Question

Maria has $1 she can invest in two assets, A and B. A dollar invested in A has a 50-50 chance of returning 16 or 0 and a dollar invested in B has a 50-50 chance of returning 9 or 0. Marias utility over wealth is given by the function U(w) = ln(w + 1).

(A) Suppose the assets returns are perfectly negatively correlated. That is, when A has a positive return, B returns nothing, and vice versa.

1) Show that Maria is better off investing half her money in each asset now than when the assets returns were independent.

2) If she can choose how much to invest in each, how much does she invest in A? (again, at least the FOC)

Answer 1

a.

Independently:

For A:

Expected return = 0.5ln(16+1) + 0.5ln(0+1) = 1.416

For B:

Expected return = 0.5ln(9+1) + 0.5ln(0+1) = 1.151

Total = 1.416 + 1.151 = 2.567

If asset returns are perfectly negatively correlated

Expected return:

= 0.5ln(8+1) + 0.5ln(0+1) + 0.5ln(0+1) + 0.5ln(4.5+1)

= 3.9019.
Since 3.9019>2.567

Which shows Maria is better of investing half her money in each asset now when asset returns where independent.

b.

However say X proportion is invested in A, then (1-X) proportion is invested in B

Expected return (ER) = 0.5ln(16X+1) + 2xln(0+1) + 0.5ln(9(1-X)+1)

Max:

On solving for x
x=1.7952, 0.2534

X cannot be greater than 1, so X is o.2534

So Maria is better of investing 0.2534$ in A.