Question

Consider the decision problem of investing an amount of wealth WW into a risky asset with return

R={0.1with probabilityp−0.05with probability1−pR={0.1with probabilityp−0.05with probability1−p

and into a risk-less asset with risk-free interest rate r=2%r=2%. You are a risk averse investor with utility function

U(WT)=10+ln(WT)U(WT)=10+ln⁡(WT)

where WTWT is the amount of wealth at the end of the investment.

a) Find the optimal allocation in risky and risk-less assets as a function of the probability pp and the initial amount of wealth WW invested.
b) Compute the optimal allocation for a probability value p=0.5p=0.5.
c) What is the lower bound of the probability pp for investing a positive share into the risky asset?
d) What is the lower bound of pp for starting to borrow money at the risk-less interest rate and invest an amount larger than WW in the risky asset?

Answer 1

a) Let W be the tota initial amount invested.

Let x be the proportion of wealth invested in risky asset, hence (1-x) is the proportion of wealth invested in risk free asset.

Using the given information, the expected total wealth at the end of time t is given by,

Expected Amount in risky asset + amount in risk free aset

= [p*1.1*x*W + (1-p)*0.95*x*W] + (1-x)*1.02*W

Hence, utility after time t is

U(Wt) = 10 + ln{ 1.1*x*W + (1-x)*1.02*W } with probability p

i.e. U(Wt) = 10 + ln{ 0.08*x*W + 1.02*W } with probability p

U(Wt) = 10 + ln{ 0.95*x*W + (1-x)*1.02*W } with probability 1-p

i.e. U(Wt) = 10 + ln{ - 0.07*x*W + 1.02*W } with probability 1-p

Expected utility = p* [10 + ln{ 0.08*x*W + 1.02*W }] + (1-p) * [10 + ln{ - 0.07*x*W + 1.02*W }]

10 + p* ln{ 0.08*x*W + 1.02*W } + (1-p) * ln{ - 0.07*x*W + 1.02*W }

For finding optimum value of expected utility, differentiate wrt x and equate to zero.

If we take double derivative of expected utility wrt x, and equalte it at the above x, we will see that it is less than zero.

Hence expected utility is maximum when

x = (15.3p-7.14)/0.56

The proportion is not deoendent on amount W invested.

b) p=0.5

x = (7.65-7.14)/0.56 = 0.9107

i.e. 91.07% wealth should be invested in risky asset and 8.93% in risk free asset.

c) Now 0<x<1, i.e. positive allocation in both risky and riskfree asset.

Hence 0<15.3p-7.14<0.56

(7.14/15.3)<p

i.e. p>0.4666

Lower bound of p is 46.66%

d) Now, we can have negative allocation to risk free asset, i.e. borrow at risk free rate to invest more than 100% in risky asset.

If x>1, 15.3p-7.14>0.56

i.e. p>7.7/15.3 = 0.5033

i.e. Lower bound of p for borrowing at risk free rate to invest more than 100% in risky asset is 50.33%