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On the magnitude of relative risk aversion
Economics Letters 18 (1985) 125-128 Jorth-Holland
DN THE MAGNITUDE
125
OF RELATIVE RISK AVERSION
E.K. CHOI and C.F. MENEZES University of Missouri, Columbia, MO 65211, USA Received 16 October 1984
rhis paper shows how the exact value of relative risk aversion can be found if it is constant. lower bound can be determined using a new risk aversion measure called probability markup.
For general utility functions a
1. Introduction
In recent years many economists have found that the qualitative content of their expected utility maximization models crucially depends on relative risk aversion (RRA). For example, authors such as Hahn (1970), and Rothschild and Stiglitz (1971) state their propositions in terms of whether RRA exceeds, is equal to, or falls short of unity, but do not advance a conjecture about which possibility is most likely to prevail. ’ The opposing hypotheses - RR4 > 1 and RRA < 1 - have been widely used in theoretical models [e.g., Champemowne (1969) and Azariadis (1978)]. Empirical studies have not been conclusive, since estimates of RR4 range from 0.05 to over 1000 [e.g., Binswanger (1981) and Schluter and Mount (1976)]. This paper first shows how knowledge about Arrow’s probability premium can be used to determine the exact value of RIU for constant relative risk aversion (CR&!) utility functions. We then define a new risk aversion measure called probability markup and use its properties to obtain a lower bound for RRA for general utility functions. 2. Relative risk aversion for CRRA utility functions
Arrow (1971) has formulated a general measure of risk aversion for the class of binary prospects (w + z), where w is the decision maker’s initial income and z = [h, -h; p, 1 - p] is a bet to gain or lose a fixed amount h E (0, w] with probability p and (1 - p) respectively. Let j be the probability such that the individual is indifferent between the status quo and the risk 2 = [h, -h; b, 1 -$I. The value of fi is given by the equation @4(w+h)+(l-B)u(w--h)=u(w), where u( .) is a strictly increasing and concave von Neumann-Morgenstem
(1) utility function. An
’ As Arrow has observed, boundedness of utility function implies that RR4 cannot tend to a limit below (above) one as income approaches infinity (zero). Stiglitz (1969) has noted that this mathematical proposition does not provide any meaningful constraints on the magnitude of RR4 in any empirically relevant range of income. Arrow (1971, p. 98) conjectured that ‘broadly speaking the relative risk aversion must hover around one’.
0165-1765/85/$3.30
Q 1985, Elsevier Science Publishers B.V. (North-Holland)
126
E. K. Choi, C. F. Menezes / Magnitude of relative risk aversion
intuitively meaningful measure of risk aversion is given by the probability premium defin’ - l/2. Many empirical studies assume CRRA and infer its value from observed consumption, invesl or portfolio choices. These studies have produced a wide range of estimates for RRA. The ch CRRA utility functions is defined by p =j
U(W) = wi-a,
O
=ln w, =
-_w
I-R
,
R>l.
Let s = h/w. The probability premium for CRRA utility function is then given by 1 -(l/2)(1 P(S,
+ s)‘-R -(l/2)(1
- s)‘+
R# 1,
R)=
(1 +s)‘-R-(l = (l/2)
ln(1 - s) -t-(l/2) ln(l-s)-ln(l+s)
-s)‘-R
’
ln(1 + s) ’
R= 1.
Since there is only one parameter R to determine, knowledge of the probability of wir $3 p + l/2 for any value of s is sufficient to ascertain the exact value of R. 2 For example, the 1 of R corresponding to jr(O.01) = 0.5025 is unity. That is, an individual who demands a probabili winning $(O.Ol) = 50$% to be indifferent between a gamble to win or lose 1% (s = 0.01) o income and the status quo must have a coefficient of relative risk aversion R = 1. Similarly, I individual demands p(O.01) = 0.711090, then his RRA is 90, and conversely.
3. Lower bounds for RRA for general utility functions Obviously, eq. (3) cannot be used to determine the value of RR4 for non-constant relative aversion utility functions. However, a lower bound for RRA for general utility functions cal found using a property of a new risk aversion measure called probability markup. Although there one-to-one correspondence between the probability premium and the probability markup, in the. the former is linear but the latter is concave in (relative) gamble size. This concavity of probat markup is crucial to finding a lower bound for RRA from knowledge about the probabilit winning. Arrow’s formulation implies an additive decomposition of the probability of winning, j? = p + We define the probability markup using a different decomposition of a. Specifically, j=1/(2-A). For a risk neutral individual A = 0 since he is indifferent between the status quo and a one-in. chance of gaining a fixed amount h. In contrast, a risk averter would insist on at least a
2 A scanning program is required to obtain the value of R for a given value of j(O.01). For CRRA utility functions, a showing the values of R corresponding to various probabilities of winning for s = 0.01 can be obtained from the au1 upon request.
E. K. Choi, CF. Menezes / Magnitude of relative
127
riskaversion
bne-in-(2 - X) chance of gaining h. Let c = 1 - p. Since h = ( j - c)/j is the percentage margin of he probability of gain over the probability of loss, we call A the probability markup. Consider a relative gamble 2 = [ w(1 + s), w(1 7 s); q, 1 - q], where s = (h/w) E (0, 11. Let 4 be Iefined by iU[W(l +s)] +(1-a)u[w(1
-s)]
=u(w).
(5)
Vote that since s = h/w, d( w, s) = fi( w, h), where i, is given by (1). To obtain the counterpart of (4), .et ~(w, s) be defined by 4 = l/(2 - p).
(6)
The function ~(w, s) is the probability markup for relative risk, and is related to X(w, h) in an obvious way, i.e., X(w, h) = X( w, ws) = p( w, s). We now present the convergence properties of the probability markup. Proposition 1. L-et u(w) be three times differentiable von Neumann-Morgenstern let A( w, h) and ~(w, s) be defined by (4) and (6). Then, F$h( w, h)/h = A( w)
and
utility function, and
~III~(W, s)/s=R(w).
(7) 0)
Eq. (6) defines a class of binary prospects whose expected utility is equal to the utility of the initial income. Let U(s, CL)= (l/2 - p)u[w(l + s)] + [(l - ~)/(2 - p)]u[w(l - s)] denote the expected utility of the risk Z = [w(l + s), w(1 - s); (l/2 - p), (1 - ~)/(2 - k)] for a giuen value of w. The locus of points (s, p) such that U(s, CL)= u(w) defines an indifference curve Z, = {(s, cc) : U(s, p) = u(w)}. Note that since Q(O) is not unique, ~(0) is not uniquely defined. However, since lim, I a4 = l/2, it follows that lim slo~(w, s)= 2 - (l/lim,1,6)= 0. This implies that the indifference curve Z, emanates from the origin (s, ~_r)= (0,O). We now present two important properties of the indifference curve. Proposition 2.
Let ~(w, s) be the probability markup for relative risks as defined in (6). Then,
p$P,(W,s) = R(w)
and
$p,,(w,
s)= -[R(w)]‘cO.
(9)00)
We have shown that the indifference curve emanates from the origin, is initially concave and that its slope at the origin is the RRA, i.e., R(w) = lim s 1,+/s. Let R(s) = p/s s (24 - l)@. It is the slope of the ray from the origin to the point (s, 11)on the indifference curve. Clearly, R(s) is a lower bound for the unknown value of R(w), given that the indifference curve remains concave in (0, s). Recall that an individual with a CRRA utility function has R = 90 if he demands a probability of winning Q(O.01)= 0.711090. If this individual’s RRA is not constant, then we find that his RR4 is at least 59.371. For R(w) to be less than unity, the value of Q(O.01) must be between 50% and 50f%, implying that the individual is ‘almost’ risk neutral. Ret erences
Arrow,K.J., 1971, Essays in the theoryof risk-bearing(Markham,Chicago,IL). Azariadis, C., 1978, Escalator clauses and the allocation of cyclical risks, Journal of Economic Theory 18,119-15%
128
E. K. Choi, CF. Menezes / Magnitude of relative risk aversion
Binswanger, H.P., 1981, Attitudes toward risk: Theoretical implications of an experiment in rural India, Economic Jour 867-890. Champemowne, D.G., 1969, Uncertainty and estimation in economics, Vol. 3 (Holden-Day, San Francisco, CA). Hahn, F., 1970, Savings and uncertainty, Review of Economic Studies 37, 21-24. Rothschild, M. and J. Stiglitz, 1971, Increasing risk II: Its economic consequences, Journal of Economic Theory 3, 66Schluter, M.G. and T.D. Mount, 1976, Some management objectives of risk aversion in the choice of cropping patterns district, India, Journal of Development Studies 12, 246-261. Stiglitz, J., 1969, Book review of: K.J. Arrow, Aspects of the theory of risk bearing (Yrjo Jahnsson Lectures) Econometr 741-743.
DN THE MAGNITUDE
125
OF RELATIVE RISK AVERSION
E.K. CHOI and C.F. MENEZES University of Missouri, Columbia, MO 65211, USA Received 16 October 1984
rhis paper shows how the exact value of relative risk aversion can be found if it is constant. lower bound can be determined using a new risk aversion measure called probability markup.
For general utility functions a
1. Introduction
In recent years many economists have found that the qualitative content of their expected utility maximization models crucially depends on relative risk aversion (RRA). For example, authors such as Hahn (1970), and Rothschild and Stiglitz (1971) state their propositions in terms of whether RRA exceeds, is equal to, or falls short of unity, but do not advance a conjecture about which possibility is most likely to prevail. ’ The opposing hypotheses - RR4 > 1 and RRA < 1 - have been widely used in theoretical models [e.g., Champemowne (1969) and Azariadis (1978)]. Empirical studies have not been conclusive, since estimates of RR4 range from 0.05 to over 1000 [e.g., Binswanger (1981) and Schluter and Mount (1976)]. This paper first shows how knowledge about Arrow’s probability premium can be used to determine the exact value of RIU for constant relative risk aversion (CR&!) utility functions. We then define a new risk aversion measure called probability markup and use its properties to obtain a lower bound for RRA for general utility functions. 2. Relative risk aversion for CRRA utility functions
Arrow (1971) has formulated a general measure of risk aversion for the class of binary prospects (w + z), where w is the decision maker’s initial income and z = [h, -h; p, 1 - p] is a bet to gain or lose a fixed amount h E (0, w] with probability p and (1 - p) respectively. Let j be the probability such that the individual is indifferent between the status quo and the risk 2 = [h, -h; b, 1 -$I. The value of fi is given by the equation @4(w+h)+(l-B)u(w--h)=u(w), where u( .) is a strictly increasing and concave von Neumann-Morgenstem
(1) utility function. An
’ As Arrow has observed, boundedness of utility function implies that RR4 cannot tend to a limit below (above) one as income approaches infinity (zero). Stiglitz (1969) has noted that this mathematical proposition does not provide any meaningful constraints on the magnitude of RR4 in any empirically relevant range of income. Arrow (1971, p. 98) conjectured that ‘broadly speaking the relative risk aversion must hover around one’.
0165-1765/85/$3.30
Q 1985, Elsevier Science Publishers B.V. (North-Holland)
126
E. K. Choi, C. F. Menezes / Magnitude of relative risk aversion
intuitively meaningful measure of risk aversion is given by the probability premium defin’ - l/2. Many empirical studies assume CRRA and infer its value from observed consumption, invesl or portfolio choices. These studies have produced a wide range of estimates for RRA. The ch CRRA utility functions is defined by p =j
U(W) = wi-a,
O
=ln w, =
-_w
I-R
,
R>l.
Let s = h/w. The probability premium for CRRA utility function is then given by 1 -(l/2)(1 P(S,
+ s)‘-R -(l/2)(1
- s)‘+
R# 1,
R)=
(1 +s)‘-R-(l = (l/2)
ln(1 - s) -t-(l/2) ln(l-s)-ln(l+s)
-s)‘-R
’
ln(1 + s) ’
R= 1.
Since there is only one parameter R to determine, knowledge of the probability of wir $3 p + l/2 for any value of s is sufficient to ascertain the exact value of R. 2 For example, the 1 of R corresponding to jr(O.01) = 0.5025 is unity. That is, an individual who demands a probabili winning $(O.Ol) = 50$% to be indifferent between a gamble to win or lose 1% (s = 0.01) o income and the status quo must have a coefficient of relative risk aversion R = 1. Similarly, I individual demands p(O.01) = 0.711090, then his RRA is 90, and conversely.
3. Lower bounds for RRA for general utility functions Obviously, eq. (3) cannot be used to determine the value of RR4 for non-constant relative aversion utility functions. However, a lower bound for RRA for general utility functions cal found using a property of a new risk aversion measure called probability markup. Although there one-to-one correspondence between the probability premium and the probability markup, in the. the former is linear but the latter is concave in (relative) gamble size. This concavity of probat markup is crucial to finding a lower bound for RRA from knowledge about the probabilit winning. Arrow’s formulation implies an additive decomposition of the probability of winning, j? = p + We define the probability markup using a different decomposition of a. Specifically, j=1/(2-A). For a risk neutral individual A = 0 since he is indifferent between the status quo and a one-in. chance of gaining a fixed amount h. In contrast, a risk averter would insist on at least a
2 A scanning program is required to obtain the value of R for a given value of j(O.01). For CRRA utility functions, a showing the values of R corresponding to various probabilities of winning for s = 0.01 can be obtained from the au1 upon request.
E. K. Choi, CF. Menezes / Magnitude of relative
127
riskaversion
bne-in-(2 - X) chance of gaining h. Let c = 1 - p. Since h = ( j - c)/j is the percentage margin of he probability of gain over the probability of loss, we call A the probability markup. Consider a relative gamble 2 = [ w(1 + s), w(1 7 s); q, 1 - q], where s = (h/w) E (0, 11. Let 4 be Iefined by iU[W(l +s)] +(1-a)u[w(1
-s)]
=u(w).
(5)
Vote that since s = h/w, d( w, s) = fi( w, h), where i, is given by (1). To obtain the counterpart of (4), .et ~(w, s) be defined by 4 = l/(2 - p).
(6)
The function ~(w, s) is the probability markup for relative risk, and is related to X(w, h) in an obvious way, i.e., X(w, h) = X( w, ws) = p( w, s). We now present the convergence properties of the probability markup. Proposition 1. L-et u(w) be three times differentiable von Neumann-Morgenstern let A( w, h) and ~(w, s) be defined by (4) and (6). Then, F$h( w, h)/h = A( w)
and
utility function, and
~III~(W, s)/s=R(w).
(7) 0)
Eq. (6) defines a class of binary prospects whose expected utility is equal to the utility of the initial income. Let U(s, CL)= (l/2 - p)u[w(l + s)] + [(l - ~)/(2 - p)]u[w(l - s)] denote the expected utility of the risk Z = [w(l + s), w(1 - s); (l/2 - p), (1 - ~)/(2 - k)] for a giuen value of w. The locus of points (s, p) such that U(s, CL)= u(w) defines an indifference curve Z, = {(s, cc) : U(s, p) = u(w)}. Note that since Q(O) is not unique, ~(0) is not uniquely defined. However, since lim, I a4 = l/2, it follows that lim slo~(w, s)= 2 - (l/lim,1,6)= 0. This implies that the indifference curve Z, emanates from the origin (s, ~_r)= (0,O). We now present two important properties of the indifference curve. Proposition 2.
Let ~(w, s) be the probability markup for relative risks as defined in (6). Then,
p$P,(W,s) = R(w)
and
$p,,(w,
s)= -[R(w)]‘cO.
(9)00)
We have shown that the indifference curve emanates from the origin, is initially concave and that its slope at the origin is the RRA, i.e., R(w) = lim s 1,+/s. Let R(s) = p/s s (24 - l)@. It is the slope of the ray from the origin to the point (s, 11)on the indifference curve. Clearly, R(s) is a lower bound for the unknown value of R(w), given that the indifference curve remains concave in (0, s). Recall that an individual with a CRRA utility function has R = 90 if he demands a probability of winning Q(O.01)= 0.711090. If this individual’s RRA is not constant, then we find that his RR4 is at least 59.371. For R(w) to be less than unity, the value of Q(O.01) must be between 50% and 50f%, implying that the individual is ‘almost’ risk neutral. Ret erences
Arrow,K.J., 1971, Essays in the theoryof risk-bearing(Markham,Chicago,IL). Azariadis, C., 1978, Escalator clauses and the allocation of cyclical risks, Journal of Economic Theory 18,119-15%
128
E. K. Choi, CF. Menezes / Magnitude of relative risk aversion
Binswanger, H.P., 1981, Attitudes toward risk: Theoretical implications of an experiment in rural India, Economic Jour 867-890. Champemowne, D.G., 1969, Uncertainty and estimation in economics, Vol. 3 (Holden-Day, San Francisco, CA). Hahn, F., 1970, Savings and uncertainty, Review of Economic Studies 37, 21-24. Rothschild, M. and J. Stiglitz, 1971, Increasing risk II: Its economic consequences, Journal of Economic Theory 3, 66Schluter, M.G. and T.D. Mount, 1976, Some management objectives of risk aversion in the choice of cropping patterns district, India, Journal of Development Studies 12, 246-261. Stiglitz, J., 1969, Book review of: K.J. Arrow, Aspects of the theory of risk bearing (Yrjo Jahnsson Lectures) Econometr 741-743.