Solutions to Homework Problems: Value of Information

1. The following sequence of questions is based on the example presented in class.

* Smith is deciding whether to invest $50,000 in a small local company that may go public within the year. If that happens S will double [quadruple] her money, but if it doesn't she'll lose 5%[20%].

* The other option is to take out a certificate of deposit that pays 10%. The buzz around town is that the chances are 50/50 that it will go public.

a) Calculate the expected utility (EU) of each option (i.e., stock vs. CD) in each scenario (i.e., double vs. quadruple).

	Goes public	Doesn’t go public
Invest	+100% $100,000	-5% $47,500
CD	+10% $55,000

EU: (100,000)(.5) + (47,500)(.5) = $73,750

EU: = $55,000

	Public	Not
Invest	$200,000	$40,000
CD	+10% $55,000

EU: (200,000)(.5) + (40,000)(.5) = $20,000

EU: = $55,000

b) Let the probability that the stock goes public be P. (In both the above scenarios P is 0.5.) For each scenario (double and quadruple), use algebra to find the value of P such that the EUs of the two options in that scenario are equal.

($100,000)(P) + ($47,500)(1-P) = $55,000

P = .1429 when double

($200,000)(P) + ($40,000)(1-P) = $55,000

P = .09375 when quadruple

Suppose now a new rumor floats around town that the decision has already been made and that for a price one can get a hold of the insider information.  Consider the meta-decision of whether to pay for info or not. Let us assume (unrealistically) that if we get inside info it will be perfectly accurate.

c) For each of the two original scenarios calculate the EU of the option of buying insider information.

	double
	public	not
Pay	0	Save $7,500
Don’t	-

(0)(.5) + (7,500)(.5) = $3,750

	quadruple
	public	not
Pay	0	Save $15,000
Don’t	-

(0)(.5) + (15,000)(.5) = $7,500

d) Now change the probability of going public from 0.5 to 0.1. Using this new probability calculate the new EU of the option of buying insider information for each scenario.

First, calculate the EUs with the new probability for both double and quadruple. NOTE that in the double case, buying the CD now looks best!

Double: EU: (100,000)(.1) + (47,500)(.9) = $52, 750

EU: = $55,000

Quadruple: EU: (200,000)(.1) + (40,000)(.9) = $56, 000

EU: = $55,000

	double
	public	not
Pay	Make $45,000	0
Don’t	-

(45,000)(.1) + (0)(.9) = $4,500

	quadruple
	public	not
Pay	0	Save $15,000
Don’t	-

(0)(.1) + (15,000)(.9) = $13,500

2. The following problem is also about the value of additional information, but it is presented in more artificial terms so that our intuitions furnish less guidance.

A closet contains 800 type A urns and 200 type B urns. The urns appear identical but A-type urns contain six blue balls and four red ones; B-type urns contain one blue ball and nine red ones. An urn is drawn at random from the closet and you must bet on the type of the urn. If you bet on type A and it is an A-type, you win $20; otherwise you lose $20. If you bet on type B and it is a B-type, you win $80; otherwise you lose $10. Assume that you maximize expected monetary values.

e) Set up a decision matrix for the choice between the EUs of the two bets. Which should you choose?

	A(.8)	B(.2)
Choose A	+$20	-$20	EV= 20(.8) – 20(2) = $12
Choose B	-$10	+80	EV= -10(.8) – 80(.2) = $8

f) Prior to actually making your bet, what is the maximum amount you should pay to learn the type of the urn?

	URN A	URN B
Pay	.8 $0	.2 $100
Don’t Pay

100(.2) = $20

g) Suppose that in the process of choosing the urn at random from the closet a ball accidentally tumbles out and you see that it is blue before it is restored to the urn. How would this bit of information affect the amount that you would pay to learn the type of the urn prior to making you bet, increase it, decrease it, make no difference? (If you say increase or decrease you need not calculate how much more or less.)

It would decrease the amount I would pay because the ratio of blue balls in URN A is significantly higher than URN B & since you would pick URN A w/o any insider info, there wouldn’t be much of a point to buy information that tells you to pick the urn you were going to pick anyway.

h) What difference would it make, if any, if the ball which tumbled out were red?

If the ball that tumbled out was red I would most likely pay more to buy the information because the ratio of red to blue is way higher in URN B, so if URN B is indeed chosen you would want to know so that you could change your URN “pick”.