Risk Aversion in the Small and in the large TORIo John w. pratt Econometrica, Vol 32, No. 1/2(Jan -Apr, 1964), pp. 122-136 Stable url: http://links.jstor.org/sici?sici=0012-9682%028196401%2f04%2932%03a1%2f2%3c122%03araitsa%3e2.0.c0%3b2-w conometrica is currently published by The Econometric Societ Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://wwwjstor.org/journals/econosoc.html Each copy of any part of a JSTOR transmission must contain the same copy tice that appears on the screen or printed page of such transmission jStOR is an independent not-for-profit organization dedicated to creating and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor. org ttp://www.jstor.org Thu mar907:04:542006
RISK AVERSION IN THE SMALL AND IN THE LARGEI BY JoHN W. pra This paper concerns utility functions for money. A measure of risk aversion in the small, the risk premium or insurance premium for an arbitrary risk, and a natural concept of decreasing risk aversion are discussed and related to one another. Risks are also considered as a proportion of total assets 1. SUMMARY AND INTRODUCTION LET u(x) BE a utility function for money. The function r(x)=-u'(x)/u(x)will be interpreted in various ways as a measure of local risk aversion(risk aversion in the small); neither u(x)nor the curvature of the graph of u is an appropriate measure No simple measure of risk aversion in the large will be introduced. Global risks will, however, be considered and it will be shown that one decision maker has greater local risk aversion r(x)than another at all x if and only if he is globally more risk-averse in the sense that, for every risk, his cash equivalent (the amount for which he would exchange the risk) is smaller than for the other decision maker Equivalently, his risk premium(expected monetary value minus cash equivalent) is always larger, and he would be willing to pay more for insurance in any situation From this it will be shown that a decision maker's local risk aversion r(x)is a de creasing function of x if and only if, for every risk, his cash equivalent is larger the larger his assets, and his risk premium and what he would be willing to pay for insurance are smaller. This condition, which many decision makers would sub- cribe to, involves the third derivative of u, as r'so is equivalent to u"u'u It is not satisfied by quadratic utilities in any region. All this means that some natural ways of thinking casually about utility functions may be misleading Except for one family, convenient utility functions for which r(x)is decreasing are ot so very easy to find. Help in this regard is given by some theorems showing that certain combinations of utility functions, in particular linear combinations with positive weights, have decreasing r(x) if all the functions in the combination have ecreasing r(x) The related function r*(x)=xr(x) will be interpreted as a local measure of aver- sion to risks measured as a proportion of assets, and monotonicity of r*(x)will be proved to be equivalent to monotonicity of every risk's cash equivalent measured as a proportion of assets, and similarly for the risk premium and insurance These results have both descriptive and normative implications. Utility functions for which r(x) is de the behavior of people who, one feels, might generally pay less for insurance against I This research was supported by the National Science Foundation(grant NSF-G24035) Reproduction in whole or in part is permitted for any purpose of the United States Govern
RISK AVERSION 123 a given risk the greater their assets. And consideration of the yield and riskiness per investment dollar of investors' portfolios may suggest, at least in some contexts description by utility functions for which r*(x)is first decreasing and then increas- Normatively, it seems likely that many decision makers would feel they ought to pay less for insurance against a given risk the greater their assets. Such a decision maker will want to choose a utility function for which r(x) is decreasing, adding this condition to the others he must already consider(consistency and probably concavity) in forging a satisfactory utility from more or less malleable pre- liminary preferences. He may wish to add a further condition on r*(x) We do not assume or assert that utility may not change with time Strictly speak- ing, we are concerned with utility at a specified time(when a decision must be nade) for money at a(possibly later)specified time. Of course, our results pertain also to behavior at different times if utility does not change with time. For instance, a decision maker whose utility for total assets is unchanging and whose assets are increasing would be willing to pay less and less for insurance against a given risk as time progresses if his r(x)is a decreasing function of x. Notice that his actual expenditure for insurance might nevertheless increase if his risks are increasing along with his assets. The risk premium, cash equivalent, and insurance premium are defined and re- lated to one another in Section 2. The local risk aversion function r(x)is introduced and interpreted in Sections 3 and 4. In Section 5, inequalities concerning global risks are obtained from inequalities between local risk aversion functions. Section 6 deals with constant risk aversion, and Section 7 demonstrates the equivalence local and global definitions of decreasing(and increasing) risk aversion. Section 8 shows that certain operations preserve the property of decreasing risk aversion Some examples are given in Section 9. Aversion to proportional risk is discussed in Sections 10 to 12. Section 13 concerns some related work of Kenneth J. Arrow. 2 Throughout this paper, the utility u(x)is regarded as a function of total assets ather than of changes which may result from a certain decision, so that x=0 is luivalent to ruin, or perhaps to loss of all readily disposable assets. ( This is essen- tial only in connection with proportional risk aversion. ) The symbol indicates that two functions are equivalent as utilities, that is, u,(x)wu2(x)means there exist constants a and b(with b>0) such that u,(x)=a+ bu2(x) for all x. The utility functions discussed may, but need not, be bounded. It is assumed, however, that they are sufficiently regular to justify the proofs; generally it is enough that they be twice continuously differentiable with positive first derivative, which is already re- The importance of the function r(x) was discovered independently by Kenneth J. Arrow and by Robert Schlaifer, in different contexts. The work presented here was, unfortunately, essentially ompleted before I learned of Arrow's related work. It is, however, a pleasure to acknowledge Schlaifer's stimulation and participation throughout, as well as that of John Bishop at certain
JOHN W. PRATT quired for r(x) to be defined and continuous. a variable with a tilde over it such as i, is a random variable. The risks i considered may, but need not, have"objective probability distributions In formal statements, i refers only to risks which are not degenerate, that is, not constant with probability one, and interval refers only to an interval with more than one point. Also, increasing and decreasing mean nonde- creasingand nonincreasing respectively; if we mean strictly increasing or decreasing we will say so 2. THE RISK PREMIUM Consider a decision maker with assets x and utility function u. We shall be inter ested in the risk premium T such that he would be indifferent between receiving a risk i and receiving the non-random amount e(2-t, that is, t less than the actuarial value E(2). If u is concave, then I20, but we dont require this. The risk premium depends on x and on the distribution of z, and will be denoted I(x, 2). (It is not, as this notation might suggest, a function n (x, z)evaluated at a randomly selected value of z, which would be random. By the properties of utility, (1)(x+E(2)-x(x,2)=E{(x+2)} We shall consider only situations where Equ(x+2) exists and is finite. Then T(x, a)exists and is uniquely defined by (1), since u(x+e()-)is a strictly de creasing, continuous function of T ranging over all possible values of u. It follows immediately from (1)that, for any constant u 丌(x,2)=(x+H,E-1) By choosing u=E(2)(assuming it exists and is finite), we may thus reduce consider ation to a risk i-u which is actuarially neutral, that is, E(Z-a)=0. Since the decision maker is indifferent between receiving the risk i and receiving for sure the amount I (x, 2)=E(2)-I(x, 2), this amount is sometimes called the cash equivalent or value of z. It is also the asking price for i, the smallest amount for which the decision maker would willingly sell i if he had it. It is given by )u(x+2(x,2)=E{u(x+2)} It is to be distinguished from the bid price T,(x, 2), the largest amount the decision maker would willingly pay to obtain i, which is given by (3b)(x)=E{u(x+2-n(x,2)} For an unfavorable risk i, it is natural to consider the insurance premium I,x, 2)such that the decision maker is indi between facing the risk i and paying the non-random amount I (x, 2). Since paying I, is equivalent to receiving 丌r, we have (3c)(x,2)=-(x,2)=丌(x,2)-E(2)
RISK AVERSION If i is actuarially neutral, the risk premium and insurance premium coincide The results of this paper will be stated in terms of the risk premium but could equally easily and meaningfully be stated in terms of the cash equivalent or insur- ance premium. 3. LOCAL RISK AVERSION To measure a decision maker's local aversion to risk, it is natural to consider risk premium for a small, actuarially neutral risk i. We therefore consider r(x, 2) for a risk i with E(2)=0 and small variance o2; that is, we consider the behavior of r(x, 2)as 020. We assume the third absolute central moment of i is of smaller order than 02. (Ordinarily it is of order a2)Expanding u around x on both sides of (1), we obtain under suitable regularity conditions (x-)=(x)-u(x)+O(x2), (4b)E{(x+2)}=E{u(x)+i(x)+12"(x)+O(23)} Setting these expressions equal, as required by(1), then gives (x,2)=1a2r(x)+o(G2), where u"(x) log u(x) Thus the decision maker's risk premium for a small, actuarially neutral risk i is approximately r(x) times half the variance of i; that is, r(x)is twice the risk pre- mium per unit of variance for infinitesimal risks. A sufficient regularity condition for(5)is that u have a third derivative which is continuous and bounded over the range of all i under discussion. The theorems to follow will not actually depend on (5), however. If i is not actuarially neutral, we have by (2), with u=E(2), and (5) r(x,2)=2r(x+E(2)+0(2) Thus the risk premium for a risk i with arbitrary mean e() but small variance is approximately r(x+ E(2))times half the variance of z. It follows also that the risk premium will just equal and hence offset the actuarial value E(Z)of a small risk (2); that is, the decision maker will be indifferent between having Z and not having it when the actuarial value is approximately r(x) times half the variance of z. Thus r(x) 3 In expansions, o( )means"terms of order at most "and o( )means"terms of smaller
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
128 OHN W. PRATT premiums for the same probability distribution of risk but for two different utilities This does not mean that when Theorem 1 is applied to two decision makers, they must have the same personal probability distributions, but only that the notation is imprecise. The theorem could be stated in terms of T (x, i,)and I2(x, i2) where the distribution assigned to i, by the first decision maker is the same as that assigned to i2 by the second decision maker. This would be less misleading, but also less onvenient and less suggestive, especially for later use. More precise notation would be, for instance, I,(x, F)and I2(, F), where Fis a cumulative distribution nction THEOREM 1: Let r(x), I(x, 2), and(x) be the local risk aversion, risk premium, ponding to the utility function ui,i=1, 2 following conditions are equivalent, in either the strong form(indicated in brackets) or the weak form(with the bracketed material omitted) (a) rI(x)2r2(x) for all x [and >for at least one x in every intervall (b) T,(x, 2)2[>]I2(x, 2) for all x and i (c) PI(r, h)2[>lp2(x, h) for all x and all h>0 (d) u1u2(t))is a [strictly] concave function of n4ts[<23()-n2(x) (e)l1(y)-1(x) u2(w)-u,vor anD, w,x, y with v<wsx<y The same equivalences hold if attention is restricted throughout to an interval, that is, f the requirement is added that x, x+i, x+h, x-h, u2(O), D, w, and y, all lie in a PROOF: We shall prove things in an order indicating somewhat how one might discover that(a) implies(b)and(c) To show that(b)follows from(d), solve(1)to obtain (13)xx,2)=x+E()-u2(E{u(x+2)}) (14)x1(x,2)-x2(x,2)=H2(E{u2(x+2)})-11(E{u1(x+2(}) where i=u2(x+2). If u(u2(t) is [strictly] concave, then(by Jensen's inequality) (15)E{u1(u2(D)}≤[<]u1(u21(E{}) Substituting(15)in(14), we obtain(b)
RISK AVERSION To show that(a)implies(d), note that (16) "1(2()=4(21(t) 2(2(t) which is [strictly] decreasing if(and only if) log ui(x)/u2(x)is. The latter follows from(a)and (17) 1(x) =r2(x)-r1(x) u2(x) That(c)is implied by(e)follows immediately upon writing( 9)in the form (18)1-p(x,h)以(x+h)-u(x) P ( x, h) u (x)-ui x-h) To show that(a)implies(e), integrate(a) from w to x, obtaining (19) n1()[>]-1og"(x)frw2(m) 1(w)-u1(U) forU<w≤x u2(x)
JoHN W. PRATT 6. CONSTANT RISK AVERSION If the local risk aversion function is constant, say r(x)=c, then by (12) (25)u(x)~eifr(x)=c0 We turn now to(i). Notice that it amounts to a definition of strictly decreasing risk aversion in a global(as opposed to local)sense. On would hope that decreasing global risk aversion would be equivalent to decreasing local risk aversion r(x) The following theorem asserts that this is indeed so. Therefore it makes sense to speak of“ decreasing risk aversion” without the qualification" local”or“"gobl:” What is nontrivial is that r(x) decreasing implies I(x, 2)decreasing, inasmuch as r(x) pertains directly only to infinitesimal gambles. Similar considerations apply to the probability premium p(x, h) THEOREM 2: The following conditions are equivalent (a)The local risk aversion function r(x)is [strictly] decreasing