<p> *** THE NATURE OF RISK PREFERENCES</p><p>* Preliminary: </p><p>. If U(a) is a concave (convex) function, then, for any , 0 <  < 1, and any two points a1 and a2,</p><p>U( a1 +(1-) a2)  () [ U(a1) + (1-) U(a2)]. </p><p>. Under differentiability, U(a) is. concave iff 2U/a2  0, a . convex iff 2U/a2  0, a. </p><p>. Jensen's inequality: If U(a) is a convex (linear, concave) function of the random variable "a", then U[E(a)]  (=, ) EU(a). </p><p>* The Risk Premium: Let U(w+a), where w = non-random initial wealth, "a" = risky monetary reward (= a random variable), and (w+a) = terminal wealth. Assume that U(w+a) is a strictly increasing function of (w+a). How could we measure the monetary value of risk? This can be done using compensation tests.</p><p>1/ The selling price of risk: Rs = the smallest sure amount of money a decision maker would be willing to receive to eliminate (sell) the risk "a" if he had it:</p><p>{w+Rs} * {w+a}. Under the EUH, this implies</p><p>U(w+Rs) = EU(w+a).</p><p>2/ The bid price of risk: Rb = the largest sure amount of money a decision maker would be willing to pay to obtain (buy) the risk "a":</p><p>{w} * {w+a-Rb}. Under the EUH, this implies</p><p>U(w) = EU(w+a-Rb).</p><p>3/ The risk premium: R = the sure amount of money that would make a decision maker indifferent between receiving the risky return "a" and receiving the non-random amount E(a)-R: {w+a} * {w+E(a)-R}. Under the EUH, this implies EU(w+a) = U(w+E(a)-R), or, given U' > 0, R = w + E(a) - U-1EU(w+a), where U-1 denotes the inverse of the function U. Note that, in general, R is a function of w and of the probability function of a. Also, from the definition of the risk premium and given that U(.) is an increasing function, maximizing EU(w+a) is equivalent to maximizing the non-random expression [w+E(a)-R]. As a result, [w+E(a)-R] has been called the certainty equivalent of EU(): it is a money metric measure of utility. </p><p>* Risk aversion: (Pratt) By definition of the risk premium, we have EU(w+a) = U(w+E(a)-R).</p><p>A Taylor series expansion of the left-hand side gives U(w+a)  U(w+E(a)) + U' [a-E(a)] + 1/2 U" [a-E(a)]2. A Taylor series expansion of the right hand side gives U(w+E(a)-R)  U(w+E(a)) - U' R.</p><p>Combining these two results with the definition of the risk premium yields U(w+E(a)) + U' E[a-E(a)] +1/2 U" E[a-E(a)]2  U(w+E(a)) - U' R, Given U' >0 and E[a-E(a)] = 0, this implies R  -1/2 (U"/U') V(a), where V(a) = E[a-E(a)]2 > 0 is the variance of "a". In the neighborhood of the riskless case, it follows that [-1/2 (U"/U') V(a)] is a local measure of the risk premium R: in the small, R is proportional to V(a). Define r = -U"/U' as the Arrow-Pratt coefficient of absolute risk aversion. Then, the risk premium R can be approximated in the small as R  1/2 r V(a).</p><p>Definition: A decision maker is said to be risk averse (risk neutral, or risk lover) if R > (= or <) 0.</p><p>From the above discussion, it follows that: - risk averse behavior (R > 0) corresponds to r > 0 and U" < 0, i.e. a concave utility function. - risk neutral behavior (R = 0) corresponds to r = 0 and U" = 0, i.e. a linear utility function. - risk loving behavior (R < 0) corresponds to r < 0 and U" > 0, i.e. a convex utility function.</p><p>Note: By definition, r = - U"/U' = - ln(U')/w. This implies r = - ln(U') + c or ln(U') = c - r or U' = ec e- r or U = ec  e- r + K. Since U is defined up to a positive linear transformation, it follows that the utility function U( ) can always be expressed exactly as U(  ) =  e - r. In other words, the Arrow-Pratt coefficient of absolute risk aversion r = -U'/U' provides all the information about the underlying preference function U().</p><p>* Constant Absolute Risk Aversion: (CARA) (Pratt) By definition, CARA corresponds to r = -U"/U' being a constant. Under CARA, . r > 0 (risk aversion) corresponds to the utility function U = - e-r (w+a) . r = 0 (risk neutrality) corresponds to the utility function U = w + a . r < 0 (risk loving) corresponds to the utility function U = e-r (w+a)</p><p>Note: Since e-r (w+a) (= e-rw e-ra) is proportional to e-ra for any w, it follows that changing initial wealth w does not affect economic decisions. In other words, CARA implies zero "wealth effects".</p><p>Note: Consider the definition of the risk premium R under CARA: -e-r [w+E(a)-R] = E[-e-r [w+a]]. This can be written alternatively as -e-r [E(a)-R] = E[-e-ra], which implies that the risk premium R is not a function of w. Thus, under CARA, a change in initial wealth does not affect the willingness to insure (as measured by the risk premium R). Note: The case of CARA preferences under a normal distribution for "a": a ~ N(A, V) where "a" has mean A and variance V.</p><p>rw (aA)2 r (w+a) e ra  E[ e ] = [ e e 2V ]da 2V 2 rw (aA+rV) r 2V2+2ArV e  = [ e 2V ]da 2V 2 r 2V2rA (aA+rV) rw 1  = e [ e 2 ]  [e 2V ]da 2V r(rV2A) rw = e [ e 2 ]</p><p> which is a monotonic increasing function of [A - r/2 V]. It follows that maximizing EU() is equivalent to maximizing [A - r/2 V]. In other words, [A - r/2 V] is the certainty equivalent, where R = r/2 V is a global measure of the risk premium. Thus, an additive mean-variance analysis can be justified under CARA and normality.</p><p>* Decreasing (or Increasing) Absolute Risk Aversion: (Pratt) Definition: Risk preferences exhibits decreasing (increasing) absolute risk aversion if R(w,.) is a decreasing (increasing) function of w.</p><p>Theorem 1: Consider two decision makers facing the terminal wealth (w+a), each with utility function </p><p>U1(w+a) and U2(w+a). Let ri = - Ui"/Ui', and Ri = the risk premium for individual i, i = 1, 2. Then, the following statements are equivalent:</p><p>. R1 (w) < (>) R2(w) for all w. . r1 (w) < (>) r2(w) for all w.</p><p>Proof: By definition, Ui[w+E(a)-R] = EUi[w+a] implies that -1 Ri = w + E(a) -Ui EUi(w+a) or -1 -1 R1-R2 = U2 EU2(w+a) - U1 EU1(w+a).</p><p>-1 Let t = U2(w+a), or (w+a) = U2 (t). It follows that -1 -1 -1 R1-R2 = U2 E(t) - U1 EU1(U2 (t)). (1) Note that -1 [U1(U2 (t))]/t = U1'/U2' = exp[ln(U1'/U2')] = an increasing function of [ln(U1'/U2')].</p><p>2 Also, ln(U1'/U2')/w = [U1"/U2' - U2"U1'/(U2') ]/(U1'/U2') = r2 - r1. This implies: -1 r2  () r1  [U1[U2 (t)]]/t is increasing (decreasing) in t -1  U1(U2 (t)) is convex (concave) in t.</p><p>By Jensen's inequality, -1 -1 r2  () r1  EU1(U2 (t))  () U1(U2 (E(t))).</p><p>From (1), this implies that -1 -1 -1 R1 - R2 = U2 E(t) - U1 EU1(U2 (t)) -1 -1 -1  () U2 E(t) - U1 U1(U2 (E(t))) = 0 as r2  () r1. Q.E.D.</p><p>Theorem 2: The following two statements are equivalent: . R(w) is an increasing (decreasing) function of w for all w. . r(w) is an increasing (decreasing) function of w for all w.</p><p>Proof: Follows directly from theorem 1, after choosing Ui(w) = U(wi). Q.E.D. Note: From theorem 2, under decreasing absolute risk aversion (DARA), r(w) = -U"/U' is a decreasing function of w, implying that r/w = - U"'/U' + (U"/U')2  0. This implies that U"'/U'  (U"/U')2  0. Given U' > 0, it follows that U"'  0 under DARA.</p><p>Note: The case of quadratic utility functions: U(x) = α + β x + c/2 x2. . Quadratic utility function can provide a second order approximation to any differentiable utility function. . They can exhibit risk aversion (c <0), risk neutrality (c = 0) or risk loving behavior (c > 0). . Since EU(x) = α + β E(x) + c/2 E(x2) = α + β E(x) +c/2 [(E(x))2 + V(x)], quadratic utility functions imply that EU(x) = U(E(x), V(x)). Thus, quadratic utility functions can be used to justify a mean-variance analysis. . But quadratic utility functions exhibit U"' = 0. Thus they cannot exhibit strictly decreasing absolute risk aversion. And in the case of departure from risk neutrality, they necessarily exhibit increasing absolute risk aversion (IARA).</p><p>Note: Examples of utility functions that exhibit DARA: . U(w) = (w + d)q, d  0, 0 < q < 1. . U(w) = ln(w + d), d  0.</p><p>* Relative Risk Aversion : (Pratt) Let x = w + a = terminal wealth, w = non-random initial wealth, and "a" = random return.</p><p>Definition: The relative risk premiumR is the proportion of terminal wealth x a decision maker is willing to pay to make him indifferent between facing the risky terminal wealth x versus receiving [(1- R) E(x)]: EU(x) = U[(1-R)E(x)]. (2)</p><p>Note that the following relationship exists between the relative risk premiumR and the absolute risk premium R: R = R/E(x).</p><p>Definition: A decision maker is said to exhibit increasing, constant, or decreasing relative risk aversion if R increases, is constant, or decreases with a proportional increase in (w+a).</p><p>Let a = a/E(x) = relative risk, with V(a) = V(x) = [E(x)]2V(a). Using the local measure for R derived above, it follows that R = R/E(x)  -1/2 U"/U' V(x)/E(x) = -1/2 (U"/U') E(x) V(a). In the neighborhood of the riskless case, this suggests that a local measure of the relative risk premium is given by R  1/2 r V(a) where r = -(U"/U') x = the relative risk aversion coefficient. Theorem 3: The following two statements are equivalent: . r(x) is increasing (decreasing) in x for all x. . R increases (decreases) with a proportional increase in x for all x.</p><p>Proof: Follows from theorem 1. See Pratt. Q.E.D.</p><p>Constant Relative Risk Aversion (CRRA): From theorem 3, CRRA corresponds to the relative risk aversion coefficient r being a constant. Examples of CRRA utility functions: . U(x) = x1-r for r < 1 . U(x) = ln(x), corresponding to r = 1 . U(x) = -x1-r for r > 1.</p><p>Note: CRRA implies that r/x = -U”’x/U’ + (U”/U’)2x - U”/U’ = 0. Thus, under risk aversion (where U’ > 0 and U” < 0), CRRA implies that (r/w)x  U”’x/U’ + (U”/U’)2 x = U”/U’ < 0. Thus, risk aversion and CRRA always imply DARA. </p><p>Note: A DARA utility function can exhibit constant relative risk aversion (CRRA), decreasing relative risk aversion (DRRA) or increasing relative risk aversion (IRRA).</p><p>Common hypotheses: . Most people seem to be averse to risk: they have a positive willingness to pay to insure against risk (R > 0). . Most people seem to exhibit DARA: the willingness to pay for insurance seems to decrease with private wealth (R/w < 0). In other words, private wealth accumulation seems to be a substitute for external insurance. . Arrow has argued in favor of IRRA, the percentage of wealth spent on insurance increasing with a proportional rise in (w+a). However, there seems to be only weak empirical support for IRRA.</p><p>* Partial Relative Risk Aversion: (Menezes and Hanson); x = w + a. Define the partial relative risk premium R = R/E(a), where R is the absolute risk premium defined above. R is a measure of the willingness to pay for insurance relative to the expected pay-off E(a). Define the relative risk  = a/E(a), with V(a) = V()[E(a)]2. From the local measure of the risk premium R, it follows that R = R/E(a)  -1/2 (U"/U') V(a)/E(a) = -1/2 (U"/U') E(a) V(). In the neighborhood of the riskless case, this suggests the following local approximation to the partial relative risk premium: R  1/2 r V(), where r = -(U"/U') a = the partial relative risk aversion coefficient.</p><p>Theorem 4: The following two statements are equivalent: . R increases (decreases) with a proportional increase in "a" for all w . r increases (decreases) with a proportional increase in "a" for all w.</p><p>Example: Under increasing partial relative risk aversion (IPRRA), a proportional increase in "a" would generate an increase in the partial relative risk premium R = R/E(a), i.e. a more than proportional increase in the risk premium R. * Preferences with respect to Moments: U(x), x = w+a. A Taylor series expansion of U(x) evaluated at x = E(x) gives i i i U(x)  U(E(x)) + i1 1/(i!)  U/x [x-E(x)]</p><p>Taking the expectation and assuming that the moments exist, this gives i i i EU(x)  U(E(x)) + i1 1/(i!)  U/x E[x-E(x)] i i = U(E(x)) + i1 1/(i!) U M</p><p> i i i i where U =  U/x , and Mi = E[x-E(x)] is the i-th central moment of x, i = 1, 2, .... Recall that EU(x) = U(E(x) - R)  U(E(x)) -(U/x) R, where R is the risk premium. Noting that M1 = 0, and combining these two results yields i 1 R  i2 -[1/(i!)] (U /U ) Mi</p><p>= i2 Ri Mi (3)</p><p> i 1 where Ri = -[1/(i!)] (U /U ) is a measure of the marginal contribution of the i-th moment of x to the risk premium R.</p><p>. for i = 2, M2 = V(x) is the variance of x, and R2 = -1/2 U"/U' = r/2, r being the Arrow-Pratt absolute risk aversion coefficient.</p><p>. for i = 3, M3 = measure of the skewness of x, and R3 = -1/6 (U"'/U'). Note that skewness to the left (M3 < 0) is associated with "down-side risk", a decrease in M3 implying an increase in down-side risk. Define downside risk aversion as corresponding to a positive willingness to pay to avoid down-side risk. </p><p>The above result indicates that downside risk aversion corresponds to R3 < 0 or U"' > 0, implying that a rise in down-side risk (a decrease in M3) would tend to increase the willingness to pay for risk (as measured by the risk premium R). But we have shown that U"'  0 under DARA. It follows that DARA preferences in general imply "downside risk aversion". Thus, if DARA characterizes the risk preferences of many people, this would imply that most people are also adverse to downside risk. . for i = 4, etc...</p><p>Note: Recall that maximizing EU(x) is equivalent to maximizing the certainty equivalent [E(x) - R], where R is the risk premium. Assuming that the moments exists, (3) implies that the objective function of a decision maker can always be approximated by </p><p>[E(x) - i2 Ri Mi].</p><p>This general formulation has several advantages: it does not require a full specification of U(x); and it allows to go beyond a simple mean-variance analysis (e.g. by including skewness, kurtosis, ..., in the investigation of risk behavior). </p>