Generalisation of the Arrow-Pratt Risk Premium

5. Generalisation of the Arrow-Pratt Risk Premium

In this section we consider a risk averse decision maker equipped with the generalised behavioural utility function and who generally distorts the objective probability distri- bution. Recall that we say that the decision maker is risk averse if a fair in the objective world gamble decreases the utility of the decision maker. More formally, for a fair gamble x such that E[x] = 0 E [U W + x] U W . We want to derive the expressions for the certainty equivalent, Cx, and the risk premium, 13 π x, which are now defined by the following indifference condition E [U W + x] = U W + Cx = U W + E[x] − πx . 20 Theorem 4. If either utility functions U − and U + are at most quadratic in wealth or the risk of a fair gamble is small, then for a risk averse decision maker the certainty equivalent and the risk premium of the fair gamble are given by Cx = 1 λ U P M 1 x, 0 − 1 2 γ + U P M 2 x, 0 − LP M 1 x, 0 + 1 2 γ − LP M 2 x, 0 , 21 π x = λ − 1 λ E[x] + LPM 1 x, 0 + 1 2 γ + λ U P M 2 x, 0 + γ − LP M 2 x, 0 . 22 Proof. If either utility functions U − and U + are at most quadratic in wealth or the risk of a fair gamble is small under the probability distortion, then, according to Theorem 1, the decision maker’s preferences can be represented by the piecewise-quadratic utility 18. Consequently, the expected utility of W = W + x is given by E [U W + x] = UPM 1 x, 0 − 1 2 γ + U P M 2 x, 0 − λ LP M 1 x, 0 + 1 2 γ − LP M 2 x, 0 . 23 For a risk averse decision maker the certainty equivalent of a fair gamble is negative. Consequently, U W + Cx = λ Cx − 1 2 γ − C 2 x . 24 Since the risk is small, C 2 x ≪ Cx which means that the term with C 2 x can be disregarded. Thus, the use of 23 and 24 in the indifference condition 20 gives the expression for the certainty equivalent 21. To derive the expression for the risk premium, we use π x = E[x] − Cx. 13 We consider the standard risk premium only. There is also so-called ‘behavioural risk premium’ considered in Davies and Satchell 2007. The behavioural risk premium is obtained by considering a small fair gamble from the perspective of the decision maker, that is, a gamble for which E[x] = 0. C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd Corollary 5. In the EUT framework, the risk premium 22 reduces to the Arrow-Pratt risk premium 6. Proof. For a von Neumann-Morgenstern utility function λ = 1 and γ − = γ + = γ . This gives π x = 1 2 γ LPM 2 x, 0 + γ U PM 2 x, 0 = 1 2 γ Var[x]. 25 Observe that since a decision maker with a von Neumann-Morgenstern utility exhibits no loss aversion, both infinitesimal losses and gains are treated similarly as clearly seen from equation 25. Consequently, the Arrow-Pratt measure of risk aversion is really a measure of aversion to variance, or a measure of aversion to uncertainty or dispersion of the probability distribution. In other words, when risk is small, a decision maker with a von Neumann-Morgenstern utility exhibits only aversion to uncertainty. In addition, the risk premium is fully characterised by a measure of uncertainty aversion and the variance, which is a measure of uncertainty. A utility function with loss aversion and different functions for losses and gains allows a much richer and detailed characterisation of risk aversion. According to equation 21 a decision maker exhibits three different types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. The loss is measured by the expected loss, and the uncertainties in gains and losses are measured by the second upper partial moment of x and the second lower partial moment of x respectively. Observe that losses and gains have different weights in the computation of the risk premium. In particular, for a loss averse decision maker, losses are λ times more important than gains. Finally, a brief comparative static analysis of the expressions for the certainty equivalent and the risk premium is presented by means of the following corollaries. Corollary 6. The certainty equivalent decreases as the decision maker’s loss aversion increases. Conversely, the risk premium increases as the decision maker’s loss aversion increases. Proof. The first-order derivatives of the certainty equivalent and risk premium with respect to λ ∂ Cx ∂λ = − ∂π x ∂λ = − 1 λ 2 U P M 1 x, 0 − 1 2 γ + U P M 2 x, 0 . Consider the sign of B = UPM 1 x, 0 − 1 2 γ + U P M 2 x, 0. Note that U PM 1 x, 0 0 and U PM 2 x, 0 0. Obviously, if γ + ≤ 0, then the sign of B is positive. However, even if γ + 0 the sign of B is positive since in this case we need to impose the upper limit maxx 1 γ + to ensure that the utility function is increasing 14 for all outcomes of x. Therefore B = 1 γ+ y − 1 2 γ + y 2 dQ x y 0, 14 To motivate for this, consider the piecewise-quadratic utility function 18. If γ + 0, then the function U W = W − W − 1 2 γ + W − W 2 is increasing for W − W 1 γ + . C 2009 The Authors Journal compilation C 2009 Blackwell Publishing Ltd since y − 1 2 γ + y 2 0 for all y 1 γ + . Finally we obtain ∂ Cx ∂λ = − ∂π x ∂λ 0. Corollary 7. The certainty equivalent decreases as the decision maker’s risk aversion in the domain of either gains or losses increases. Conversely, the risk premium increases as the decision maker’s risk aversion in the domain of either gains or losses increases. Proof. The first-order derivatives of the certainty equivalent and risk premium with respect to γ + ∂ Cx ∂γ + = − ∂π x ∂γ + = − 1 2λ U P M 2 x, 0 0, since U PM 2 x, 0 0. The first-order derivatives of the certainty equivalent and risk premium with respect to γ − ∂ Cx ∂γ − = − ∂π x ∂γ − = − 1 2 LP M 2 x, 0 0, since LPM 2 x, 0 0.

6. Impact of the Decision Maker’s Skewness Preferences on Risk Measurement