Question

$1M is available to invest in S or B. The percentage yield on each investment depends on whether the econ has a good or bad year.

Econ has a Good year    Econ has a Bad year

Yield on S    22% of 1M 10% of 1M

   (i.e. $220,000)    ($100,000)

Yield on B 16% of 1M 14% of 1M

($160,000) ($140,000)

It is equally likely (50%) that the econ will have a good or bad year.

For $10,000, a firm can be hired to forecast the state of the econ. The firm's forecasts have the following probabilities:

p(Good forecast | Econ is good) = .8

p(GF | EIB) = .2

It is equally likely (50%) for EIG & EIB to occur

a) Calculate the following:

p(EIG | GF) =

p(EIB | GF) =

p(EIG | BF) =

p( EIB | BF) =

b) Draw a decision tree to determine to invest in S or B to maximize expected profits. Should the firm be hired?

c) What are the values of the EVSI and EVPI?

Answer 1

a) We know the following probabilities

Let EIG be the event that the economy is good and EIB be the event that the economy is bad.

P(EIG) =0.50

P(EIB)=0.50

Let GF be the event that the firm forecasts that the economy is good, and BF be the event that the firm forecasts that the economy is bad.

We know the following conditional probabilities

The marginal probability of getting a good forecast is

The marginal probability of getting a bad forecast is

P(BF) = 1- P(GF) = 1-0.50=0.50

Now we get the probabilities that we need

b) Now the decision tree

Moving from the right to the left, we have

Chance node 6: Hire the firm, ignore the cost to hire for now and invest in S if good forecast

The expected value is

Chance node 7: Hire the firm, ignore the cost to hire for now and invest in B if good forecast

The expected value is

Chance node 8: Hire the firm, ignore the cost to hire for now and invest in S if bad forecast

The expected value is

Chance node 9: Hire the firm, ignore the cost to hire for now and invest in B if bad forecast

The expected value is

Chance node 10: Do not Hire the firm, and invest in S

The expected value is

Chance node 11: Do not Hire the firm, invest in B

The expected value is

Decision node 3: Hire the firm, get a good forecast.

We have to decide between investing in S (expected value=$196,000) vs invest in B (Expected value = $156,000).

the profit maximizing decision is to invest in S.

The expected value of node 3 is

EV(3) = $196,000

Decision node 4: Hire the firm, get a bad forecast.

We have to decide between investing in S (expected value=$124,000) vs invest in B (Expected value = $144,000).

the profit maximizing decision is to invest in B.

The expected value of node 4 is

EV(4) = $144,000

Decision node 5: Do not Hire the firm

We have to decide between investing in S (expected value=$160,000) vs invest in B (Expected value = $150,000).

the profit maximizing decision is to invest in S.

The expected value of node 5 is

EV(5) = $160,000

Chance node 2: Hire a firm, ignore the cost to hire for now

The expected value for this node is

Finally at node 1, we have 2 options

1. Hire the firm at an expected value of $170,000, but spend $10,000 hire the firm and hence the net expected value of hiring the firm is $160,000
2. Do not hire and make an expected value of $160,000

Since hiring a firm at $10,000 and not hiring a firm yield the same expect value, we should be indifferent between the 2 decisions.

That means it does not matter, if we hire (at $10,000) or not hire the firm. It makes sense to hire the firm if it charges are less than $10,000

c) From part b, the expected value when the firm is hired is $170,000

This is the expected value with sample information

EVwSI = $170,000

The expected value when we do not hire is the expected value without sample information

EVwoSI=$160,000

the expected value of sample information is

EVSI = EVsSI - ECwoSI = $170000-$160000 = $10,000

The payoff table that we have is

	State of nature
Decision alternatives	Econ has a Good year	Econ has a bad year
Invest in S	$220,000	$100,000
Invest in B	$160,000	$140,000

If we know the state of nature, as Econ has a Good year, then we know that we would invest in S for a payoff of $220,000

If we know the state of nature, as Econ has a bad year, then we know that we would invest in B for a payoff of $140,000

That means if we have the perfect information about the economy then we would invest in S when the economy is good and invest in B when the economy is bad.

the expected value with perfect information is

The expected value without the perfect information is the expected value when we do not hire.

EVwoPI = $160,000

The expected value of Perfect Information is

EVPI = EVwPI - EVwoPI = $180,000-$160,000=$20,000