How Investors Face Financial Risk: Loss Aversion and Wealth Allocation

Transcription

1 41 JCC by 1 Ph. D. in Economics. Université Catholique de Louvain, Belgium Assistant Professor, Fordham University, USA 2 Ph.D. in Economics. J. W. Goethe University Frankfurt am Main, Germany Bayerngas GmbH Abstract Keywords: Introduction procedure: 3 The second step, the so-called asset allocation decision, expertise, time, or any combination of these factors to to rely on the help of professional portfolio managers in 4 of the paper is on the decision process of nonprofessional illustrates more sophisticated details, such as choosing

2 42 5 Second, to split this money optimally among different of their portfolio to professional managers who dispose 6 7 8, by explicitly accounting for the formation of what we and is formulated in line with the extended prospect of expected returns, as such presenting a more complete The empirical contributions of this paper come from the portfolio performance more often than once a year, which The following section presents the main theoretical illustrate the implementation of our theoretical model and Theoretical Model setting on the portfolio selection model of Campbell et

3 43 Optimal portfolio selection with exogenous VaR 9 VaR ex VaR ex nonprofessional client is disposed to bear, communicated account for how VaR ex They consider VaR ex exogenous to the optimization problem the optimal weights w o tt p B t The latter stands for the optimal sum to be borrowed (B t > B t and yields the following expression: q t (w o tt p distribution of portfolio gross returns R (w t at the VaR t is the portfolio VaR 11 at time 1 can be written as follows: S t+1 =(W t +B t R t+1 where W t mainly concerned with how to split their money between The individual loss level VaR* VaR ex in the 12 The value function v 13 S t Z t x t = R 1 R ft S t Z t cushion, and to z t = Z t / S t Z t S t (Z t > S t z t t > 1). and can appear as follows 13 :

4 44 where and t (R t+1 R ft S t Z t. The derivation of VaR* loss, individual, and maximum [x t +1] = [R t +1] - R ft denotes the expected that corresponds to VaR ex B t B t B t S t /W t VaR* under the traditional approach 16 on past performance k and yields the following 17 : The prospective value of risky investments results from the perception of the utility generated by 18

5 45 V 19 : into two parts, the expressions of which are apparent in StEt[ 1], S t Z t, which we yields the following : 21 The impact of portfolio evaluation frequency narrow framing 22 emerges 23 S t and B t V, and short durations, which we considered more plausible in Theoretically, we could study the direct dependence the current returns, we can discard the direct effect respect to expected returns, where the direct impact The application section will present the analysis of a t was optimal in the sense that it led to the most relaxed 24 Application

6 46 Table 1 3-month T-bill Note. - t f general, we smoothed the outlier by replacing it with the = k = 3 29 last date before the decision time The VaR* and the evolution of the risky investment performance of their portfolios once a year and employ affected perceptions, but we were also interested in how different ways of assessing the cushion contribute to de- tions: myopic and dynamic cushions 31 - Z t = S t-1, so that the myopic cushion expression yields S t - S t-1 ticular Z t Z t-1 R S t S t Z t-1 R return 34 t- t t x t+1 + (1 z t ft +1 - B t of borrowing (B t B t

7 47 S t t t, at different portfolio Table 2 Portfolio return Portfolio return t t 1 year 6 months 4 months 3 months 1 month 1 day Note. S t /W t S t S t-1 and dynamic S t Z t-1 Rt with 5 degrees of freedom distributed portfolio gross returns R t k R = mean (R t series of past performances, become extremely loss Figure 1. Note. of the index performance encompassed by the myopic cushion S t S t-1, and the resulting yearly wealth percentages S t /W t ex R t, [R t+1 ] = mean [R s s=0,...,t

8 48 distributions 35, we focused on the case with normally cushions 36 by a series of past gains or losses, and the resulting yearly The evolution of the prospective value of risky investments cushion or the probabilities of past and current gains and Figure 2. Note. of the index encompassed by the myopic cushion S t S t-1, and the resulting daily wealth percentages S t t ex, [R t+1 ] = mean [R s s=0,...,t

9 49 Figure 3. Note. two components, the PT effect and the cushion effect, which S t S t-1 t ~ [R t+1 ] = mean [R s s=0,...,t 37 and its two components again but now as functions of the Figure 4. Note. V V sentation of V V - of V - myopic cushion S t S t-1, R t, [R t+1 ] = mean [R s ], s=0,...,t -

10 8 years, where we considered monthly increments of up to 1 year and yearly increments thereafter or 2 years appears to deteriorate the perception of the V be accurate year 41 The evolution of the loss attitude we studied the indirect transmission mechanism of the 1 month to 8 years 42 can be ambiguous in a context considering both direct and indirect transmission mechanisms, the indirect mechanism

11 51 perceptions and decisions, indicate that, under practical A comparison with the exogenous portfolio optimization framework basis of the corresponding formulas and estimates in Figure 5. Note. - other than expected returns, such as the cushion, the probabilities - S t S t-1, R t, [R t+1 ] = mean [R s s=0,...,t at time t Bt Bt

12 52 Table 3 Table 4 Portfolio return Portfolio return t t 1 year 6 months 4 months 3 months 1 month 1 day Note. - t S t-1 S t Z t-1 R t with 5 degrees of freedom distributed portfolio gross returns R t k = 3, R = mean[r t Portfolio-equivalent indices of loss aversion t+1 47 Portfolio return Portfolio return t t 1 year 6 months 4 months 3 months 1 month 1 day - - Table 5 Portfolio return Portfolio return t t 1 year 6 months 4 months 3 months 1 month 1 day Note. S t S t-1 t with 5 degrees of freedom distributed portfolio gross returns R t k

13 53 4 are still much higher than are those in Table 2, where can conclude that the traditional portfolio optimization to become more acute for more relaxed assumptions 5 for myopic cushions are much lower than the empirical 48 k Summary and Conclusions References Mathematical Finance, 9, Preferences with frames: A The loss aversion/narrow framing approach to stock market pricing and participation puzzles Quarterly Journal of Economics, 116,

14 54 Review of Financial Studies, 14, Quarterly Journal of Economics, 110, Journal of Banking and Finance, 33, Review of Economics and Statistics, 86, Investments The American Economic Review, 98 Journal of Banking and Finance, 25, Correlation and dependence in risk management: Properties and pitfalls. Journal of Finance, 58, Quarterly Journal of Economics, 102, Journal of Monetary Economics, 49 An experimental analysis. Journal of Finance, 60, Econometrica, 47, risk loss aversion and wealth allocation with two-dimensional individual utility: A VaR application. The Journal of Risk, 2, Journal of Banking and Finance, 26, Quarterly Journal of Economics, 102, Journal of Risk and Uncertainty, 4 Footnotes referees for their comments and suggestions that allowed us who contributes to a pension fund and whose main decision The process described here is nothing more than the twofund separation theorem of classical portfolio optimization - - w t opt and VaR t portfolio optimization in the strict sense, as performed by managers, but also the earlier decision of nonprofessional

15 55 terested in the formal details and most of these concepts are R 1 z t R ft = x 1 + (1+z t R ft - - Accordingly, gains continue to be considered as possible - narrow framing would be better suited to describe the un k - portfolio gross returns were computed as a zero-mean pro so that Z t = Z 1 = S 0 Z t = Z t-1 R t - -1 R, which corresponds to a more - -1 R we could not apply the simultaneous estimation procedure of results with R ] and R ] are almost that it does not satisfy one of the four properties for coherent - t -

16 more wary of the possibility of registering current losses for - - closely to the predicted maximum point of 1 year and be The ambiguity of the total transmission mechanism reported ion effect, which is highly dependent on returns, distorts the 45 The statement is now consistent with both the data and the is more complex, so second-order polynomials are necessary - 49 Clearly, a i,t = w i,t (W t + B t p i,t - lio formation and enters the portfolio optimization problem an output of this optimization and measures the actual maximum loss possible at time t the obtained optimal portfolio w t opt * Correspondence concerning this article should be di-

17 57 Appendix Optimal portfolio selection with exogenous VaR +1 portfolio consists of i = 1,,n, prices p i,t, and portfolio weights w i,t t 49 folio optimization problem as follows: Such that, [R 1 (w t for the corresponding expected B t (B t B t R f procedure P t t where q t (w t R t+1 (w t P t [R 1 (w t t (w t 1 ex la VaR t = W t (q t (w t opt, 1, ex 51 t = P t (z t

18 58 t = P t (x 1 z t > t = P t (x 1 z t R ft z t x t+1 z t R ft x t+1, [loss value t ] = t t t ] + ( S t t R ft t k(s t Z t [loss value t ] = t ] + ( t t R ft + k S t t and [loss value 2 1 ] = t t t ] + ( S t t R ft 2 t k(s t Z t 2 [loss value 2 1 ] = ( t 2 + ( t t R ft + k k ] t ] (S t Z t Var t [loss value 1 ] = [loss value 2 1 E2 t [loss value2 1 ] Var t [loss value 1 ] = t t R ft 2 (S t Z t S t t of 1, that is t t VaR * t+1 = [loss value 1 [loss value 1 ] VaR * t+1 = t k S t Z t t t t t t R ft t S t Z t VaR * t+1 = t t R ft t k S t Z t [x t+1 t t+1 notation t = t t t t t S t S t Z t * 1 VaR*

19 59 The prospective value function V = t [ t S t t (S t Z t R ft t S t ]+ t t ]+k(z t t V = ( t t t +( t ( t S t ]( t ( t R ft k ](S t Z t S t - P t S t Z t when current losses follow past gains, which occurs with the joint probability t (1 t R ft t (1 t k gra t = ( t t w t + ( t S t k(s t Z t - S t Z t probability (1 t S t Z t