present value of a dollar received in the
future is less than $1.
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, starting in 1 year?
$10.
(c) How much would it cost to buy a perpetuity that paid $50,000 a year forever, starting in one year?
$500,000.
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 �nd the present value of Archie’s contract by subtracting the
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 as a lump sum of $
500,000
to be received all at once, exactly 20 years from now.(e) If the interest rate is and will remain at 10%, what is the present value of $50,000 per year forever, starting 21 years from now?
$75,000.
(Hint: The present value of $1 to be paid in 20 years is 1/(1+ r)20 = .15.)(f) Now calculate the present value of Archie’s contract.
8.50 ×
50, 000 = $425, 000.
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 in�nite series
S = 1 1 + r + 1 (1 + r)2 + 1 (1 + r)3 + . . . ,
but he can’t remember the simpli�ed formula for this sum. All he knows is that if the �rst 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 su�ers from a similar memory lapse. He can’t remember the formula for S either. But, he knows that the present value of $1 a year forever, starting right now has to be 1 + r times 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. This equation is S + 1 =
(1 + r)S.
Solving this equation, they �nd that S =1/r.
(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 =
S(K)(1 + r).
Solving this equation for S(K), they �ndthat the formula for S(K) is
S(K) = (1−
(1+r)1 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 t years 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 than 5% until the trees reach
75
years of age. (Hint: It follows 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 Mr. Bunyan let the trees get?
75 years.
(b) At what age do the trees have the greatest market value?
100
years.
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 $x for 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 $x at the market interest rate?
It has cost .02x.
(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 x at which you would be just willing to buy the painting
.02x = 100.
(c) How much should you be willing to pay to buy the painting?
$5,000.
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 A today. But if Wine A is allowed to mature for one year, it will improve. In fact wine drinkers will be willing to pay $220 a case to drink this wine one year from today. After that, the wine gradually deteriorates and becomes less valuable every year.
• From now until one year from now, Wine B is indistinguishable from Wine A. But instead of deteriorating after one year, Wine B will improve. In fact the amount that wine drinkers would be willing to pay to drink Wine B will be $220 a case in one year and will rise by $10 per case per year for the next 30 years.
• Wine drinkers would be willing to pay $100 per case to drink Wine C right now. But one year from now, they will be willing to pay $250 per case to drink it and the amount they will be willing to pay to drink it will rise by $50 per case per year for the next 20 years.
(a) What is the most Ashley would be willing to pay per case for Wine
A?
$200.
(b) What is the most Ashley would be willing to pay per case for Wine B?
$200.
(Hint: When will Wine B be drunk?)(c) How old will Wine C be when it �rst becomes worthwhile for investors to sell o� their holdings and for drinkers to drink it?
6 years.
(Hint: When does the rate of return on holding wine get to 10%?) (d) What will the price of Wine C be at the time it is �rst drunk?$500 per case.
(e) What is the most that Ashley would be willing to pay today for a case of Wine C? (Hint: What is the present value of his investment if he sells it to a drinker at the optimal time?) Express your answer in exponential notation without calculating it out.
$500/1.1
6.
11.7 (0) Fisher Brown is taxed at 40% on his income from ordinary bonds. Ordinary bonds pay 10% interest. Interest on municipal bonds is not taxed at all.
(a) If the interest rate on municipal bonds is 7%, should he buy municipal
bonds or ordinary bonds?
Brown should buy municipal
bonds.
(b) Hunter Black makes less money than Fisher Brown and is taxed at only 25% on his income from ordinary bonds. Which kind of bonds should
he buy?
Black should buy ordinary bonds.
(c) If Fisher has $1,000,000 in bonds and Hunter has $10,000 in bonds, how much tax does Fisher pay on his interest from bonds?
0.
How much tax does Hunter pay on his interest from bonds?$250.
(d) The government is considering a new tax plan under which no interest income will be taxed. If the interest rates on the two types of bonds do not change, and Fisher and Hunter are allowed to adjust their portfolios, how much will Fisher’s after-tax income be increased?$30,000.
How much will Hunter’s after-tax income be increased?$250.
(e) What would the change in the tax law do to the demand for municipal bonds if the interest rates did not change?It would reduce
it to zero.
(f) What interest rate will new issues of municipal bonds have to pay
in order to attract purchasers?
They will have to pay
(g) What do you think will happen to the market price of the old mu- nicipal bonds, which had a 7% yield originally?
The price of
the old bonds will fall until their yield
equals 10%.
11.8 (0) In the text we discussed the market for oil assuming zero production costs, but now suppose that it is costly to get the oil out of the ground. Suppose that it costs $5 dollars per barrel to extract oil from the ground. Let the price in period t be denoted by pt and let r be the
interest rate.
(a) If a �rm extracts a barrel of oil in period t, how much pro�t does it make in period t?
p
t− 5.
(b) If a �rm extracts a barrel of oil in period t + 1, how much pro�t does it make in period t + 1?
p
t+1− 5.
(c) What is the present value of the pro�ts from extracting a barrel of oil in period t+1?
(p
t+1−5)/(1+r)
t+1.
What is the present value of pro�t from extracting a barrel of oil in period t?(p
t−5)/(1+r)
t.
(d) If the �rm is willing to supply oil in each of the two periods, what must be true about the relation between the present value of pro�ts from sale of a barrel of oil in the two periods?The present
values must be equal.
Express this relation as an equation.pt+1−5
(1+r)t+1
=
pt−5
(1+r)t
.
(e) Solve the equation in the above part for pt+1 as a function of pt and
r.
p
t+1= (1 + r)p
t− 5r.
(f) Is the percentage rate of price increase between periods larger or
smaller than the interest rate?