CHAPTER 13

RISK ANALYSIS

1. A firm is considering two business projects. Project A will return a loss of $45 if conditions are poor, a profit of $35 if conditions are good, and a profit of $155 if conditions are excellent. Project B will return a loss of $100 if conditions are poor, a profit of $60 if conditions are good, and a profit of $300 if conditions are excellent. The probability distribution of conditions follows:

Conditions:          Poor                    Good                     Excellent

Probabilities:    40 percent           50 percent           10 percent

 (i) Calculate the expected value of each project and identify the preferred project according to this criterion.

(ii) Calculate the standard deviation of each project and identify the project that has the higher level of risk.

(iii) Calculate the coefficient of variation for each project and identify the preferred project according to this criterion.

Solution:

(i) Expected value of A: (0.4)(−45) + (0.5)(35) + (0.1)(155) = 15

Expected value of B: (0.4)(−100) + (0.5)(60) + (0.1)(300) = 20

Project B is preferred.

(ii) Variance of A: (0.4)(−602)+(0.5)(202)+(0.1)(1402)=3,600 so the standard

deviation of A is 60.

Variance of B: (0.4)(−1202) + (0.5)(402) + (0.1)(2802) = 14,400 so the standard

deviation of B is 120.

Project B has the higher level of risk.

(iii) Coefficient of variation of A: 60/15 = 4

Coefficient of variation of B: 120/20 = 6

Project A is preferred.

2. A firm is considering two business projects. Project A will return a loss of $5 if

conditions are poor, a profit of $35 if conditions are good, and a profit of $95 if

conditions are excellent. Project B will return a loss of $15 if conditions are poor, a profit of $45 if conditions are good, and a profit of $135 if conditions are excellent. The probability distribution of conditions follows:

Conditions:          Poor                    Good                     Excellent

Probabilities:    40 percent           50 percent           10 percent

(i) Calculate the expected value of each project and identify the preferred project according to this criterion.

(ii) Calculate the standard deviation of each project and identify the project that has the higher level of risk.

(iii) Calculate the coefficient of variation for each project and identify the preferred project according to this criterion.

Solution:

(i) Expected value of A: (0.4)(−5) + (0.5)(35) + (0.1)(95) = 25

Expected value of B: (0.4)(−15) + (0.5)(45) + (0.1)(135) = 30

Project B is preferred.

(ii) Variance of A: (0.4)(−302) + (0.5)(102) + (0.1)(702) = 900 so the standard

deviation of A is 30.

Variance of B: (0.4)(−452) + (0.5)(152) + (0.1)(1052) = 2,025 so the standard

deviation of B is 45.

Project B has the higher level of risk.

(iii) Coefficient of variation of A: 30/25 = 1.20

Coefficient of variation of B: 45/30 = 1.50

Project A is preferred.

3. A firm is considering two business projects. Project A will return a profit of zero if conditions are poor, a profit of $16 if conditions are good, and a profit of $49 if conditions are excellent. Project B will return a profit of $4 if conditions are poor, a profit of $9 if conditions are good, and a profit of $49 if conditions are excellent. The probability distribution of conditions follows:

Conditions:          Poor                    Good                     Excellent

Probabilities:    40 percent           50 percent           10 percent

(i) Calculate the expected value of each project and identify the preferred project according to this criterion.

(ii) Assume that the firm has determined that its utility function for profit is equal to the square root of profit. Calculate the expected utility of each project and identify the preferred project according to this criterion.

(iii) Is the firm risk averse, risk neutral, or risk seeking? How can you tell?

Solution:

(i) Expected value of A: (0.4)(0) + (0.5)(16) + (0.1)(49) = 12.90

Expected value of B: (0.4)(4) + (0.5)(9) + (0.1)(49) = 11.0

Project A is preferred.

(ii) Expected utility of A: (0.4)(0) + (0.5)(4) + (0.1)(7) = 2.70

Expected utility of B: (0.4)(2) + (0.5)(3) + (0.1)(7) = 3.0

Project B is preferred.

(iii) The firm is risk averse because the utility function increases at a decreasing rate; that is, the marginal utility of profit diminishes.

4. A firm is considering two business projects. Project A will return a profit of zero if conditions are poor, a profit of $4 if conditions are good, and a profit of $8 if conditions are excellent. Project B will return a profit of $2 if conditions are poor, a profit of $3 if conditions are good, and a profit of $4 if conditions are excellent. The probability distribution of conditions follows:

Conditions:          Poor                    Good                     Excellent

Probabilities:    40 percent           50 percent           10 percent

(i) Calculate the expected value of each project and identify the preferred project according to this criterion.

(ii) Assume that the firm has determined that its utility function for profit is as

follows:

U(X) = X − 0.05X2

Calculate the expected utility of each project and identify the preferred project

according to this criterion.

(iii) Is the firm risk averse, risk neutral, or risk seeking? How can you tell?

Solution:

(i) Expected value of A: (0.4)(0) + (0.5)(4) + (0.1)(8) = 2.80

Expected value of B: (0.4)(2) + (0.5)(3) + (0.1)(4) = 2.70

Project A is preferred.

(ii) Expected utility of A: (0.4)(0) + (0.5)(3.2) + (0.1)(4.8) = 2.08

Expected utility of B: (0.4)(1.80) + (0.5)(2.55) + (0.1)(3.20) = 2.315

Project B is preferred.

(iii) The firm is risk averse because the utility function increases at a decreasing rate; that is, the marginal utility of profit diminishes.

5. A firm is considering three business projects. Project A will return a profit of $5 if conditions are poor, $10 if conditions are good, and $15 if conditions are excellent. Project B will return a profit of $12 if conditions are poor, $8 if conditions are good, and &4 if conditions are excellent. Project C will return a profit of $3 if conditions are poor, $20 if conditions are good, and $7 if conditions are excellent.

(i) Use the maximin criterion to determine the preferred project. Show how you arrived at your solution.

(ii) Calculate the regret matrix.

(iii) Use the minimax regret criterion to determine the preferred project. Show how you arrived at your solution.

Solution:

(i) The minimum profit for A is $5, for B is $4, and for C is $3. The maximin

solution ($5) is Project A.

(ii)              Poor            Good           Excellent

         A          7                  10                    0

         B          0                  12                   11

         C          9                    0                      8

(iii) The maximum regret for A is $10, for B is $12, and for C is $9. The minimax

regret solution ($9) is Project C.

6. A firm is considering three business projects. Project A will return a profit of $1 if conditions are poor, $5 if conditions are good, and $8 if conditions are excellent. Project B will return a profit of $2 if conditions are poor, $2 if conditions are good, and $6 if conditions are excellent. Project C will return a profit of $10 if conditions are poor, $1 if conditions are good, and $2 if conditions are excellent.

(i) Use the maximin criterion to determine the preferred project. Show how you arrived at your solution.

(ii) Calculate the regret matrix.

(iii) Use the minimax regret criterion to determine the preferred project. Show how you arrived at your solution.

Solution:

(i) The minimum profit for A is $1, for B is $2, and for C is $1. The maximin

solution ($2) is Project B.

(ii)               Poor            Good           Excellent

         A          9                   0                      0

         B          8                   3                       2

         C          0                   4                       6

               

(iii) The maximum regret for A is $9, for B is $8, and for C is $6. The minimax regret solution ($6) is Project C.

7. A firm is considering three business projects. Project A will return a profit of $1 if conditions are poor, $7 if conditions are good, and $8 if conditions are excellent. Project B will return a profit of $1 if conditions are poor, $4 if conditions are good, and $11 if conditions are excellent. Project C will return a profit of $8 if conditions are poor, $3 if conditions are good, and $5 if conditions are excellent.

(i) Use the maximin criterion to determine the preferred project. Show how you arrived at your solution.

(ii) Calculate the regret matrix.

(iii) Use the minimax regret criterion to determine the preferred project. Show how you arrived at your solution.

Solution:

(i) The minimum profit for A is $1, for B is $1, and for C is $3. The maximin

solution ($3) is Project C.

(ii)                Poor            Good           Excellent

         A          7                  0                      3

         B          7                  3                      0

         C          4                  4                      6

(iii) The maximum regret for A is $7, for B is $7, and for C is $6. The minimax regret solution ($6) is Project C.

8. A firm is considering three business projects. Project A will return a loss of $5 if conditions are poor, a profit of $4 if conditions are good, and a profit of $8 if conditions are excellent. Project B will return a profit of zero if conditions are poor, a profit of $2 if conditions are good, and a profit of $7 if conditions are excellent. Project C will return a profit of $1 if conditions are poor, $3 if conditions are good, and $5 if conditions are excellent.

(i) Use the maximin criterion to determine the preferred project. Show how you arrived at your solution.

(ii) Calculate the regret matrix.

(iii) Use the minimax regret criterion to determine the preferred project. Show how you arrived at your solution.

Solution:

(i) The minimum profit for A is −$5, for B is $0, and for C is $1. The maximin

solution ($1) is Project C.

(ii)               Poor            Good           Excellent

         A          6                  0                      0

         B          1                  2                      1

         C          0                  1                      3

(iii) The maximum regret for A is $6, for B is $2, and for C is $3. The minimax regret solution ($2) is Project B.