Page 3/4 3. An investor has $1,000 to invest into two types of share. If he inve

Question

Page 3/4 3. An investor has $1,000 to invest into two types of share. If he invests Sm in Share A, he will invest $(1,000-m) in Share B. An investment in Share A has 0.65 chance of doubling in value and a 0.35 chance of being lost altogether. An investment in Share B has 0.72 chance of doubling in value and 0.28 chance of being lost altogether. The chances associated with Shares A & B are independent. a. Find all outcomes in this investment. b. Determine the optimal value of m, if decision maker's utility function for a gain is e)- logx+3000)2 c. what would be the optimal value of", if his utility function was instead i(x)-(x + 3000)

Explanation / Answer

a.

Expected outcome of share A = $2m * 0.65 + $0 * 0.35 = $1.3m

Expected outcome of share B = $2(1000-m) * 0.72 + $0 * 0.28 = $2000 - $1.44m

Total expected outcome of investment = $1.3m + $2000 - $1.44m = $2000 - $0.14m

As, m can vary between $0 and $1000, the range of expected outcomes is ($1860, $2000)

b.

Expected outcome of share A = log(2m+3000) * 0.65 + log(0+3000) * 0.35 = 0.65log(2m+3000) + 0.35log(3000)

Expected outcome of share B = log(2(1000-m)+3000) * 0.72 + log(0+3000) * 0.28 = 0.72log(5000-2m) + 0.28log(3000)

Total expected outcome, E = 0.65log(2m+3000) + 0.72log(5000-2m) + 0.63log(3000)

Differentiating wrt m and equating with 0, we get

dE/dm = [2*0.65 / (2m+3000) ] - [2*0.72 / (5000-2m) ] = 0

1.3 (5000 - 2m) - 1.44 (2m + 3000) = 0

m (2.6 + 2.88) = 1.3 * 5000 - 1.44 * 3000

m = 397.8102

So, the optimal value of m is $397.8102

c.

Expected outcome of share A = (2m+3000)2 * 0.65 + (0+3000)2 * 0.35 = 0.65(2m+3000)2 + 0.35(3000)2

Expected outcome of share B = (2(1000-m)+3000)2 * 0.72 + (0+3000)2 * 0.28 = 0.72(5000-2m)2 + 0.28(3000)2

Total expected outcome, E = 0.65(2m+3000)2 + 0.72(5000-2m)2 + 0.63(3000)2

Differentiating wrt m and equating with 0, we get

dE/dm = 2*2*0.65(2m+3000) - 2*2*0.72(5000-2m) = 0

1.3m + 1950 - 3600 + 1.44m = 0

2.74m = 1650

m = 602.1898

So, the optimal value of m is $602.1898

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