). In this study, we develop an algorithm of numerical quadrature which converts Eq. (2) and (3) into the following form (see Appendix for details).
∑m
mW1-g = s: W £ s
psWs1-g + A
W
∑m
s: Ws >
psWs1-g
W
Pr (W £ mw ) + A Pr (W > mw )
∑m
s: Ws £
psWs-g ( exp ( ys ) - exp ( r ) ) + A w
∑m
s: Ws
0.677 in all cases), leading to relatively low exposure to the equity market (e.g. the optimal weight turns into negative for the case of Denmark at intermediate levels of DA). Finally, the results in Panel C confirm the negative relationship between the level of DA and equity exposure for the case of non-European countries and also for
Vulnerability, Uncertainty, and Risk
1184
Vulnerability, Uncertainty, and Risk ©ASCE 2014
1185
Downloaded from ascelibrary.org by Columbia University on 06/04/15. Copyright ASCE. For personal use only; all rights reserved.
portfolios constructed on the basis of global /European equity indexes. It is also worthy to note that investors in Australia exhibit the lowest level of A*=0.566 from all cases considered, which drives the high exposure to the equity market (within our hypothetical DA levels, its optimal weights never become negative). Also, Japanese investors maintain very conservative equity holdings for most of DA levels but they start to purchase stocks just after A reaches 0.656, showing a greater tolerance than investors from Canada (0.665) and New Zealand (0.685). Table 1. Optimal Portfolio Weights under Different DA levels Panel A: Optimal Weights for countries from the Euro-zone A
Belgium
Finland
France
Germany
Ireland
Italy
Netherlands
Spain
0.6
-0.351
-0.118
-0.077
-0.103
-0.341
-0.120
-0.183
-0.343
0.65
-0.222
-0.016
0.056
-0.008
-0.208
-0.014
-0.041
-0.205
0.7
-0.101
0.079
0.180
0.082
-0.084
0.086
0.092
-0.076
0.75
0.014
0.168
0.296
0.166
0.033
0.180
0.218
0.047
0.8
0.122
0.253
0.406
0.246
0.144
0.268
0.336
0.162
0.85
0.225
0.333
0.509
0.322
0.249
0.352
0.447
0.271
0.9
0.322
0.408
0.606
0.393
0.348
0.431
0.552
0.374
0.95
0.414
0.480
0.698
0.461
0.442
0.506
0.651
0.472
1
0.501
0.548
0.785
0.525
0.531
0.577
0.745
0.565
A*
0.744
0.658
0.629
0.654
0.736
0.657
0.665
0.731
ERP
0.028
0.055
0.057
0.057
0.030
0.055
0.047
0.031
S.D.
0.236
0.304
0.235
0.322
0.231
0.290
0.218
0.222
Panel B: Optimal Weights for European countries outside the Euro-zone A
Denmark
Norway
Sweden
Switzerland
UK
0.6
-0.416
-0.291
-0.232
-0.350
-0.236
0.65
-0.269
-0.180
-0.097
-0.193
-0.080
0.7
-0.132
-0.076
0.029
-0.046
0.066
0.75
-0.002
0.023
0.148
0.092
0.203
0.8
0.120
0.117
0.260
0.222
0.333
Vulnerability, Uncertainty, and Risk
Downloaded from ascelibrary.org by Columbia University on 06/04/15. Copyright ASCE. For personal use only; all rights reserved.
Vulnerability, Uncertainty, and Risk ©ASCE 2014
1186
0.85
0.236
0.205
0.366
0.345
0.454
0.9
0.345
0.289
0.466
0.461
0.569
0.95
0.449
0.369
0.561
0.571
0.678
1
0.548
0.444
0.651
0.676
0.781
A*
0.751
0.738
0.688
0.716
0.677
ERP
0.027
0.029
0.042
0.033
0.042
S.D.
0.209
0.273
0.229
0.197
0.199
Panel C: Optimal Weights for non-European countries
A
Australia
Canada
Japan
New Zealand
South Africa
USA
World
Europe
0.6
0.125
-0.233
-0.113
-0.261
-0.037
-0.114
-0.190
-0.300
0.65
0.297
-0.052
-0.010
-0.104
0.102
0.040
-0.014
-0.156
0.7
0.458
0.118
0.087
0.044
0.232
0.185
0.151
-0.022
0.75
0.608
0.277
0.179
0.183
0.354
0.320
0.305
0.105
0.8
0.748
0.426
0.265
0.313
0.468
0.447
0.450
0.224
0.85
0.879
0.567
0.346
0.436
0.575
0.567
0.587
0.337
0.9
1.003
0.699
0.423
0.552
0.677
0.680
0.716
0.443
0.95
1.120
0.825
0.496
0.662
0.772
0.787
0.837
0.544
1
1.229
0.943
0.566
0.766
0.863
0.888
0.952
0.640
A*
0.566
0.665
0.656
0.685
0.613
0.637
0.654
0.708
ERP
0.065
0.041
0.056
0.040
0.062
0.052
0.044
0.036
S.D.
0.182
0.172
0.298
0.197
0.225
0.202
0.177
0.215
Taken together, the following inferences can be drawn so far. First, the highest stock holdings are always associated with the case of no DA (A = 1), with the equity market being incredibly attractive in countries such as Australia, Canada, United States and South Africa. Second, the optimal weights are significantly depressed when the level of DA increases. Third, when the DA reaches very high levels, equity weights may even drop below zero, which means that an optimal investment strategy involves shorting (rather than holding) equities.
Vulnerability, Uncertainty, and Risk
Vulnerability, Uncertainty, and Risk ©ASCE 2014
Downloaded from ascelibrary.org by Columbia University on 06/04/15. Copyright ASCE. For personal use only; all rights reserved.
4. Conclusion and Further Directions Drawing upon the seminal study of Ang et al. (2005), we incorporate disappointment aversion (that is, aversion to outcomes that are worse than prior expectations) within a simple theoretical portfolio-choice model. Based on the results of this model, we then empirically address the portfolio allocation problem of an investor who chooses between a risky and a risk-free asset using international data from 19 countries. Our findings strongly support the view that disappointment aversion leads investors to reduce their exposure to the stock market (i.e., disappointment aversion significantly depresses the portfolio weights on equities in all cases considered). Overall, it appears that disappointment aversion plays an important role in explaining the equity premium puzzle around the world. In spite of its desirable perspective, procedures of formally measuring the parameters within DA models are barely available. The most important reason for this neglect might be that DA is derived by an axiomatic approach, rather than experimental investigations. In recent literatures, discussions and conclusions are often made on pre-determined parameters without any theoretical guidance (Ang et al, 2005; Fielding and Stracca, 2006; Routledge and Zin, 2010; Saltari and Travaglini, 2010). How these parameters can be correctly chosen to the corresponding markets remains blank. Therefore, a quantitative measurement of DA is not only crucial for the descriptive rigorous, but also for further applications under a dynamic sense. Without specifying the initial values of preference setting, any approach designated to multiple-period studies will have nowhere to begin with. For above concerns, in our further works we aim to provide a normative method and its derivatives to discuss the quantitative properties of the DA parameters. Within a typical asset allocation framework that maximizes agents’ utility, we will examine (1) theoretical restrictions on the risk aversion parameters required by its optimal process (2) the possible intervals of DA corresponding to reasonable combinations of risk aversions. Regarding to point one, the results should confirm a milder concave-convex appearance across elations and disappointments. Moreover, it is expected that the diminishing effect is more pronounced for elations than disappointments. On the other hand, the magnitudes of disappointments might be strongly diverse from country to country. Namely, a high degree of DA probably exists for those markets which associate the better performance. Reference Abel, A. B. (1990). ‘Asset prices under habit formation and catching up with the joneses. A.E.R.’ Papers and Proceedings, 80: 38–42. Allais, M. (1953). Le Comportement de l’Homme Rationnel devant le Risque, Critique des Postulats et Axiomes de l’Ecole Americaine, Econometrica 21: 503–546. Ang, A., Bekaert, G. and Liu, J. (2005). ‘Why stock may disappoint?’, Journal of Financial Economics, 76: 471–508.
Vulnerability, Uncertainty, and Risk
1187
Vulnerability, Uncertainty, and Risk ©ASCE 2014
Barro, R. 2006. Rare disasters and asset markets in the twentieth century. Quarterly Journal of Economics, 121: 823–866. Benartzi, S., and Thaler, H. (1995). ‘Myopic Loss Aversion and the Equity Premium Puzzle’, The Quarterly Journal of Economics, 110(1): 73–92.
Downloaded from ascelibrary.org by Columbia University on 06/04/15. Copyright ASCE. For personal use only; all rights reserved.
Campbell, J.Y. (2001). ‘Asset pricing at the millennium.’ Journal of Finance, 55: 1515–1567. Cochrane, J.H., and J. Campbell. (1999). ‘By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior.’ Scholarly Articles 3119444, Harvard University Department of Economics. Cohen, A., and Einav, L. (2007). ‘Estimating Risk Preferences from Deductible Choice.’ American Economic Review, 97: 757–788. Constantinides, G., J.B. Donaldson, R. Mehra. (2002). ‘Junior can’t borrow: a new perspective on the equity premium puzzle.’ Quarterly Journal of Economics, 17(1): 269–296. Dimson, E., P. Marsh, and M. Staunton. (2006a). Global Investment Returns Yearbook 2006. ABN AMRO/London Business School. Dimson, E., P. Marsh, and M. Staunton. (2006b). DMS Global Returns data module. Ibbotson Associates, Chicago (a subsidiary of Morningst Inc.). Dimson, E., P. Marsh, and M. Staunton. (2006c).The Worldwide Equity Premium: A Smaller Puzzle. EFA 2006 Zurich Meetings Paper; AFA 2008 New Orleans Meetings Paper. Edwards, W. (1955). ‘The prediction of decisions among bets’. Journal of Experimental Psychology, 50: 201–214. Edwards, W. (1962). Subjective probabilities inferred from decisions, Psychological Review, 69: 109–135. Epstein, L. G., and S.E. Zin. (1991). ‘Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis.’ Journal of Political Economy, 99: 263–286. Fama, E.F., and K.R. French. (2002). ‘The Equity Premium.’ The Journal of Finance, LVII, 2: 637–659. Fennema, H. and van Assen, M.A.L.M. (1998). ‘Measuring the utility of losses by means of the trade-off method.’ Journal of Risk and Uncertainty 17: 277–295. Fielding, D. and Stracca, L. (2006). ‘Myopic Loss Aversion, Disappointment Aversion, and the Equity Premium Puzzle’, Journal of Economic Behavior & Organization, 64(2): 250–268. Gul, F. (1991). ‘A theory of Disappointment Aversion.’ Econometrica, 59: 667 – 686. Mehra, R. (2008). The Handbook of the Equity Risk Premium. Edited by Rajnish Mehra, Elsevier. Amsterdam.
Vulnerability, Uncertainty, and Risk
1188
Vulnerability, Uncertainty, and Risk ©ASCE 2014
Mehra, R. (2007). The equity premium in India: K. Basu (ed.): Oxford Companion to Economics in India. Oxford University Press.
Downloaded from ascelibrary.org by Columbia University on 06/04/15. Copyright ASCE. For personal use only; all rights reserved.
Odean, T. (1999). ‘Do Investors Trade Too Much?’ American Economic Review, 89(5): 1775–1798. Rabin, M. (2000a). ‘Diminishing marginal utility of wealth cannot explain risk aversion.’ In D. Kahneman & A. Tversky (Eds.). Choices, values, and frames (202– 08), New York: Cambridge University Press. Rabin, M. (2000b). ‘Risk aversion and expected-utility theory: A calibration theorem.’ Econometrica, 68: 1281–1292. Routledge, B.R., Zin, S.E. (2010). ‘Generalized Disappointment Aversion and Asset Prices.’ The Journal of Finance, LXV. 4: 1303–1332. Saltari, E., and Giuseppe. T. (2010). ‘Behavioral Portfolio Choice and Disappointment Aversion: An Analytical Solution with "Small" Risks.’ Nonlinear Dynamics in Economics: 295-311 Tversky, A., and Kahneman, D. (1979). ‘Prospect Theory: An Analysis of Decision under Risk.’ Econometrica, 47(2): 263–292. Tversky, A., and Kahneman, D. (1992). ‘Advances in Prospect Theory: Cumulative Representation of Uncertainty.’ Journal of Risk and Uncertainty, 5(4): 297–323. Wakker, P.P., Köbberling., and Schwieren, C. (2007). ‘Prospect-theory’s diminishing sensitivity versus economics’ intrinsic utility of money: how the introduction of the euro can be used to disentangle the two empirically.’ Theory and Decision, 63: 205–231. Weil, P. (1993). ‘The Equity Premium Puzzle and the Risk-free Rate Puzzle.’ Journal of Monetary Economics, 24: 401–421. Xie, Y., Pantelous, A.A., and Florackis, C. (2013), Disappointment Aversion and the Equity Premium Puzzle: New International Evidence, The European Journal of Finance (submitted).
Vulnerability, Uncertainty, and Risk
1189
