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EMIS 7300

Use a decision tree to determine the optimal investment decision between bonds and stocks based on expected monetary value.

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EMIS 7300

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  1. EMIS 7300 Decision Trees Updated 2 December 2005

  2. STATE OF NATURE No Growth (0.65) Rapid Growth (0.35) DECISIONALTERNATIVE BondsStocks $500-$200 $100$1100 Example 1 There is a 0.65 probability of no growth in the investment climate and 0.35 probability of rapid growth. The payoffs are $500 for a bond investment in a no-growth state, $100 for a bond investment in a rapid-growth state, -$200 for a stock investment in a no-growth state, and a $1100 payoff for a stock investment in a rapid-growth state.

  3. EMV Criterion The expected monetary value for the bonds decision alternative is EMV(bonds) = $500(0.65) + $100(0.35) = $360 The expected monetary value for the stocks decision alternative is EMV(stocks) = -$200(0.65) + $1100(0.35) = $255 Select bonds under the EMV criterion.

  4. No Growth (0.65) 1 Rapid Growth (0.35) 2 Example 1 as a Decision Tree Payoffs $500 Decision Node Bonds $100 No Growth (0.65) $-200 Stocks Rapid Growth (0.35) $1100 State-of-Nature Node

  5. EMV for Node 1 = $360 No Growth (0.65) 1 Rapid Growth (0.35) 2 EMV for Node 2 = $255 Calculate EMV for Each Node Payoffs $500 Bonds $100 No Growth (0.65) $-200 Stocks Rapid Growth (0.35) $1100

  6. No Growth (0.65) 1 Rapid Growth (0.35) 2 Completed Decision Tree Payoffs $500 $360 Bonds $100 $360 Select Bonds No Growth (0.65) $-200 Stocks $255 Rapid Growth (0.35) $1100

  7. Expected Value with Perfect Information • Suppose that before we invest, we can consult an oracle who knows with certainty which state of nature will occur. Our investment policy will be: • If the oracle predicts no growth, then invest in bonds and receive a payoff of $500. • If the oracle predicts rapid growth, then invest in stocks and receive a payoff of $1,100. • EVwPI = (0.65)($500) + (0.35)($1,100) = $710.

  8. Example 2 Banana Computer Company manufactures memory chips in batches of ten chips. From past experience, Banana knows that 80% of all batches contain 10% (1 out of 10) defective chips, and 20% of all batches contain 50% (5 out of 10) defective chips. If a good (that is, 10% defective) batch of chips is sent on to the next stage of production, processing costs of $1000 are incurred, and if a bad batch (50% defective) is sent on to the next stage of production, processing costs of $4000 are incurred. Banana also has the option of reworking a batch at a cost of $1000 before sending it to the next stage of production. A reworked batch is sure to be a good batch.

  9. Example 2 Continued • Develop a decision tree for this problem and use it to determine a policy for minimizing Banana’s expected total cost per batch.

  10. 1 -$2,000 Rework every batch Send Directly to next stage Good Batch (0.8) -$1,000 2 Bad Batch (0.2) -$4,000 Decision Node State of Nature Node Decision Tree EMV(2)=(0.8)(-$1,000)-(0.2)($4,000) = -$1,600

  11. Optimal Policy • Send every batch directly to the next stage • EMV = -$1,600

  12. Example 3 • For a cost of $100, Banana can test one chip from each batch in an attempt to determine whether the batch is defective. • Develop a decision tree for this problem and use it to determine a policy for minimizing Banana’s expected total cost per batch. • What is the expected value of the sample information (EVSI) obtained from testing chips? • What is the expected value of perfect information (EVPI) for this problem?

  13. Policies to Consider • Rework every batch before sending it Stage 2. This policy will cost $2000 for every batch: $1000 to rework it and $1000 to process the reworked batch at stage 2. • Send every batch to Stage 2 without reworking it. E[cost] = (0.8)($1000) + (0.2)($4000) = $1600 • Test a chip from each batch and decide what to do based on the result.

  14. Expected Cost of Policy 3 • Case 1: The chip tested is good (not defective). • The expected cost of sending the batch to stage 2 is ($1000)P(GB|GC) + ($4000)P(BB|GC). • Case 2: The chip is defective. • The expected cost of sending the batch to stage 2 is ($1000)P(GB|DC) + ($4000)P(BB|DC).

  15. Banana Computer Solution

  16. Good Batch (?) Direct to Stage 2 -$1,100 4 Bad Batch (1-?) -$4,100 Good Chip (0.82) Rework -$2,100 Direct to Stage 2 Good Batch (??) -$1,100 3 5 Bad Batch (1-??) -$4,100 Defective Chip (0.18) Rework -$2,100 Test a Chip 1 -$2,000 Rework every batch Send Directly to next stage Good Batch (0.8) -$1,000 2 Bad Batch (0.2) -$4,000 Decision Node State of Nature Node Decision Tree

  17. Banana Computer Solution: Posterior Probabilities

  18. Banana Computer Solution: Posterior Probabilities

  19. Good Batch (0.88) Direct to Stage 2 -$1,100 4 Bad Batch (0.12) -$4,100 Good Chip (0.82) Rework -$2,100 Direct to Stage 2 Good Batch (0.44) -$1,100 3 5 Bad Batch (0.56) -$4,100 Defective Chip (0.18) Rework -$2,100 Test a Chip 1 -$2,000 Rework every batch Send Directly to next stage Good Batch (0.8) -$1,000 2 Bad Batch (0.2) -$4,000 Decision Node State of Nature Node Decision Tree EMV(4)=(0.88)(-$1,100)-(0.12)($4,100) = -$1,460 EMV(5)=(0.44)(-$1,100)-(0.56)($4,100) = -$2,780 EMV(3)=(0.82)(-$1,460)-(0.18)($2,100) = -$1575.2 EMV(2)=(0.8)(-$1,000)-(0.2)($4,000) = -$1,600

  20. Optimal Policy • Test one chip • If the chip is good, then send the batch directly to the next stage • If the chip is defective, then rework the batch before sending it to the next stage • EMV = -$1,575.25

  21. Expected Value of Sample Information (EVSI) • The optimal decision is to test a chip. • The expected payoff is -$1,575.25. • If the test were free, the expected payoff would be -$1,475.25 • The optimal decision without the test is to send all batches directly to the next stage • The expected payoff is -$1,600. • The expected value of the test (EVSI) is given by (-$1,475.25) – (-$1,600) = $124.75 • If the test cost more than $124.75 we wouldn’t use it.

  22. EVPI • The optimal decision is to pay to test the chip.The expected payoff is -$1,575.25 • The optimal decision with perfect information is • If the batch is good, then send it to the next stage. • If the batch is bad, then rework it before sending it to the next stage • EMVwPI = (-$1,000)(0.8)+(-2,000)(0.2) = -$1,200. • The expected value of perfect information is -$1,200 – (- $1,575.25) = $375.25

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