JOHN W. PRATT may also be interpreted as twice the actuarial value the decision maker requires per unit of variance for infinitesimal risks Notice that it is the variance, not the standard deviation, that enters these for- mulas. To first order any(differentiable)utility is linear in small gambles. In this sense, these are second order formulas Still another interpretation of r(x) arises in the special case i=+h, that is where the risk is to gain or lose a fixed amount h>0. Such a risk is actuarially neutral if +h and -h are equally probable, so P(i=h)-P(=-h)measures the probability premium of i. Let p(r, h) be the probability premium such that the de- cision maker is indifferent between the status quo and a risk i=+h with (8)P(E=h)-P(2=-h)=p(x,h) P(E=h)=i[l+P(x, h)l, P(i=-h)=2[l-P(r, h)], and p(x, h)is defin When u is expanded around x as before, (9)becomes h)(x)+h2u'(x)+O(h3) Solving for p(x, h), we find (11)p(x,h)=1hr(x)+O(h2) Thus for small h the decision maker is indifferent between the status quo and a risk of th with a probability premium of r(x) times th; that is, r(x)is twice the prob. ability premium he requires per unit risked for small risks In these ways we may interpret r(x)as a measure of the local risk aversion or local propensity to insure at the point x under the utility function u; -r(x)would measure locally liking for risk or propensity to gamble. Notice that we have not introduced any measure of risk aversion in the large. Aversion to ordinary (as opposed to infinitesimal) risks might be considered measured by n (x, 2), but r is a much more complicated function than r. Despite the absence of any simple measure of risk aversion in the large, we shall see that comparisons of aversion to risk can be made simply in the large as well as in the small By(6), integrating -r(x) gives log u(x)+c; exponentiating and integrating again then gives ecu()+d. The constants of integration are immaterial because ecu(x)+du(x).(Note ec>0. )Thus we may write and we observe that the local risk aversion function r associated with any utility function u contains all essential information about u while eliminating everything arbitrary about u. However, decisions about ordinary(as opposed to"small") risks are determined by r only through u as given by(12), so it is not cor entirely to eliminate u from consideration in favor of
RISK AVERSION 127 4. CONCAVITY The aversion to risk implied by a utility function u seems to be a form of con cavity, and one might set out to measure concavity as representing aversion to risk. It is clear from the foregoing that for this purpose r(x)=-u(x)u()can be con- sidered a measure of the concavity of u at the point x. A case might perhaps be made for using instead some one-to-one function of r(x), but it should be noted that u(x)or -u()is not in itself a meaningful measure of concavity in utility theory, nor is the curvature(reciprocal of the signed radius of the tangent circle)u(x)(1+ u(x)]2)-3/2. Multiplying u by a positive constant, for example, does not alter A more striking and instructive example is provided by the function u(x)=-e-r As x increases, this function approaches the asymptote u=0 and looks graphically less and less concave and more and more like a horizontal straight line, in accord ance with the fact that u(x)=e- and u(x)=e-x both approach 0. As a utility function, however, it does not change at all with the level of assets x, that is, the behavior implied by u(x)is the same for all x, since u(k+x)=-e-k-wufx In particular, the risk premium I(, 2)for any risk z and the probability premium p(x, h) for any h remain absolutely constant as x varies. Thus, regardless of the appearance of its graph, u(x)=-e-x is just as far from implying linear behavior at x=oo as at x=o or x=-oo. all this is duly reflected in r(x), which is constant One feature of u"(x) does have a meaning, namely its sign, which equals that of r(x). A negative(positive)sign at x implies unwillingness(willingness) to accept small, actuarially neutral risks with assets x. Furthermore, a negative(positive) sign for all x implies strict concavity(convexity) and hence unwillingness(willing ness)to accept any actuarially neutral risk with any assets. The absolute magnitude of u(x) does not in itself have any meaning in utility theory, however 5. COMPARATIVE RISK AVERSION Let u, and u2 be utility functions with local risk aversion functions r1 and r respectively. If, at a point x, r,(x)>r2(), then u, is locally more risk-averse than u2 at the point x; that is, the corresponding risk premiums satisfy (r, 2)>72(x, 2) for sufficiently small risks i, and the corresponding probability premiums satisfy P1(x, h)>P2(x, h) for sufficiently small h>0. The main point of the theorem we are about to prove is that the corresponding global properties also hold. For instance, if r1(x)>r2(x)for all x, that is, u, has greater local risk aversion than u2 everywhere, then I,(x, 2)>2(x, 2) for every risk i, so that u, is also globally more risk-averse in a natural sense It is to be understood in this section that the probability distribution of z, which determines I,(x, i)and I2(x, i), is the same in each. We are comparing the risk