978-0393123982 Chapter 11 Solution Manual Part 1

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Chapter 11 NAME

Asset Markets

Introduction. The fundamental equilibrium condition for asset markets

is that in equilibrium the rate of return on all assets must be the same.

Thus if you know the rate of interest and the cash ﬂow generated by an

asset, you can predict what its market equilibrium price will be. This

condition has many interesting implications for the pricing of durable

assets. Here you will explore several of these implications.

Example: A drug manufacturing firm owns the patent for a new medicine.

The patent will expire on January 1, 1996, at which time anyone can pro-

duce the drug. Whoever owns the patent will make a profit of $1,000,000

per year until the patent expires. For simplicity, let us suppose that prof-

its for any year are all collected on December 31. The interest rate is

5%. Let us figure out what the selling price of the patent rights will be

on January 1, 1993. On January 1, 1993, potential buyers realize that

owning the patent will give them $1,000,000 every year starting 1 year

from now and continuing for 3 years. The present value of this cash ﬂow

$1,000,000

(1.05) +1,000,000

(1.05)2+1,000,000

(1.05)3∼$2,723,248.

Nobody would pay more than this amount for the patent since if you put

$2,723,248 at 5% interest, you could collect $1,000,000 a year from the

bank for 3 years, starting 1 year from now. The patent wouldn’t sell for

less than $2,723,248, since if it sold for less, one would get a higher rate

of return by investing in this patent than one could get from investing in

anything else. What will the price of the patent be on January 1, 1994?

At that time, the patent is equivalent to a cash ﬂow of $1,000,000 in 1

year and another $1,000,000 in 2 years. The present value of this ﬂow,

viewed from the standpoint of January 1, 1994, will be

$1,000,000

(1.05) +1,000,000

(1.05)2∼$1,859,310.

A slightly more difficult problem is one where the cash ﬂow from an

asset depends on how the asset is used. To find the price of such an asset,

one must ask what will be the present value of the cash ﬂow that the asset

yields if it is managed in such a way as to maximize its present value.

Example: People will be willing to pay $15 a bottle to drink a certain wine

this year. Next year they would be willing to pay $25, and the year after

that they would be willing to pay $26. After that, it starts to deteriorate

and the amount people are willing to pay to drink it falls. The interest

rate is 5%. We can determine not only what the wine will sell for but

also when it will be drunk. If the wine is drunk in the first year, it would

have to sell for $15. But no rational investor is going to sell the wine for

148 ASSET MARKETS (Ch. 11)

$15 in the first year, because it will sell for $25 one year later. This is a

66.66% rate of return, which is better than the rate of interest. When the

interest rate is 5%, investors are willing to pay at least $25/1.05 = $23.81

for the wine. So investors must outbid drinkers, and none will be drunk

this year. Will investors want to hold onto the wine for 2 years? In 2

years, the wine will be worth $26, so the present value of buying the wine

and storing it for 2 years is $26/(1.05)2= $23.58. This is less than the

present value of holding the wine for 1 year and selling it for $25. So, we

conclude that the wine will be drunk after 1 year. Its current selling price

will be $23.81, and 1 year from now, it will sell for $25.

11.0 Warm Up Exercise. Here are a few problems on present val-

ues. In all of the following examples, assume that you can both borrow

and lend at an annual interest rate of rand that the interest rate will

remainthesameforever.

one year from now, because if you put the dollar in the bank, then one

deposited in the bank right now would enable you to withdraw principal

and interest worth $1.

(d) The present value of an obligation to pay $Xone year from now is

(e) The present value of $X, to be received 2 years from now, is

(f) The present value of an asset that pays Xtdollars tyears from now

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(g) The present value of an asset that pays $X1one year from now, $X2in

(h) The present value of an asset that pays a constant amount, $Xper

year forever can be computed in two different ways. One way is to figure

out the amount of money you need in the bank so that the bank would

give you $Xper year, forever, without ever exhausting your principal.

getting $Xa year forever.

(i) Another way to calculate the present value of $Xa year forever is to

evaluate the infinite series ∞

(j) If the interest rate is 10%, the present value of receiving $1,000 one

value of receiving $1,000 a year forever, will be, to the nearest dollar,

(k) If the interest rate is 10%, what is the present value of an asset that

requires you to pay out $550 one year from now and will pay you back

11.1 (0) An area of land has been planted with Christmas trees. On

December 1, ten years from now, the trees will be ready for harvest. At

that time, the standing Christmas trees can be sold for $1,000 per acre.

The land, after the trees have been removed, will be worth $200 per acre.

There are no taxes or operating expenses, but also no revenue from this

land until the trees are harvested. The interest rate is 10%.

(a) What can we expect the market price of the land to be?

150 ASSET MARKETS (Ch. 11)

(b) Suppose that the Christmas trees do not have to be sold after 10

years, but could be sold in any year. Their value if they are cut before

they are 10 years old is zero. After the trees are 10 years old, an acre of

trees is worth $1,000 and its value will increase by $100 per year for the

next 20 years. After the trees are cut, the land on which the trees stood

can always be sold for $200 an acre. When should the trees be cut to

maximize the present value of the payments received for trees and land?

11.2 (0) Publicity agents for the Detroit Felines announce the signing

of a phenomenal new quarterback, Archie Parabola. They say that the

contract is worth $1,000,000 and will be paid in 20 installments of $50,000

per year starting one year from now and with one new installment each

year for next 20 years. The contract contains a clause that guarantees he

will get all of the money even if he is injured and cannot play a single game.

Sports writers declare that Archie has become an “instant millionaire.”

(a) Archie’s brother, Fenwick, who majored in economics, explains to

Archie that he is not a millionaire. In fact, his contract is worth less than

half a million dollars. Explain in words why this is so.

Archie’s college course on “Sports Management” didn’t cover present

values. So his brother tried to reason out the calculation for him. Here

is how it goes:

(b) Suppose that the interest rate is 10% and is expected to remain at

10% forever. How much would it cost the team to buy Archie a perpetuity

that would pay him and his heirs $1 per year forever,startingin1year?

In the last part, you found the present value of Archie’s contract

if he were going to get $50,000 a year forever. But Archie is not going

to get $50,000 a year forever. The payments stop after 20 years. The

present value of Archie’s actual contract is the same as the present value

of a contract that pays him $50,000 a year forever, but makes him pay

back $50,000 each year, forever, starting 21 years from now. Therefore

you can find the present value of Archie’s contract by subtracting the

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present value of $50,000 a year forever, starting 21 years from now from

the present value of $50,000 a year forever.

(d) If the interest rate is and will remain at 10%, a stream of payments

of $50,000 a year, starting 21 years from now has the same present value

years from now.

(e) If the interest rate is and will remain at 10%, what is the present value

(Hint: Thepresentvalueof$1tobepaidin20yearsis1/(1+r)20 =.15.)

11.3 (0) Professor Thesis is puzzling over the formula for the present

value of a stream of payments of $1 a year, starting 1 year from now and

continuing forever. He knows that the value of this stream is expressed

by the infinite series

S=1

1+r+1

(1 + r)2+1

(1 + r)3+…,

but he can’t remember the simplified formula for this sum. All he knows

is that if the first payment were to arrive today, rather than a year from

now, the present value of the sum would be $1 higher. So he knows that

S+1=1+ 1

(1 + r)+1

(1 + r)2+1

(1 + r)3+….

Professor Antithesis suffers from a similar memory lapse. He can’t

remember the formula for Seither. But, he knows that the present value

of $1 a year forever, starting right now has to be 1 + rtimes as large as

the present value of $1 a year, starting a year from now. (This is true

because if you advance any income stream by a year, you multiply its

present value by 1+r.) That is,

1+ 1

(1 + r)+1

(1 + r)2+1

(1 + r)3+…=(1+r)S.

(a) If Professor Thesis and Professor Antithesis put their knowledge to-

gether, they can express a simple equation involving only the variable S.

152 ASSET MARKETS (Ch. 11)

(b) The two professors have also forgotten the formula for the present

value of a stream of $1 per year starting next year and continuing for K

years. They agree to call this number S(K) and they see that

S(K)= 1

(1 + r)+1

(1 + r)2+…+1

(1 + r)K.

Professor Thesis notices that if each of the payments came 1 year earlier,

the present value of the resulting stream of payments would be

1+ 1

(1 + r)+1

(1 + r)2+…+1

(1 + r)K−1=S(K)+1−1

(1 + r)K.

Professor Antithesis points out that speeding up any stream of payments

by a year is also equivalent to multiplying its present value by (1 + r).

Putting their two observations together, the two professors noticed an

equation that could be solved for S(K). This equation is S(K)+1−

(1+r)K)/r.

Calculus 11.4 (0) You are the business manager of P. Bunyan Forests, Inc., and

are trying to decide when you should cut your trees. The market value of

the lumber that you will get if you let your trees reach the age of tyears

is given by the function W(t)=e.20t−.001t2. Mr. Bunyan can earn an

interest rate of 5% per year on money in the bank.

The rate of growth of the market value of the trees will be greater

from elementary calculus that if F(t)=eg(t), then F(t)/F (t)=g(t).)

(a) If he is only interested in the trees as an investment, how old should

11.5 (0) You expect the price of a certain painting to rise by 8% per

year forever. The market interest rate for borrowing and lending is 10%.

Assume there are no brokerage costs in purchasing or selling.

(a) If you pay a price of $xfor the painting now and sell it in a year, how

much has it cost you to hold the painting rather than to have loaned the

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(b) You would be willing to pay $100 a year to have the painting on your

walls. Write an equation that you can solve for the price xat which you

11.6 (2) Ashley is thinking of buying a truckload of wine for investment

purposes. He can borrow and lend as much as he likes at an annual

interest rate of 10%. He is looking at three kinds of wine. To keep our

calculations simple, let us assume that handling and storage costs are

negligible.

•Wine drinkers would pay exactly $175 a case to drink Wine Atoday.

But if Wine Ais allowed to mature for one year, it will improve. In fact

•From now until one year from now, Wine Bis indistinguishable

from Wine A. But instead of deteriorating after one year, Wine Bwill

•Wine drinkers would be willing to pay $100 per case to drink Wine

Cright now. But one year from now, they will be willing to pay $250

(a) What is the most Ashley would be willing to pay per case for Wine

(b) What is the most Ashley would be willing to pay per case for Wine

(Hint: When does the rate of return on holding wine get to 10%?)

(d) What will the price of Wine Cbe at the time it is first drunk?