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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
is
$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?
(c) How much would it cost to buy a perpetuity that paid $50,000 a year
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
(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
(c) How much should you be willing to pay to buy the painting?
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
(c) How old will Wine Cbe when it first becomes worthwhile for investors
(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?
