Capital Asset Pricing under Loss Aversion

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Capital Asset Pricing under Loss Aversion

In determining asset prices, investors examine not only expected return but downside risk. That is, investors exhibit loss aversion and their portfolio decisions attempt to limit the probability of returns falling below a desired minimum level. In finance, the model of asset pricing under loss aversion was put forth by E. Arzac and V. Bawa building on earlier work by A. D. Roy and Lester Telser on Roy's safety-first criterion. Since its creation, the model has been further developed and applied by D. W. Jansen et al., R. Campbell et al., C. Gourieroux et al., H. Shefrin and M. Statman, M-H Broihanne, et al., S. G. Rhee and F. Wu, M. van Oord and C. Zhou and others. Campbell et al. used the model to provide the economic foundation of value at risk (VaR) management required by the Basel Accords and used in macroprudential regulation.

Loss aversion asset pricing has received renew interest because of the financial crises that have taken place in the new millennium and the awareness of the relevance of loss aversion to investment decisions. Its application to real-life left-tail probability events has permitted researchers to achieve further understanding of investor behavior, including the assessment of the gains of adding Asian and Latin American equity markets to diversified portfolios while limiting the losses associated with infrequent and catastrophic events, the construction of safety-first portfolios to protect investors' net worth from the financial risk of extreme events like terrorist attacks, and the construction of optimal portfolios for precious metals, oil and stocks. N. Hyung and C. de Vries showed that portfolio optimization under loss aversion with fat tails lowers diversification much below the level prescribed by traditional portfolio theory and thus explains the low diversification puzzle. Y. Atilgan, et al. documented that investors underestimate the persistence of left-tail risk and overprice stocks with large recent losses.

Safety-first portfolio selection

A number of departures from the mean-standard deviation approach of the standard capital asset pricing model (CAPM) have been proposed in the literature, starting with Roy's safety-first criterion which requires investors’ portfolios to minimize a probability of disaster, defined as a return falling short of some level s, also referred to as the value at risk (VaR) limit. In symbols:

min Pr(X ≤ s), where X is the random return.

In turn, Lester Telser proposed a more general criterion consisting in maximizing expected return subject to the constraint that the probability of return does not fall below a safety level α, that is,

max E(X) subject to Pr(X ≤ s) ≤ α, where E is the expectation operator,

and E. Arzac and V. Bawa proposed the following lexicographic preferences form of the safety-first principle:

max (π, μ ), where π =1 if P = Pr {X ≤ s} ≤ α, and π = 1 – P otherwise, and μ = E(X).

This criterion orders two portfolios lexicographically according to π and μ. The portfolio with the higher probability of meeting the safety-first constraint is the preferred one and the ordering of any two portfolios with the same π is made according to their expected values μ. It can be shown that this criterion satisfies first-degree stochastic dominance. Stochastic dominance is violated by the criteria of Roy and Telser. Lexicographic safety-first investors exhibit risk aversion in the sense of Kenneth Arrow, exhibit decreasing absolute risk aversion, and their relative risk aversion can be increasing (for s < 0), constant (for s = 0) or decreasing (for s > 0). Also, their portfolio problem satisfies the separation property in the sense that it can be separated into two problems: The choice of the optimal risk assets proportions, and the choice of the scale of the risky portfolio and the amount to borrow. This property extends the class of preferences admitting separability beyond those satisfying expected utility theory.

Asset pricing under loss aversion

The separation property exhibited by lexicographic safety-first investors permits solving the portfolio problem maximizing the ratio

(E(R) – R_f)/(R_f – q_p) were R_f is the risk-free interest rate and q_p is the α fractile of the return distribution.

R. Campbell et al., noted that this is a performance index like the Sharpe ratio that measures the reward to return loss that can be incurred with probability α, with the denominator representing the potential opportunity loss of investing in risky assets. Under the assumption that investors hold homogenous beliefs and α level loss aversion, the equilibrium rate of return of asset j is

E(R_j) = R_f +β_j (E(R_m) – R_f)

where:

$\beta _{j}$ = (R_f – q_j)/(R_f – q_m)

q_j is the α fractile of the return distribution of asset j, that is, Pr(R_j ≤ q_j) = α

q_m is the α fractile of the return distribution of the market, that is, Pr(R_m ≤ q_m) = α.

$\beta _{j}$ is the ratio of the return loss that asset j would incur with probability α to the return loss that the market can incur with probability α. In other words, it is the contribution of asset j to the to the admissible return loss of the market. This measure of systematic risk has a similar meaning to the beta coefficient of the standard CAPM but the α fractile is a better measure of loss aversion than the second moment because it is based upon leftward deviations only.

Only in the cases in which the distributions of returns are normal or stable Paretian (stable distribution), or information is limited to the first two moments and require the use of a Tchebycheff inequality,

$\beta _{j}$ = (R_f – q_j)/(R_f – q_m) = cov(R_j, R_m)/Var(R_m)

as in the standard CAPM. In this sense, the latter allows for the existence of lexicographic safety-first investors under its distribution assumptions, and asset pricing under loss aversion with lexicographic safety-first preferences generalizes the CAPM.

Loss aversion efficient frontier

Portfolio optimization and the computation of the efficient frontier in the loss aversion model can be made with parametric methods or kernel (statistics). N. Hyung and C. Vries and C. Gourieroux et al. showed that the loss aversion efficient frontier is convex. D. Jansen et al. used extreme value theory to calculate the optimal value-at-risk efficient portfolios accounting for the fact that asset returns are fat-tailed.

Lexicographic preferences and their psychological foundations

Lexicographic preferences represent a departure from Von Neumann-Morgenstern utility theorem in that they are not compatible with the axioms of continuity and independence and allow behavior not consistent with the expected utility hypothesis as in Allais paradox and Prospect theory. The psychological foundations of loss aversion choices were laid out in the pioneering work of Kurt Lewin, the experimental test of Sidney Siegel of the level of aspiration theory, and the framework of behavioral economics developed by Daniel Kahneman and Amos Tversky.

References

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