Graduate Program Finance and Monetary Economics Summer 2005 Asset Pricing Problem Set 1. Solutions.

Transcription

1 Graduate Program Finance and Monetary Economics Summer 2005 Asset Pricing Problem Set 1. Solutions. Problem 1. Consider a two-period economy with a representative agent that has the following utility functions: Uc 1, c 2 ) = lnc 1 ) + βe [ln c 2 )] Assume that at date 2 two states of nature are possible and the endowment structure is: t=1 t=2 s = 1 s = 2 Endowment y t s) The state probabilities are π1) = 0.6 and π2) = 0.4 for state s = 1 and s = 2 respectively. The discount factor is β = 0.9. There is a competitive market in A.-D. securities for every state s. Suppose the agent starts out with zero assets. Let b 2 s) be the individual s net purchase of state s A.-D. securities on date 1. The prices of A.-D. securities are: p1) = 0.8, p2) = 0.6. a) What is the agent s consumption across time and states? b) What is the agent s purchase of the A.-D. securities? 1

2 Problem 1. Solution. The agent s problem is: s.t. max ln c 1) + β [π1) ln c 2 1)) + π2) ln c 2 2))] c 1,c 2 1),c 2 2) c 1 + c 2 1)p1) + c 2 2)p2) = y 1 + y 2 1)p1) + y 2 2)p2) c 1 0, c 2 1) 0, c 2 2) 0 Plugging the constraint into the objective and replacing c 1 the agent s problem becomes: max ln y 1 p1)b1) p2)b2))+β [π1) ln y 2 1) + b 2 1)) + π2) ln y 2 2) + b 2 2))] b 2 1),b 2 2) Substituting the respective values, the FOCs are: b 2 1) : b 2 2) : = 3 0.8b 2 1) 0.6b 2 2) 6 + b 2 1) = 3 0.8b 2 1) 0.6b 2 2) 7 + b 2 2) Together with market clearing conditions y 2 1) = c 2 1) b 2 1) = 6; y 2 2) = c 2 2) b 2 2) = 7 ) these FOCs give the following solution to our maximization problems: The agent s purchase of the A.-D. securities: The agent s consumption across time and states: t=1 t=2 θ = 1 θ = 2 Consumption b 2 1) = b 2 2) =

3 Problem 2. Consider the following utility functions defined over wealth W ): 1) UW ) = 1 W 2) UW ) = lnw ) 3) UW ) = W γ 4) UW ) = exp γw ) 5) UW ) = W γ γ 6) UW ) = αw βw 2 a) Check that they are well-behaved U > 0, U < 0), or state restrictions on the parameters so that they are utility functions 1)-6)). For utility function 6), take positive α and β, and give the range of wealth over which the utility function is well behaved. b) Compute the absolute and relative risk aversion coefficients. c) What is the effect of the parameter γ when relevant)? d) Classify them as increasing/decreasing risk aversion utility functions both absolute and relative). Problem 2. Solution. Answers to a), b), c) and d) are given together here: 1) UW ) = 1 W U W ) = 1 W 2 > 0 U W ) = 2 W 3 < 0 R A = 2 W R R = 2 W = 2 W 2 < 0 3

4 2) UW ) = lnw ) U W ) = 1 W > 0 U W ) = 1 W 2 < 0 R A = 1 W R R = 1 W = 1 W 2 < 0 3) UW ) = W γ 4) UW ) = exp γw ) U W ) = γw γ 1 > 0 γ > 0 U W ) = γγ + 1)W γ 2 < 0 R A = γ + 1 W R R = γ + 1 W = γ + 1 W 2 < 0 5) UW ) = W γ γ U W ) = γ exp γw ) > 0 γ > 0 U W ) = γ 2 exp γw ) < 0 R A = γ R R = γw R R W = γ > 0 4

5 U W ) = W γ 1 > 0 6) UW ) = αw βw 2, α > 0, β > 0 U W ) = γ 1)W γ 2 < 0 γ < 1 R A = 1 γ W R R = 1 γ W = γ 1 W 2 < 0 U W ) = α 2βW > 0 W < α 2β U W ) = 2β < 0 R A = 2β α 2βW > 0 R R = 2β α 2βW W > 0 W = 4β 2 α 2βW ) 2 > 0 R R W = R AW ) W = W W + R A > 0 Note that γ controls for the degree of risk aversion. We check it with the derivative of R A and R R w.r.t. γ. UW ) = W γ γ = 1 W R R γ = 1 UW ) = exp γw ) γ = 1 R R γ = 1 W UW ) = W γ γ γ = 1 R R W γ = 1 In the last utility function, we should better use γ = 1 θ, so that UW ) = W γ γ = W 1 θ 1 θ θ = 1 W R R θ = 1. After this change, every derivative w.r.t. θ is positive. If we increase θ, we increase the level of risk aversion both absolute and relative). 5

6 Problem 3. There is an individual with a well-behaved utility function, and initial wealth W. Let a lottery offer a payoff of G with probability π and a payoff of B with probability 1 π. a) If the individual already owns this lottery denote the minimum price he would sell it for by P s. Write down the expression P s has to satisfy. b) If he does not own it, write down the expression P b the maximum price he would be willing to pay for it) has to satisfy. c) Assume now that π = 1 2, W = 10, G = 26, B = 6, and the utility function is UW ) = W 1/2. Find buying and selling prices. Are they equal? What about the case when the agent is risk-neutral? Problem 3. Solution a) If he already owns the lottery, P s must satisfy or UW + P s ) = πuw + G) + 1 π)uw + B) P s = U 1 πuw + G) + 1 π)uw + B)) W b) If he does not own the lottery, the maximum he would be willing to pay, P b, must satisfy : UW ) = πuw P b + G) + 1 π)uw P b + B) c) Assume now that π = 1 2, W = 10, G = 26, B = 6, and the utility function is UW ) = W 1/2. Find buying and selling prices. Are they equal? What about the case when the agent is risk-neutral? P b satisfies: With UW ) = W 1/2 : U10) = 1 2 U10 P b + 26) U10 P b + 6) 6

7 10 1/2 = P b) 1/ P b) 1/ = 36 P b ) 1/ P b ) 1/2 P b 13.5 P s satisfies: 10 + P s ) 1/2 = )1/ )1/2 2 = = P s ) 1/2 = P s ) = 25 Clearly, P b < P s. If the agent were risk neutral, To check it out, assume Ux) = x P s = 15 P b = P s = πg + 1 π)b P s : UW + P s ) = πuw + G) + 1 π)uw + B) W + P s = πw + G) + 1 π)w + B) = W + πg + 1 π)b P s = πg + 1 π)b P b : UW ) = πuw P b + G) + 1 π)uw P b + B) W = πw P b + G) + 1 π)w P b + B) P b = πg + 1 π)b 7

8 Problem 4. Suppose that a risk-averse individual can only invest in two risky securities A and B, whose future returns are described by identical but independent probability distributions. How should he allocate his given initial wealth normalized to 1 for simplicity) among these two assets so as to maximize the expected utility of next period wealth? Problem 4. Solution. Let R A and R B denote the gross returns on the two assets. E U a R A + 1 a) R )) B ae > E U ) )) au RA + 1 a)u RB = )) )) RA + 1 a)e U RB = )) E U RA = )) E U RB since the utility function is concave and gross asset returns are identically distributed. This result means that the investor is going to invest in both securities it is never optimal in this situation to invest only in one of the two securities. He thus chooses to diversify. Moreover, if the investor cares only about the first two moments he will invest equal amounts in the assets to minimize variance. To show this note that the expected return on the portfolio is constant independently of the chosen allocation. A meanvariance investor will thus choose a to minimize the variance of the portfolio. The latter is a 2 var R A +1 a) 2 var R B = var R A 2a a ) = var R A 1 + 2a 2 a) ) Minimizing the portfolio variance thus amounts to minimizing the quantity a 2 a) which requires selecting a =

9 Problem 5. Consider an agent with a well-behaved utility function who must balance his portfolio between a riskless asset and a risky asset. The first asset, with price has the certain payoff z 1 while the second asset, with price p 2 pays off z 2 which is a random variable. The agent has an initial wealth W 0 and he only cares about next period. Assuming that he holds x 1 units of the riskless asset and x 2 units of risky asset; a) Write down his expected utility function in terms of W 0, x 1, x 2, z 1, and z 2. Write down his budget constraint as well. Find the equation the FOC) that the optimal demand for the risky asset has to satisfy. b) Assume now that his utility function is of the form : UW ) = a be AW, with a, b > 0. Show that the demand for the risky asset is independent of the initial wealth. Explain intuitively why this is so. Problem 5. Solution. a) max x 1,x 2 EU x 1 z 1 + x 2 z 2 ) s.t. x 1 + p 2 x 2 W 0 Since we assume U ) > 0, x 1 + p 2 x 2 = W 0, thus x 1 = W 0 p 2 x 2. The problem then can be written as: max EU x 2 The necessary and sufficient F.O.C. is: [ ) ] W0 p 2 x 2 z 1 + x 2 z 2 [ ) ] EU W0 p 2 x 2 z 1 + x 2 z 2 z 2 p ) 2 z 1 = 0 9

10 b) Suppose UW ) = a be AW. The above equation becomes: equivalently, { [ AbE exp A W0 p 2 x 2 ) ] z 1 Ax 2 z 2 z 2 p )} 2 z 1 = 0 { [ AbE exp A W ] [ 0z 1 exp A p ] 2x 2 z 1 Ax 2 z 2 z 2 p )} 2 z 1 = 0 The first term which contains W 0 can be eliminated from the equation. The intuition for this result is in fact that the stated utility is CARA, that is, the rate of absolute risk aversion is constant and independent of the initial wealth. 10