4. (35 pts) An investor has $1,000 to invest into two types of share. If he inve

Question

4. (35 pts) An investor has $1,000 to invest into two types of share. If he invests Sm in Share A, he will invest S(1,000- m) in Share B. An investment in Share A has 0.7 chance of doubling in value and a 0.3 chance of being lost altogether. An investment in Share B has 0.6 chance of doubling in value and 0.4 chance of being lost altogether. The chances associated with Shares A & B are independent. 1) Find all outcomes in this investment. 2) Determine the optimal value of m, if decision maker's utility function for a gain is u) log(x3000)? 3) What would be the optimal value ofn, if his utility function was instead u(x)-(x + 3000)

Explanation / Answer

1)

Expected outcome of share A = $2m * 0.7 + $0 * 0.3 = $1.4m

Expected outcome of share B = $2(1000-m) * 0.6 + $0 * 0.4 = $1200 - $1.2m

Total expected outcome of investment = $1.4m + $1200 - $1.2m = $1200 + $0.2m

As, m can vary between $0 and $1000, the range of expected outcomes is ($1200, $1400)

b.

Expected outcome of share A = log(2m+3000) * 0.7 + log(0+3000) * 0.3 = 0.7log(2m+3000) + 0.3log(3000)

Expected outcome of share B = log(2(1000-m)+3000) * 0.6 + log(0+3000) * 0.4 = 0.6log(5000-2m) + 0.4log(3000)

Total expected outcome, E = 0.7log(2m+3000) + 0.6log(5000-2m) + 0.7log(3000)

Differentiating wrt m and equating with 0, we get

dE/dm = [2*0.7 / (2m+3000) ] - [2*0.6 / (5000-2m) ] = 0

1.4 (5000 - 2m) - 1.2 (2m + 3000) = 0

m (2.4 + 2.8) = 1.4 * 5000 - 1.2 * 3000

m = 3400 / 5.2 = 653.8462

So, the optimal value of m is $653.8462

c.

Expected outcome of share A = (2m+3000)2 * 0.7 + (0+3000)2 * 0.3 = 0.7(2m+3000)2 + 0.3(3000)2

Expected outcome of share B = (2(1000-m)+3000)2 * 0.6 + (0+3000)2 * 0.4 = 0.6(5000-2m)2 + 0.4(3000)2

Total expected outcome, E = 0.7(2m+3000)2 + 0.6(5000-2m)2 + 0.7(3000)2

Differentiating wrt m and equating with 0, we get

dE/dm = 2*2*0.7(2m+3000) - 2*2*0.6(5000-2m) = 0

1.4m + 2100 - 3000 + 1.2m = 0

2.6m = -900

m = -346.1538

As, optimal value of m is negative, which is not feasible. So, the optimal solution does not exist for this utility function.

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