MEASURES OF RISK-AVERSION

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1 Department of Economics, Universit of California, Davis Professor Giacomo Bonanno Ecn 03 Economics of Uncertaint and Information Lecture MESURES OF RISK-VERSION Can we answer the question: Which of two risk-averse individuals is the more riskaverse? Recall that a vnm utilit function transformation: if U is unique onl up to an affine U is a VNM utilit function representing an individual s preferences when V ( ) au ( ) b with a 0 and b 0 represents the same preferences. It follows that no meaning can be attached to the absolute value of utilit or even to its first derivative ( Marginal Utilit ) or second derivative, as all of these can be arbitraril changed b an appropriate choice of the constants a and b above. Thus, we cannot compare the first or second derivative of the utilit function of two individuals and draw conclusions about who is more risk averse. Nevertheless, the above is a sensible question and one with a read answer. First we define the risk-premium. Given a lotter with monetar prizes L, denote b E L the expected value of L. If U is the individual s vnm utilit-of-mone function, EU(L) denotes the expected utilit of L and U(E L ) the utilit of the expected value. Suppose that the individual is risk-averse. Then U(E L ) > EU(L). The risk premium associated with lotter L is that amount of mone r such that Let L be the lotter E pw p W W L p U(E L r) = EU(L). W. Then the expected value of L is p L ( ). The expected utilit of L is EU L pu W p U W ( ) ( ) ( ) ( ). B risk-aversion, EU ( L) U( E ), that is, pu ( W ) ( p) U ( W ) U pw ( p) W L Page of 7

2 U(W) U(W) U(exp value of L) EU(L) U(W) risk premium r W W W U(ELr) = EU(L) expected value of L, EL = pw+(-p)w For example, suppose that John s vnm utilit of mone function is m U ( m) 0m 00 (for 0 m,000) m m Then U ( m) , 000 risk averse. Page of 7 and U( m) 0 thus John is 5,000 $40 $600 Consider the lotter L 3. The expected value of this lotter is 4 4 EL = The expected utilit is EU ( L) U (40) U (600) 0(40) 0(600) while the utilit of the expected value is U(460) = 0(460) , 00 greater than the expected utilit (as expected, since he is risk averse). What is the risk premium associated with this lotter for John? It is the solution (with respect to r) to the following equation: The solution is r = U(460 r) = 9,7.96 [=EU(L)]

3 Note that, tpicall, the degree of risk aversion of an individual varies with her level of wealth. Consider, for example, a risk-averse individual who has the following utilit-of-mone function: U ( m) m. Suppose that the individual currentl has $m and is faced with the lotter that with probabilit takes awa $50 from her and with probabilit gives him $50: m 50 m 50. The expected value of this lotter is $m. If given a choice, the individual will prefer to keep her $m and not pla the lotter. What is the risk premium associated with this lotter? The answer is: it depends on what sum of mone she starts with. Suppose that she starts with m = 50. Then the lotter she faces 0 00 whose expected value is 50 and expected utilit is is EU The risk premium is the solution to 50 r 5 which is r = $5. This means that forcing her to pla the lotter is the same as taking awa $5 from her. Equivalentl, she would be willing to pa up to $5 to avoid the lotter. Suppose that she starts with m =,000. Then the lotter she faces is 950, 050 whose expected value is,000 and expected utilit is EU 950, The risk premium is the solution to, 000 r 3.63 which is r = $0.65. This means that forcing her to pla the lotter is the same as taking awa 6 cents from her. Equivalentl, she would be willing to pa up to 6 cents to avoid the lotter. u and Now let us compare two individuals. The next figure shows the utilit functions, v, of two individuals. Clearl, v is more concave than Page 3 of 7 u and intuitivel we would associate greater risk-aversion with v. Indeed the risk premium associated with v is higher than the risk premium associated with u.

4 utilit utilit function v u(max)=v(max) expected utilit (same for u and v) utilit function u u(min)=v(min) risk premium for v risk premium for u v(elrv) = Ev(L) u(elru) = Eu(L) min expected value 0 = pmin+(-p)max max We have that r v r u : the more risk-averse individual has the higher risk-premium (is prepared to give up more to avoid the risk) and therefore places a greater value on the certaint. For example, suppose that manda has the following vnm utilit-of-mone function: U(m) = m. Then U( m) and U( m) 0 so that manda is also risk 3 m 4 m $40 $600 averse. Consider again the lotter L 3 which has an expected value of 4 4 $460. The utilit of the expected value is U (460) The expected utilit of the lotter is 3 EU ( L) , less than the utilit of the expected 4 4 value as expected from risk aversion. For manda the risk premium associated with this lotter is the solution (with respect to r) to the following equation: U(460 r) = 9.95 [=EU(L)] The solution is r = Thus manda is much more risk-averse than John. Page 4 of 7

5 Note that the risk premium associated with a lotter in invariant to an affine transformation of the utilit function. In fact, let L be a lotter with prizes x,, x n and respective probabilities p,, p n. Let E L be the expected value of L and let U be a von Neumann-Morgenstern utilit-of-mone function. Now, if r is the risk-premium associated with L and U, then r is the solution to U ( EL r) EU ( L) where EU ( L) pu ( x )... pnu ( xn ). Multipl both sides of the above equation b a > 0 and add b to both sides to obtain au ( E r) b a EU( L) b. L Define W ( x) au ( x) b. Then the LHS of the above equation is W(E L r) while the RHS is EW(L). Thus r is also the risk premium associated with lotter L and utilit function W. B inspection of U and V it is clear that at, U is more negative than V and so greater risk aversion is associated with greater concavit. Unfortunatel, we cannot simpl take the second derivative as measure of risk-aversion. We cannot simpl sa that U is more risk-averse than V because U V Remember we can alwas linearl transform U or V and therefore arbitraril change the size of the second derivative at each. For example, if we transform U according to U a bu b then U bu () So that the individual is suddenl b times more risk-averse than before! To overcomes this we sa that U is more risk-averse than V If U 0, V 0 nd U V U V () If this is true, then a linear transformation of u according to () gives Page 5 of 7

6 bu u v bu u v and () is preserved. For an individual with a utilit function risk-aversion is given b R u / u n individual is then risk-averse if and risk-preferring if is R. u the rrow-pratt measure of absolute is positive, risk-neutral if is negative. person is more risk-averse the larger is zero For instance, if U(m) = m then d m dm 3 U( m) 4 ( m) m 0. U( m) m m and m d m dm 4 3 m, so that disadvantage with is that its value depends on the units chosen for in general. Consider the following example. The utilit function is given b Hence u n 0 so that u 0 and u 0 If is measured in dollars, if measured in cents 0.0 Page 6 of 7 R. It would be preferable if a measure of risk-aversion took the form of a pure number, such as an elasticit. Happil, this is achieved b the rrow-pratt measure of relative riskaversion: u RR R u

7 In the above example, For instance, if U(m) = That and R RR m then = regardless of 's units. ( m) m and thus RR ( m) m( m). R R depend on is an indicator that the are local measures of risk-aversion. To see the importance of this we return to the last Figure and reinterpret the value 0 = 30 as an amount alread in our possession and we are given the opportunit of plaing a game in which we face a chance of gaining or losing 0. risk-averter will turn down this opportunit and keep the 30, and the risk-premium shows how much the individual will pa to avoid the gamble. Suppose we start with an amount greater than 30; are we more or less likel to avoid the gamble of gaining or losing 0? If we suppose we had a wealth of,000 alread in our possession ( 0 =,000), then a gamble involving the possible loss of 0 surel involves less of a threat! This effect is summarized as follows: risk-averter who fears fair gambles less when wealth is greater has a utilit function which exhibits decreasing absolute risk-aversion That is, R 0 lthough not a universal rule, man people appear to exhibit decreasing risk-aversion with increased wealth. The fact that an individual degree of risk-aversion ma var with justifies our notation of. Page 7 of 7