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Baylor University Hankamer School of Business Department of Finance, Insurance & Real Estate Risk Management Name: SOLUTIONS Dr. Garven Problem Set 4 Show your work and write as legibly as possible. Good luck! Miles and John are considering two mutually exclusive risky investments, 1 and 2, with payoffs given by: W 1 ,s =    $20 with probability 20% $60 with probability 50% $100 with probability 30% and W 2 ,s = $40 with probability 55% $80 with probability 45% Suppose that Miles’s initial wealth W 0 = $0, and his utility U ( W ) = W . 5 . A. Calculate Miles’s expected utility ( E ( U ( W )) for both investments. SOLUTION: E ( U ( W 1 ,s )) = . 2 √ 20 + . 5 √ 60 + . 3 √ 100 = 7 . 76 E ( U ( W 2 ,s )) = . 55 √ 40 + . 45 √ 80 = 7 . 49 B. Now suppose that John also has initial wealth W 0 = $0, but his utility function is U ( W ) = ln W . Calculate John’s expected utility ( E ( U ( W )) for both investments. SOLUTION: E ( U ( W 1 ,s )) = . 2 ln(20) + . 5 ln(60) + . 3 ln(100) = 4 . 03 E ( U ( W 2 ,s )) = . 55 ln(40) + . 45 ln(80) = 4 . 00 C. Does either investment first order stochastically dominate the other? Explain why or why not. SOLUTION: We check for first order stochastic dominance by comparing the cumulative probabilities: W s f ( W 1 ,s ) F ( W 1 ,s ) f ( W 2 ,s ) F ( W 2 ,s ) $ 20 20% 20% 0% 0% $ 40 0% 20% 55% 55% $ 60 50% 70% 0% 55% $ 80 0% 70% 45% 100% $ 100 30% 100% 0% 100% Since F ( W 1 ,s ) initially exceeds F ( W 2 ,s ) and is subsequently less than F ( W 2 ,s ), there is no first order stochastic dominance.

D. Compare these investments once again. Is there second order stochastic dominance? Explain why or why not. SOLUTION: We check for second order stochastic dominance by adding an additional column to the table shown in part A where we calculate n ∑ s =1 ( F ( W 2 ,s ) − F ( W 1 ,s )): W s f ( W 1 ,s ) F ( W 1 ,s ) f ( W 2 ,s ) F ( W 2 ,s ) F ( W 2 ,s ) − F ( W 1 ,s ) $ 20 20% 20% 0% 0% -20% $ 40 0% 20% 55% 55% 35% $ 60 50% 70% 0% 55% -15% $ 80 0% 70% 45% 100% 30% $ 100 30% 100% 0% 100% 0% n ∑ s =1 ( F ( W 2 ,s ) − F ( W 1 ,s )) = 30% Thus, we find that investment 1 second order stochastically dominates investment 2. E. Which investment should Miles choose? Explain why. SOLUTION: Since investment 1 stochastically dominates investment 2, this ensures that Miles will have higher expected utility from investment 1 than from investment 2 (as shown in Part A of this problem). Therefore, Miles prefers (and will choose) investment 1 instead of investment 2. F. Which investment should John choose? Explain why. SOLUTION: Even though John is more risk averse than Miles (since as we showed in class, logarithmic utility is more risk averse than square root utility), John prefers (and will choose) investment 1 instead of investment 2 for the very same reason as Miles. Specifically, stochastic dominance ensures that John will have higher expected utility from investment 1 than from investment 2 (as shown in Part B of this problem).

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