(4.17) [Choice, finance] Suppose an investor has the utility function U(R,s) = R

Question

(4.17) [Choice, finance] Suppose an investor has the utility function

U(R,s) = R - 0.3s^2 (Here, R denotes the expected rate of return (R) of their

investment portfolio and s denotes the risk associated with that portfolio, but this

information is not necessary to solve this problem.)

a. Find a formula for the marginal rate of substitution in consumption (MRSC) of s

for R. [Hint: This is the absolute value of the slope of the indifference curve,

when R is on the vertical axis and s is on the horizontal axis.]

According to the Capital Asset Pricing Model, if there is a risk-free asset with a return of

4 percent, and if the market return is 10 percent with a standard deviation of 20 percent,

then the investor faces a constraint of R = 4 + (10-4)/20s or R = 4 + 0.4s

b. Compute the slope of the constraint, dR/ds .

c. Compute the values of R and s that this investor will choose. [Hint: Set your

answer to (a) equal to your answer to (b) and solve for s. Then insert this into

the constraint to find R.]

Explanation / Answer

U(R,s) = R - 0.3s^2

MUs = dU/ds = -0.6s

MUR = dU/dR = 1

MRSC: dR/ds = MUs/MUR = -0.6s

R = 4 + 0.4s

DR/ds = 0.4

-0.6s = 0.4

s = -2/3 = -0.67

R = 4 + 0.4*-2/3 = 3.73

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