Question

Suppose there are two assets, one is risk-free and one is risky. The risk-free asset has a sure rate of return rj, the risky asset has a random rate of return r. Suppose the utility function of an investor is U(x) =--. The initial wealth is wo, the dollar amount invested in the risky asset is θ. r is normally distributed with mean μ and variance σ2. Based on the maximum utility framework, find the optimal investment strategy 6. (25 points)

Answer 1

Given,

Risk free assets as r_f , risky one rate of return is r and wealth = W_0

utility function = $-e^-^4^x/4$

OPTIMUM INVESTMENT STRATEGY AT θ:

Theta means decrease in the value of investment over a period of time.

thereby , assume the following pattern of X = 0,-1,-2,

U(1) = $-e^-^4^(^1^)/4$ = -0.004579

U(0) = $-e^-^4^(^0^)/4$ = -0.25

U(-1) = $-e^-^4^(^-^1^)/4$ = -13.65

u(-2) = $-e^-^4^(^-^2^)/4$ = -745.24

Mu = mean of the normal distribution table and variance is the σ^2.

as the normal distribution, lies between 0 to θ i.e 0 to -2 , mean is (1-745.24)/2 = -372.12

since the variance lies between 3 times on either of mean = ((1-745.24)/3)^2= 61543.6864

Thereby the Z value = Z= (X-μ)/σ^2 = when X= (1), W_0, then Z=1-(-374.12)/61543.6864 = 0.006095

when X=  0.367879 W_-1 , then Z =  (-13.65-(-372.12) )/61543.6864 = 0.005824

when X= 0.135335, W_2, then Z =(-745.24-(-372.12))/61543.6864 = -0.0060627

THEREBy , the optimum investment strategy is INVEST IN risk free entire value of Wealth.