Question A

You are given the option to invest in one of three uncertain investments; A, B or C the outcomes for which are given in the table below. Since all three investments result in positive profits do not consider the option of not investing.

Compare these 3 projects using a Stochastic Dominance framework and advise which, if any is the best project to invest in.  Show all working

Undertaking pairwise comparisons

1.   When comparing A and B, A dominates B by FSD as FA (x) <= FB (x) for all 40 <=x<=150

2.   When comparing Projects A and C Stochastic Dominance is inconclusive. because the

integral of the difference between the CDFs over the range turns negative. (Students need to demonstrate mathematically).

3.   When comparing Projects B and C, C dominates B by SSD because the integral of the

3 difference between the CDFs over the range remains positive.  (Students need to demonstrate mathematically).

Question B

Sydney Development Corporation (SDC) is a profit maximising property development          company that has a parcel of land on which they are planning to build a retirement village. They can choose to develop either a:

•   Small village – 30 units

•   Medium Village – 60 units

•   Large village – 90 units

Market demand is uncertain but based on experience, SDC believe there is an 80% chance demand will either be Strong (S) and a 20% chance it will be Weak (W).  The payoffs           associated with each decision are indicated in the following table.

1.   Assuming SDC, risk neutral company, what choice will they make and what is the value of that choice?

Choose that decision which maximises EMV

Small = 7.8

Medium = 12.2

Large = 14.2

Large village, 14.2

2.   If SDC was uncertain about the payoff for the large village option under strong  demand, what would the payoff need to be, to ensure the optimal decision you arrived at in Part A remains optimal, all other things being equal?

EV large >= EV medium

S x 0.8 -9 x 0.2 >= 12.2

S >= 17.5

3.   If SDC knew with absolute certainty which demand state will occur, what strategy would they employ and what would the EVPI of this approach be?

If strong choose large, if weak choose small

EMV = 20 x 0.8 + 7 x 0.2 = 17.4 so EVPI = 17.4 – 14.2 = 3.2

4.   If we assume the probability of strong demand is now some percentage “p”, what would that probability be so that SDC would be indifferent to investing in either a medium or large village development

Set EV large = EV medium and solve for p

20p – 9(1-p) = 14p + 5(1 – p)

P= 70%

Question C

Sydney Electronics produces electronic boards for the computer industry. Each board          contains 10 chips. From past experience, it is known that after the first stage of production, 80% of the boards have 1 damaged chip (“good boards”) and 20% have 5 damaged chips     (“bad boards”). The next (and final) stage of production costs $1,000 dollars for boards       containing 1 damaged chip, and $4,000 for boards containing 5 damaged chips.

There is an option to send boards for repair (before the next stage) at a cost of $1,000. After repair, all boards will contain only 1 damaged chip. There is also an option to test boards       (before repair) at a cost of $100.

The test consists of choosing one chip from a board and checking if it is damaged (the test can only be performed once on every board).

The company’s goal is to minimize the cost of production.

1.   Draw a decision tree for the decision situation.

The decision problem consists of two decisions: (1) whether to test the      boards, and (2) whether to repair the boards. If the boards are tested, the decision whether to send them to repair may depend on the outcome of  the test.

Denote the following events and their associated probabilities:

G - " Good " Board: P (G) = 0.8

B - " Bad " Board: P (B) = 1 - P (G) = 0.2

D - Damaged Chip : P (D)

ND - Non - Damaged Chip : P (ND) = 1 - P(D)

P (D |G) = 0.1; P (ND |G) = 0.9

P (D |B) = 0.5; P (ND |B) = 0.5

The decision tree is in the separate PDF

2.   What is the optimal production policy? What is the expected cost per board under this policy?

First, we must calculate the probabilities of obtaining a damaged or a non- damaged chip, and the conditional probabilities of obtaining different        boards given the outcome of the test:

Using the total probability formula: