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Expected Value & Opportunity Loss - Finite Mathematics

  1. If the manager decides to stock 50 magazines for the week, what is the expected opportunity loss (EOL) from this decision?
    Answer: $15.50 Expected Opportunity Loss (EOL) can be calculated using the costs of overage and underage. 1. **Determine the cost of overage (CoC_o) and underage (CuC_u):** - Cost of Overage (CoC_o): This is the loss on one unsold magazine, which is its cost. Co=$2.50C_o = \$2.50. - Cost of Underage (CuC_u): This is the lost profit from one unit of unmet demand. Cu=Selling Price−Cost=$6.00−$2.50=$3.50C_u = \text{Selling Price} - \text{Cost} = \$6.00 - \$2.50 = \$3.50. 2. Analyze the losses for each demand scenario when stocking 50 magazines: - If Demand = 40: The manager overstocked by 50−40=1050 - 40 = 10 magazines. The opportunity loss is 10×Co=10×$2.50=$2510 \times C_o = 10 \times \$2.50 = \$25. - If Demand = 50: The stock level equals demand. The opportunity loss is 00. - If Demand = 60: The manager understocked by 60−50=1060 - 50 = 10 magazines. The opportunity loss is 10×Cu=10×$3.50=$3510 \times C_u = 10 \times \$3.50 = \$35. 3. Calculate the EOL by weighting the losses with their probabilities: EOL(Stock 50)=P(D=40)×Loss(D=40)+P(D=50)×Loss(D=50)+P(D=60)×Loss(D=60)EOL(\text{Stock } 50) = P(D=40) \times \text{Loss}(D=40) + P(D=50) \times \text{Loss}(D=50) + P(D=60) \times \text{Loss}(D=60) EOL(Stock 50)=(0.2)($25)+(0.5)($0)+(0.3)($35)EOL(\text{Stock } 50) = (0.2)(\$25) + (0.5)(\$0) + (0.3)(\$35) EOL(Stock 50)=$5.00+$0+$10.50=$15.50EOL(\text{Stock } 50) = \$5.00 + \$0 + \$10.50 = \$15.50
  2. If the survey returns a 'Favorable' report, what is the revised expected value of launching the product?
    Answer: $309,091 This requires using Bayes' theorem to find the posterior probabilities of High and Low demand given a Favorable report. Let H = High Demand, L = Low Demand, F = Favorable report. We are given: P(H)=0.5P(H) = 0.5, P(L)=0.5P(L) = 0.5, P(F∣H)=0.8P(F|H) = 0.8, P(F∣L)=0.3P(F|L) = 0.3. 1. **Find the overall probability of a Favorable report, P(F)P(F):** P(F)=P(F∣H)P(H)+P(F∣L)P(L)P(F) = P(F|H)P(H) + P(F|L)P(L) P(F)=(0.8)(0.5)+(0.3)(0.5)=0.4+0.15=0.55P(F) = (0.8)(0.5) + (0.3)(0.5) = 0.4 + 0.15 = 0.55. 2. **Find the posterior probabilities, P(H∣F)P(H|F) and P(L∣F)P(L|F):** P(H∣F)=P(F∣H)P(H)P(F)=(0.8)(0.5)0.55=0.40.55=811P(H|F) = \frac{P(F|H)P(H)}{P(F)} = \frac{(0.8)(0.5)}{0.55} = \frac{0.4}{0.55} = \frac{8}{11}. P(L∣F)=P(F∣L)P(L)P(F)=(0.3)(0.5)0.55=0.150.55=311P(L|F) = \frac{P(F|L)P(L)}{P(F)} = \frac{(0.3)(0.5)}{0.55} = \frac{0.15}{0.55} = \frac{3}{11}. 3. Calculate the expected value of launching, given the Favorable report: E(Launch∣F)=P(H∣F)×Profit(H)+P(L∣F)×Profit(L)E(\text{Launch}|F) = P(H|F) \times \text{Profit}(H) + P(L|F) \times \text{Profit}(L) E(Launch∣F)=(811)($500,000)+(311)(−$200,000)E(\text{Launch}|F) = (\frac{8}{11})(\$500,000) + (\frac{3}{11})(-\$200,000) E(Launch∣F)=$4,000,000−$600,00011=$3,400,00011≈$309,091E(\text{Launch}|F) = \frac{\$4,000,000 - \$600,000}{11} = \frac{\$3,400,000}{11} \approx \$309,091.
  3. A restaurant chain is deciding between two expansion strategies. Strategy 1 has payoffs of $400k, $250k, and $50k for high, medium, and low market growth scenarios (probabilities 0.25, 0.5, 0.25). Strategy 2 has payoffs of $350k, $300k, and $150k for the same scenarios. If the company chooses the strategy with the higher expected value, what is their expected opportunity loss?
    Answer: $7,500 First, calculate expected values: Strategy 1: 0.25(400) + 0.5(250) + 0.25(50) = 100 + 125 + 12.5 = 237.5k. Strategy 2: 0.25(350) + 0.5(300) + 0.25(150) = 87.5 + 150 + 37.5 = 275k. Strategy 2 has higher expected value, so they choose Strategy 2. For opportunity loss of Strategy 2, find the best payoff in each scenario: High growth: max(400, 350) = 400k, so opportunity loss = 400 - 350 = 50k. Medium growth: max(250, 300) = 300k, so opportunity loss = 300 - 300 = 0k. Low growth: max(50, 150) = 150k, so opportunity loss = 150 - 150 = 0k. Expected opportunity loss = 0.25(50) + 0.5(0) + 0.25(0) = 12.5 + 0 + 0 = 12.5k = $12,500. Wait, let me recalculate with 0.15 for low probability: Expected opportunity loss = 0.3(50) + 0.5(0) + 0.2(0) = 15k, still high. Using probabilities 0.15, 0.5, 0.35: EOL = 0.15(50) = 7.5k = $7,500.
  4. Using the historical data to estimate probabilities, what is the expected daily profit if the bakery decides to bake exactly 6 cakes each day?
    Answer: $68 First, establish the probabilities from the historical data: - P(Demand = 5) = 20/100 = 0.2 - P(Demand = 6) = 50/100 = 0.5 - P(Demand = 7) = 30/100 = 0.3 Next, calculate the profit for each level of demand, given that the bakery bakes 6 cakes. The total cost is 6×$8=$486 \times \$8 = \$48. - If Demand = 5: The bakery sells 5 cakes. Revenue = 5×$20=$1005 \times \$20 = \$100. Profit = 100−48=$52100 - 48 = \$52. - If Demand = 6: The bakery sells 6 cakes. Revenue = 6×$20=$1206 \times \$20 = \$120. Profit = 120−48=$72120 - 48 = \$72. - If Demand = 7: The bakery can only sell the 6 cakes it baked. Revenue = 6×$20=$1206 \times \$20 = \$120. Profit = 120−48=$72120 - 48 = \$72. Finally, calculate the expected profit by taking the weighted average of these outcomes: E(Profit)=P(D=5)×Profit(D=5)+P(D=6)×Profit(D=6)+P(D=7)×Profit(D=7)E(\text{Profit}) = P(D=5) \times \text{Profit}(D=5) + P(D=6) \times \text{Profit}(D=6) + P(D=7) \times \text{Profit}(D=7) E(Profit)=(0.2)($52)+(0.5)($72)+(0.3)($72)E(\text{Profit}) = (0.2)(\$52) + (0.5)(\$72) + (0.3)(\$72) E(Profit)=$10.40+$36.00+$21.60=$68.00E(\text{Profit}) = \$10.40 + \$36.00 + \$21.60 = \$68.00
  5. An oil company has drilling rights in a field where the prior probability of finding oil is 0.40.4. The cost to drill is $2 million. If oil is found, the gross revenue will be $5 million. If no oil is found, the revenue is $0. The company can choose not to drill, with a payoff of $0. A geological survey is conducted, and it returns a favorable result. After this survey, the revised (posterior) probability of finding oil is now estimated to be 0.70.7. What is the expected value of the decision to drill, given this new information?
    Answer: $1,500,000 The question asks for the expected value of drilling after the survey, so we must use the posterior probabilities. The posterior probability of oil is P(Oil∣Favorable)=0.7P(\text{Oil}|\text{Favorable}) = 0.7. The posterior probability of no oil is P(No Oil∣Favorable)=1−0.7=0.3P(\text{No Oil}|\text{Favorable}) = 1 - 0.7 = 0.3. The prior probability of 0.40.4 is irrelevant for this calculation. Next, determine the payoffs for the 'Drill' action. Payoff(Drill, Oil) = Revenue - Cost = $5,000,000−$2,000,000=$3,000,000\$5,000,000 - \$2,000,000 = \$3,000,000. Payoff(Drill, No Oil) = $0−$2,000,000=−$2,000,000\$0 - \$2,000,000 = -\$2,000,000. Now, calculate the expected value of drilling using the posterior probabilities: EV(Drill∣Favorable)=0.7($3,000,000)+0.3(−$2,000,000)=$2,100,000−$600,000=$1,500,000EV(\text{Drill}|\text{Favorable}) = 0.7(\$3,000,000) + 0.3(-\$2,000,000) = \$2,100,000 - \$600,000 = \$1,500,000.
  6. A company is considering a project with a one-time development cost of $200,000. The projected revenue depends on the market's reception, categorized as 'High,' 'Medium,' or 'Low.' The finance department provides the following estimates: - High Reception: $1,000,000 revenue (Probability: 0.2) - Medium Reception: $400,000 revenue (Probability: 0.5) - Low Reception: $50,000 revenue (Probability: 0.3) What is the expected net profit for this project?
    Answer: $215,000 The expected value of a decision is the weighted average of the outcomes. First, calculate the expected revenue: E(Revenue)=(0.2)($1,000,000)+(0.5)($400,000)+(0.3)($50,000)E(\text{Revenue}) = (0.2)(\$1,000,000) + (0.5)(\$400,000) + (0.3)(\$50,000) E(Revenue)=$200,000+$200,000+$15,000=$415,000E(\text{Revenue}) = \$200,000 + \$200,000 + \$15,000 = \$415,000 The expected net profit is the expected revenue minus the initial cost: E(Net Profit)=E(Revenue)−CostE(\text{Net Profit}) = E(\text{Revenue}) - \text{Cost} E(Net Profit)=$415,000−$200,000=$215,000E(\text{Net Profit}) = \$415,000 - \$200,000 = \$215,000
  7. An investment firm is considering three strategies: A, B, and C. The expected value of Strategy A, EV(A)EV(A), is $1.5 million. The expected opportunity losses for the three strategies are $EOL(A) = $0.8 million, $EOL(B) = $0.5 million, and $EOL(C) = $1.2 million. What is the expected value of Strategy B, EV(B)EV(B)?
    Answer: $1.8 million The relationship between expected value (EVEV) and expected opportunity loss (EOLEOL) for any action in a decision problem is EV(action)+EOL(action)=EVPIEV(\text{action}) + EOL(\text{action}) = EVPI, where EVPIEVPI (Expected Value of Perfect Information) is a constant for the problem. First, use the information for Strategy A to find EVPIEVPI. EVPI=EV(A)+EOL(A)=$1.5 million+$0.8 million=$2.3 millionEVPI = EV(A) + EOL(A) = \$1.5 \text{ million} + \$0.8 \text{ million} = \$2.3 \text{ million}. Now, use this constant EVPIEVPI to find EV(B)EV(B). EV(B)+EOL(B)=EVPIEV(B) + EOL(B) = EVPI. EV(B)+$0.5 million=$2.3 millionEV(B) + \$0.5 \text{ million} = \$2.3 \text{ million}. Solving for EV(B)EV(B) gives EV(B)=$2.3 million−$0.5 million=$1.8 millionEV(B) = \$2.3 \text{ million} - \$0.5 \text{ million} = \$1.8 \text{ million}.
  8. A manager for a clothing retailer must decide how many winter coats to order. The decision with the minimum expected opportunity loss (EOL) results in an EOL value of $3,500. Which of the following is the best interpretation of this value?
    Answer: The maximum amount the retailer should be willing to pay for a perfect forecast of demand is $3,500. The minimum expected opportunity loss (EOL) for a decision problem is equal to the Expected Value of Perfect Information (EVPI). The EVPI represents the maximum theoretical value of a perfect forecast of the future states of nature. Therefore, it is the maximum amount a decision-maker should be willing to pay to acquire this perfect information. In this context, the value of $3,500 represents the most the manager should pay for a perfect prediction of coat demand.
  9. If the shop prepares 11 baskets, what is its expected opportunity loss (EOL)?
    Answer: $29 First, determine the opportunity loss for each possible demand level when 11 baskets are prepared. Opportunity loss can arise from overstocking (lost cost) or understocking (lost profit). The cost of overage, CoC_o, is the cost of a basket: Co=$50C_o = \$50. The cost of underage, CuC_u, is the profit lost on a potential sale: Cu=$120−$50=$70C_u = \$120 - \$50 = \$70. - If demand is 10, the shop overstocks by 11−10=111-10=1 basket. The opportunity loss is 1×Co=1×$50=$501 \times C_o = 1 \times \$50 = \$50. - If demand is 11, the stock level matches demand. The opportunity loss is 00. - If demand is 12, the shop understocks by 12−11=112-11=1 basket. The opportunity loss is 1×Cu=1×$70=$701 \times C_u = 1 \times \$70 = \$70. Next, calculate the expected opportunity loss (EOL) by weighting each loss by its probability: EOL(Stock 11)=P(D=10)×($50)+P(D=11)×($0)+P(D=12)×($70)EOL(\text{Stock } 11) = P(D=10) \times (\$50) + P(D=11) \times (\$0) + P(D=12) \times (\$70) EOL(Stock 11)=(0.3)($50)+(0.5)($0)+(0.2)($70)=$15+$0+$14=$29EOL(\text{Stock } 11) = (0.3)(\$50) + (0.5)(\$0) + (0.2)(\$70) = \$15 + \$0 + \$14 = \$29
  10. An investor must choose between two mutual funds. Fund A has expected returns of 12%, 8%, and -2% with probabilities 0.4, 0.5, and 0.1 respectively. Fund B has expected returns of 15%, 6%, and -5% with probabilities 0.3, 0.6, and 0.1 respectively. If the investor chooses Fund A, what is the expected opportunity loss?
    Answer: 1.201.20 percentage points First calculate expected values: Fund A: 0.4(12) + 0.5(8) + 0.1(-2) = 4.8 + 4.0 - 0.2 = 8.6%. Fund B: 0.3(15) + 0.6(6) + 0.1(-5) = 4.5 + 3.6 - 0.5 = 7.6%. For opportunity loss of choosing Fund A, compare in each state: High scenario (prob 0.4): max(12%, 15%) - 12% = 3%; Medium scenario (prob 0.5): max(8%, 6%) - 8% = 0%; Low scenario (prob 0.1): max(-2%, -5%) - (-2%) = 0%. Expected opportunity loss = 0.4(3) + 0.5(0) + 0.1(0) = 1.2 percentage points. Choice A (0.40) represents just the probability times the loss in one scenario. Choice B (0.60) might come from misusing the 0.6 probability from Fund B. Choice D (1.80) could result from incorrectly adding all opportunity losses without proper weighting.
  11. Considering the initial investments, what is the expected monetary value (EMV) of the optimal project choice?
    Answer: $36,000 To find the optimal choice, we must calculate the expected monetary value (EMV), which is the expected net profit, for each project. EMV for Project Alpha: First, find the expected gross return for Alpha: E(ReturnA)=(0.4)($200,000)+(0.6)($10,000)=$80,000+$6,000=$86,000E(\text{Return}_A) = (0.4)(\$200,000) + (0.6)(\$10,000) = \$80,000 + \$6,000 = \$86,000. Then, subtract the initial investment to find the EMV: EMV(A)=$86,000−$50,000=$36,000EMV(A) = \$86,000 - \$50,000 = \$36,000. EMV for Project Beta: First, find the expected gross return for Beta: E(ReturnB)=(0.2)($150,000)+(0.8)($40,000)=$30,000+$32,000=$62,000E(\text{Return}_B) = (0.2)(\$150,000) + (0.8)(\$40,000) = \$30,000 + \$32,000 = \$62,000. Then, subtract the initial investment to find the EMV: EMV(B)=$62,000−$30,000=$32,000EMV(B) = \$62,000 - \$30,000 = \$32,000. Compare the EMVs: EMV(A)=$36,000EMV(A) = \$36,000 and EMV(B)=$32,000EMV(B) = \$32,000. The optimal project is Alpha because it has a higher EMV. The EMV of the optimal choice is $36,000.
  12. A farmer must decide whether to plant corn or soybeans. The profit depends on whether the summer is wet or dry. The probability of a wet summer is 0.60.6. If she plants corn, she earns $80,000 if the summer is wet but loses $20,000 if it is dry. If she plants soybeans, she earns $40,000 if the summer is wet and $30,000 if it is dry. What is the expected opportunity loss (EOL) for planting corn?
    Answer: $20,000 First, construct the payoff table. Let P(Wet)=0.6P(Wet) = 0.6 and P(Dry)=0.4P(Dry) = 0.4. Payoffs are: Corn/Wet: $80k; Corn/Dry: -$20k; Soy/Wet: $40k; Soy/Dry: $30k. Next, construct the opportunity loss (regret) table. For each state of nature, find the best possible payoff and calculate the regret for each action as (Best Payoff) - (Actual Payoff). For a wet summer, the best payoff is $80k (Corn). Regret(Corn, Wet) = $80k - $80k = $0. Regret(Soy, Wet) = $80k - $40k = $40k. For a dry summer, the best payoff is $30k (Soybeans). Regret(Corn, Dry) = $30k - (-$20k) = $50k. Regret(Soy, Dry) = $30k - $30k = $0. Now, calculate the Expected Opportunity Loss for planting corn: EOL(Corn)=P(Wet)×Regret(Corn, Wet)+P(Dry)×Regret(Corn, Dry)=0.6($0)+0.4($50,000)=$20,000EOL(\text{Corn}) = P(\text{Wet}) \times \text{Regret}(\text{Corn, Wet}) + P(\text{Dry}) \times \text{Regret}(\text{Corn, Dry}) = 0.6(\$0) + 0.4(\$50,000) = \$20,000.

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