Commodities Broker Decision TreePage 764, Problem 12

Group: Maria Boyter, Saud Aldegaither, Serina Alkejek

1/17/2012

ContentsProblem Statement.....................................................................................................................................2

Solving the Problem....................................................................................................................................2

Given Information...................................................................................................................................2

Probabilities.........................................................................................................................................2

Cost and Profit.....................................................................................................................................3

Decision Tree...........................................................................................................................................3

WinQSB Input..........................................................................................................................................4

WinQSB Output.......................................................................................................................................4

WinQSB Decision Tree Graph..................................................................................................................5

Best Decision...........................................................................................................................................6

Utilities........................................................................................................................................................7

EVPI.............................................................................................................................................................8

Sensitivity Analysis......................................................................................................................................9

Report to Manager....................................................................................................................................10

Appendix A................................................................................................................................................12

How to solve using WinQSB...................................................................................................................12

Appendix B................................................................................................................................................15

Risk-Neutral...........................................................................................................................................15

Risk-averse.............................................................................................................................................18

Risk-seeking...........................................................................................................................................20

Appendix C................................................................................................................................................22

WinQSB Sensitivity Analysis...................................................................................................................22

Problem StatementEdwina, a commodities broker, has acquired an option to buy 1,000 oz of gold at $50/oz. If she takes the option and if Congress relaxes import quotas, she can sell the gold for $80/oz. If she takes the option and Congress does not relax the import quotas, however, the company will lose $10/oz. Edwina believes that there is a 50% change that the government will relax the quota. She also has the option of waiting until Congress decides whether to relax the import quota. If she adopts this strategy, however, there is a 70 % change that some other broker will have already taken the option.

a) If Edwina is risk-neutral, what should she do?b) If Edwina’s utility function for a change x in her asset position is given by

u(x )=〖(10,000+x )〗(1/2) ,What should she do?

Solving the Problem

Given Information

Probabilities

P(Congress relaxed quota) = 0.5There is a 50% chance that Congress will relax the import quota for gold whether Edwina decides to wait for the Congress’s decision or not.

P(Congress does not relax quota) = 0.5There is a 50% chance that Congress will not relax the import quota for gold.

P(Option still open │Relaxed quota) = 0.3There is a 30% chance that Edwina will be able to buy and sell the gold given that Congress has decided to relax the import quota after waiting for the decision first.

P(Option closed │Relaxed quota) = 0.7There is a 70% chance that the option will be closed to Edwina and she will not be able to buy and sell the gold given that Congress has decided to relax the import quota after waiting for the decision.

Cost and Profit

Cost of 1,000 oz of gold = $50,000

Profit of selling gold given that Congress has not relaxed import quotas = $-10,000

Profit of selling gold given that Congress has relaxed import quotas = $30,000Profit=$80,000−$50,000=$30,000

Decision Tree

Figure 1. Manual Decision Tree Diagram.

WinQSB Input

Figure 2. WinQSB input values for Decision Tree.

WinQSB Output

Figure 3. WinQSB output values for Decision Tree.

WinQSB Decision Tree Graph

Figure 4. WinQSB output Decision Tree Diagram.

Best Decision

The best decision would be to not wait for Congress and to directly buy the gold. The expected value would be $10,000. The actual values you would expect if Congress relaxes import quotas is $30,000 and if Congress chooses not to relax the quotas then there would be a loss of $10,000.

Figure 5. WinQSB output Decision Tree Diagram with highlighted decision path.

The best decision path has been highlighted in lavender on the WinQSB Decision Tree Graph.

$

Utility

Utilities

Profit Risk-Averse Risk-Neutral Risk-Seeking-10,000 0 0 0

0 0.05 0.25 0.0330,000 1 1 1

Table 1. Utility conversion for various personalities.

Figure 6. Utility function graph shows risk-averse, risk-neutral and risk-seeking personalities.

Personality Decision Expected Utility* Expected Value ($)**Risk-Averse Wait 0.57 2,996Risk-Neutral Buy 0.5 10,000Risk-Seeking Buy 0.5 17,000

Table 2. Best decision based on each personality.

*Refer to Appendix B for WinQSB input and output

** Expected value ($) is converted from expected value according to the graph

The decision changes depending on the type of personality. Table 2 shows the best decision and expected profit for each personality type.

EVPIExpected value of perfect information is used to decide what we are willing to pay in order to know in advance if Congress will either relax the import quota or not.

Quota Best Decision Profit ProbabilityRelaxed Buy $30,000 0.5

Not Relaxed Don’t Buy $0 0.5Table 3. Best outcomes depending on Congress decision.

Expected value using perfect information: 30,000(.5)+ 0(.5) = $15,000Expected value without using perfect information: $10,000

EVPI= 15,000- 10,000 = $5,000Edwina would be willing to pay up to $5,000 for perfect information.

Sensitivity Analysis

ProbablityExpected outcome WinQSB Output Decision to Make

0.4 10,000N Buy Y/N->

N.Buy Relaxed Y/N Don't wait for congress, Buy gold

0.5 10,000N Buy Y/N->

N.Buy Relaxed Y/N Don't wait for congress, Buy gold

0.6 10,000N Buy Y/N->

N.Buy Relaxed Y/N Don't wait for congress, Buy gold

0.7 10,500 W Relaxed Y/N Wait for congress before buyingTable 4. Adjustments on the probability of having the option to buy and sell gold after waiting on Congress’s decision.

WinQSB input and output values can be found in Appendix C.

The turning point of decision is between the probability of 0.6 and 0.7. After trial and error, the exact turning point is 0.67. This means that Edwina is willing to wait for congress to first decide if they are going to relax the import quotas as long as there is a 67% chance that the option will still be open for her to buy and sell the gold.

Report to Manager

Dear Manager,

After careful consideration and analysis, our best option is not to wait for the congress to

release the quota and just buy the gold right now. This option gives us an expected profit of $10,000.

Please keep in mind that expected profit is not the exact amount of profit you will receive; it is the profit

that we expect to see over time if this decision was repeated numerous times. There is a 50% chance

that the actual profit can be $30,000 if Congress decides to relax the import quota. Should Congress

decide otherwise, Edwina’s company would see a loss of $10,000.

When we incorporate different personalities into decision making, there will be different

expected values. A Risk-Averse person should decide to wait for congress to first make their decision.

The expected profit with this decision is only $2,996. Risk-Neutral and Risk-Seeking personalities will

both make the same decision each time but their expected values are still very different. This is because

the amount expected depends on that specific personality. The risk-seeker overtime will expect to get

$17,000 if this decision is made multiple times and a risk-neutral person only expects $10,000.

However, since they are both making the same decision, the outcome will either be a profit of $30,000

or a loss of $10,000 each time.

Personality Decision Expected Utility* Expected Value ($)**Risk-Averse Wait 0.57 2,996Risk-Neutral Buy 0.5 10,000Risk-Seeking Buy 0.5 17,000

If the option is available, Edwina’s company is willing to pay up to $5,000 for information

that would tell in advance what Congress will decide.

Since there is a probability involved in the outcome of the result, we performed an analysis

by changing the probability of the option availability after congress chooses to relax the quota. Based

on the probability, the decision would change.

ProbablityExpected outcome WinQSB Output Decision to Make

0.4 10,000N Buy Y/N->

N.Buy Relaxed Y/N Don't wait for congress, Buy gold

0.5 10,000N Buy Y/N->

N.Buy Relaxed Y/N Don't wait for congress, Buy gold

0.6 10,000N Buy Y/N->

N.Buy Relaxed Y/N Don't wait for congress, Buy gold

0.7 10,500 W Relaxed Y/N Wait for congress before buying

It would only be advised to wait on buying gold if the option availability changed from 30% to any value higher than 67%.

Appendix A

How to solve using WinQSBTo solve this problem using WinQSB, it is advisable that a manual tree is created that could reflect all the decision, event, and terminal nodes in the problem. All nodes should be numbered and named for communication purposes and to be entered into WinQSB.

Figure 7. Manual tree for WinQSB Input.

After creating the manual tree, information can be put into the software in an easy manner. First go into Decision Analysis and choose to open a new program. Then select the option for Decision Tree Analysis and enter the Problem Title as well as the total number of decision, event and terminal nodes.

A table will then appear that will need to following information: Node Name, Node Type, Immediate Following Node, Node Payoff, and Probability. The figure below shows the description and values that will satisfy this problem.

Yes

No

Yes

No

Yes

NoYes

No

YesNo

Yes

No

Wait

Y/N

Yes

No

N Buy Relaxe

dY/N

N Buy Y/N

W Relaxed

Y/N

N RelaxedBuy Y/N

W Relaxe

d Option

Y/N

Y Ope

n Buy Y/N0.

5

0.5

0.30.7

0.50.5

Manual Tree For WinQSB Input

1

$ 0

$ 30,000 $ 0

- $ 10,000

$ 0$

30,000 - $ 10,000 $

0

3

2

5

6

4

7

10

Terminal 2

Terminal 3

Terminal 1

Terminal 7

Terminal 8

Terminal 4

Terminal 5

Terminal 6

11

12

13

15

14

9

8

Figure 7. WinQSB input values for Decision Tree.

To find a solution, select Solve and Analyze and Solve the Problem. The Solution should look like the figure below. The expected value will be $10,000.

Figure 8. WinQSB output values for Decision Tree.

To view the decision tree, select Results and Show Decision Tree Graph. There are various options where the graph can be adjusted, one such option is to mark the box to Display the expected values of each node or event. This will show in red on the graph below each decision or event node. To visually follow the solution, simply start with the first node expected value and follow the branch that matches the same value. This will show the decisions and choices that will give the best results for that scenario.

Figure 9. WinQSB output Decision Tree Diagram.

In the case for the figure above, the solution that will provide the optimal expected value will be to buy the gold instead of waiting for congress to first decide to relax the commodities. This decision will have an expected outcome of $10,000.

Appendix B

Risk-Neutral

Figure 10. WinQSB input values for risk-neutral Decision Tree.

Figure 11. WinQSB output values for risk-neutral Decision Tree.

Figure 12. WinQSB output Decision Tree Diagram for risk-neutral personalities.

Risk-averse

Figure 13. WinQSB output values for risk-averse Decision Tree.

Figure 14. WinQSB output values for risk-averse Decision Tree.

Figure 15. WinQSB output Decision Tree Diagram for risk-averse personalities.

Risk-seeking

Figure 16. WinQSB output values for risk-seeking Decision Tree.

Figure 17. WinQSB output values for risk-seeking Decision Tree.

Figure 18. WinQSB output Decision Tree Diagram for risk-seeking personalities.

Appendix C

WinQSB Sensitivity Analysis

Probability of open option after waiting for Congress to relax import quota is changed from 0.3 to 0.4 and the probability of a closed option changed accordingly from 0.7 to 0.6.

Figure 19. WinQSB input values.

Figure 20. WinQSB output values.

Probability of open option after waiting for Congress to relax import quota is changed from 0.3 to 0.5 and the probability of a closed option changed accordingly from 0.7 to 0.5.

Figure 21. WinQSB input values.

Figure 22. WinQSB output values.

Probability of open option after waiting for Congress to relax import quota is changed from 0.3 to 0.6 and the probability of a closed option changed accordingly from 0.7 to 0.4.

Figure 23. WinQSB input values.

Figure 24. WinQSB output values.

Probability of open option after waiting for Congress to relax import quota is changed from 0.3 to 0.7 and the probability of a closed option changed accordingly from 0.7 to 0.3.

Figure 25. WinQSB input values.

Figure 26. WinQSB output values.

Probability of open option after waiting for Congress to relax import quota is changed from 0.3 to 0.67 and the probability of a closed option changed accordingly from 0.7 to 0.33.

Figure 27. WinQSB input values.

Figure 28. WinQSB output values.