. 3
( 39)


block in ¬nance.
translated into the same units”dollars today”before they can be compared or added. You
must be comfortable with the mechanics of computing the net present value of projects. You will
solve many more NPV problems throughout the book. As you will ¬nd out in later chapters,
despite its conceptual simplicity, the application of NPV in the real world is often surprisingly
di¬cult, because you must estimate cash ¬‚ows and discount factors.
¬le=constantinterest.tex: LP
28 Chapter 2. The Time Value of Money.

Let us work another example. A project costs $900 today, yields $200/year for two years, then
Work a project example.
$400/year for two years, and ¬nally requires a cleanup expense of $100. The prevailing interest
rate is 5% per annum. Should you take this project?
First you need to determine the cost of capital for tieing up money for 1 year, 2 years, 3 years,
First, determine the
multi-year costs of etc. The formula is
(1 + r0,t ) = (1 + r0,1 )t = (1 + 5%)t = 1.05t . (2.32)

So, for money right now, the cost of capital r0,0 is 1.050 ’ 1 = 0; for money in one year, it is
r0,1 is 1.051 ’ 1 = 5%; for money in two years, it is r0,2 is 1.052 ’ 1 = 10.25%; and so on. The
discount factors are one divided by one plus your cost of capital. So, a dollar in one year is
worth 1/(1 + 5%) ≈ 0.952 dollars today. A dollar in two years is worth 1/(1 + 5%)2 ≈ 0.907.
You must now multiply the promised payo¬s by the appropriate discount factor to get their
present value equivalents. Because present values are additive, you then sum up all the terms
to compute the overall net present value. Put this all into one table, as in Table 2.5, and you ¬nd
that the project NPV is $68.15. Because this is a positive value, you should take this project.
However, if the upfront expense was $1,000 instead of $900, the NPV would be negative “$31.85,
A negative version of the
same project. and you would be better o¬ investing the money into the appropriate sequence of bonds from
which the discount factors were computed. In this case, you should have rejected the project.

Table 2.5. Hypothetical Project Cash Flow Table

Project Interest Rate Discount Present
Time Cash Flow Annualized Holding Factor Value
t r r0,t PV( CFt )
1 + r0,t
Today “$900 5.00% 0.00% 1.000 “$900.00
Year +1 +$200 5.00% 5.00% 0.952 +$190.48
Year +2 +$200 5.00% 10.25% 0.907 +$181.41
Year +3 +$400 5.00% 15.76% 0.864 +$345.54
Year +4 +$400 5.00% 21.55% 0.823 +$329.08
Year +5 “$100 5.00% 27.63% 0.784 “$78.35
Net Present Value (Sum): $68.15

As a manager, you must estimate your project cash ¬‚ows. The appropriate interest rate (also called cost of capital
in this context) is provided to you by the opportunity cost of your investors”determined by the supply and demand
for capital in the broader economy, where your investors can place their capital instead. These are the two input
columns. We computed the remaining columns from these inputs.

Solve Now!
Q 2.27 Write down the NPV formula from memory.

Q 2.28 What is the NPV capital budgeting rule?

Q 2.29 If the cost of capital is 5% per annum, what is the discount factor for a cash ¬‚ow in two

Q 2.30 Interpret the meaning of the discount factor.

Q 2.31 What are the units on rates of return, discount factors, future values, and present values?
¬le=constantinterest.tex: RP
Section 2·4. Capital Budgeting.

Q 2.32 Determine the NPV of the project in Table 2.5, if the per-period interest rate is not 5%
but 8% per year. Should you take this project?

Q 2.33 A project has a cost of capital of 30%. The ¬nal payo¬ is $250. What should it cost today?

Q 2.34 A bond promises to pay $150 in 12 months. The annual applicable interest rate is 5%.
What is the bond™s price today?

Q 2.35 A bond promises to pay $150 in 12 months. The bank quotes you interest of 5%. What is
the bond™s price today?

Q 2.36 Work out the present value of your tuition payments for the next two years. Assume that
the tuition is $30,000 per year, payable at the start of the year. Your ¬rst tuition payment will
occur in six months, your second tuition payment will occur in eighteen months. You can borrow
capital at an interest rate of 6% per annum.

Q 2.37 Would it be good or bad for you, in terms of the present value of your liability, if your
opportunity cost of capital increased?

Q 2.38 The price of a bond o¬ering a ¬rm promise of $100 in 1 year is $95. What is the implied
interest rate? If the bond™s interest rate suddenly jumped up by 150 basis points, what would
the bond price be? How much would an investor gain/lose if she held the bond while the interest
rate jumped up by these 150 basis points?
¬le=constantinterest.tex: LP
30 Chapter 2. The Time Value of Money.

2·5. Summary

The chapter covered the following major points:

• Returns must not be averaged, but compounded over time.

• The time-value of money means that one dollar today is worth more than one dollar
tomorrow. Put di¬erently, the future value of one dollar is less than the present value of
one dollar.

• The discounted present value (PV) translates future cash values into present cash values.
The net present value (NPV) is the sum of all present values of a project, including the
usually negative upfront cash ¬‚ow”the investment cost”today.

• The NPV formula can be written as
NPV = CF0 + + + ... . (2.33)
1 + r0,1 1 + r0,2

In this context, r is called the discount rate or cost of capital, and 1/(1 + r ) is called the
discount factor.

• The Net Present Value Capital Budgeting Rule states that you should accept projects with
a positive NPV and reject projects with a negative NPV.

• 100 basis points are equal to 1%.

• Interest rate quotes are not interest rates”misunderstandings often reigns supreme. For
example, stated annual rates may not be the e¬ective rates that you will earn if you put
money into the bank. Or, a 3% coupon bond does not o¬er its investors a 3% rate of return.
If in doubt, ask!

• Bonds commit to payments in the future. Bank savings deposits are like a sequence of
1-day bonds, where a new interest is set daily.
A sudden increase in the prevailing economy-wide interest rate decreases the present
value of a bonds™ future payouts, and therefore decreases today™s price of the bond.
¬le=constantinterest.tex: RP
Section 2·5. Summary.

Solutions and Exercises

1. De¬nitely yes. Foregone salary is a cost that you are bearing. This can be reasonably estimated, and many
economic consulting ¬rms regularly do so. As to non-monetary bene¬ts, there is the reputation that the
degree o¬ers you, the education that betters you, and the beer consumption pleasure, if applicable.
2. In¬‚ows: Value of Implicit Rent. Capital Gain if house appreciates. Out¬‚ows: Maintenance Costs. Transaction
Costs. Mortgage Costs. Real Estate Tax. Theft. Capital Loss if house depreciates. And so on.

$1, 050 ’ $1, 000
3. r = = 5%.
$1, 000
4. r = = 2.5%.
$1, 000
5. 200.
6. 9.8%.

x ’ $250
7. r = 30% = ’ x = (1 + 30%) · $250 = $325.
8. (1 + 20%)5 ’ 1 ≈ 149%.
9. (1 + 100%)1/5 ’ 1 ≈ 14.87%.
10. $2, 000 · (1 + 25%)15 = $56, 843.42.
11. 1.0520 ’ 1 ≈ 165.3%, so you would end up with $200 · (1 + 165.3%) ≈ $530.66.

(1 + r0,0.25 )4 = (1 + r0,1 ) 1 + r0,1 ’ 1 = (1 + 50%)1/4 ’ 1 = 10.67% . (2.34)
r0,0.25 =

13. r0,2 = (1 + r0,1 ) · (1 + r1,2 ) ’ 1 = (1 + 5%) · (1 + 5%) ’ 1 = 10.25%.
14. r0,2 = (1 + r0,1 )10 ’ 1 = 62.89%.
r0,2 = (1 + r0,1 )100 ’ 1 == 130.5 = 13, 050% In words, this is about 130 times the initial investment, and
substantially more than 500% (5 times the initial investment).
16. About 22.5 years.
17. [1 + (’1/3)]5 ’ 1 ≈ ’86.83%. So about $2,633.74 is left.
First, solve for the interest rate: 1d ·(1 + r )5 = 2d ’ r = 14.87%. Therefore, in 100 years, he will have
(1 + r )100 = 1, 048, 576 denari. Of course, you can solve this in a simpler way: you have twenty 5-year
periods, in each of which the holdings double. So, the answer is 220 denari.
19. Yes, because dividends could then be reinvested, and earn extra return themselves.
20. (1 + 12%)(1/365) = 3.105bp/day.
21. The true daily interest rate, assuming 365 days, is 0.03105%. To get the true rate of return, compound this
over 7 days: (1 + 0.03105%)7 ≈ 1.0021758. So, your $100,000 will grow into $100,217.58. You can compute
this di¬erently: (1 + 12%)( 1/52.15).
22. 12%/365 = 3.288bp/day.
23. (1 + 0.03288%)7 ≈ 1.003116. So, your $100,000 will grow into $100,230.36.
24. (1 + 0.12/365)365 ≈ $112, 747.46.
25. The bank quote of 6% means that it will pay an interest rate of 6%/365= 0.0164384% per day. This earns an
actual interest rate of (1 + 0.0164384%)365 = 6.18% per year: Each invested $100 earns $6.18 over the year.
26. The bank quote of 8% means that you will have to pay an interest rate of 8%/365= 0.021918% per day. This
earns an actual interest rate of (1 + 0.021918%)365 = 8.33% per year: Each borrowed $100 requires $108.33
in repayment.

27. If you cannot do this by heart, do not go on until you have it memorized.
28. Accept if NPV is positive. Reject if NPV is negative.
29. 1/[(1 + 5%) · (1 + 5%)] = 0.9070.
30. It is today™s value in dollars for one future dollar, i.e., at a speci¬c point in time in the future.
31. The ¬rst two are unit-less, the latter two are in dollars (though the former is dollars in the future, while the
latter is dollars today).
¬le=constantinterest.tex: LP
32 Chapter 2. The Time Value of Money.

32. ’$900 + $200/(1 + 8%)1 + $200/(1 + 8%)2 + $400/(1 + 8%)3 + $400/(1 + 8%)4 ’ $100/(1 + 8%)5 ≈ $0.14. The
NPV is positive, therefore this is a worthwhile project.
$250 ’ x
33. r = 30% = ’ x = $250/(1 + 30%) ≈ $192.31.
34. $150/(1 + 5%) ≈ $142.86.
35. $150/(1 + 5%/365)365 ≈ $142.68.
The ¬rst tuition payment is worth $30, 000/(1 + 6%)1/2 ≈ $29, 139. The second tuition payment is worth
$30, 000/(1 + 6%)3/2 ≈ $27, 489. Thus, the total present value is $56,628.
37. Good. Your future payments would be worth less in today™s money.
38. The original interest rate is $100/$95 ’ 1 = 5.26%. Increasing the interest rate by 150 basis points is 6.76%.
This means that the price should be $100/(1 + 6.76%) = $93.67. A price change from $95 to $93.67 is a rate
of return of ’1.4%.

(All answers should be treated as suspect. They have only been sketched, and not been checked.)
More Time Value of Money

Quick Formulas for Perpetuities and Annuities
last ¬le change: Feb 23, 2006 (14:11h)

last major edit: Mar 2004, Nov 2004

This chapter is a natural extension of the previous chapter. We remain in our world of constant
interest rates, perfect foresight, and perfect markets. We cover two important but di¬erent
questions that remain:

1. What should in¬‚uence your investment decisions?
In particular, if you have a need to have cash early on, would this make you value short-
term projects more? And should you value companies which grow faster more?

2. Are there any shortcut formulas that can speed up your PV computations?

The answer to the ¬rst question will be that nothing other than NPV should matter for project
valuation, so neither your need to have cash nor the growth pattern of the ¬rm should make
any di¬erence
The answer to the second question will be that there are indeed valuation formulas for projects
that have peculiar cash ¬‚ow patterns”perpetuities and annuities. Their values are easy to
compute when interest rates are constant. This often makes them useful “quick-and-dirty”
tools for approximations. They are not only in wide use, but also are necessary to compute
cash ¬‚ows for common bonds (like mortgages) and to help you understand the economics of
corporate growth.

¬le=perpetuities.tex: LP
34 Chapter 3. More Time Value of Money.

3·1. Separating Investment Decisions and Present Values From
Other Considerations

There are two philosophical issues left in our perfect world. First, does it matter when we
need or generate money (i.e., who we are and how much cash we have), or does NPV subsume
everything important that there is to know about our projects? Second, are fast-growing ¬rms
better investments than slow-growing ¬rms?

3·1.A. Does It Matter When You Need Cash?

The answer is no”the value of a project is its net present value. It does not depend on who
Who owns a project is
not important in a owns it, and/or on when the owner needs cash. Here is why. You already know about the
perfectly competitive
time value of money, the fact that cash today is worth more than cash tomorrow. (A positive
zero-friction capital
time value of money means nothing more than that the interest rate is positive.) If you do
not agree”that is, if you value money tomorrow more than you value money today”then just
give it to me until you need it back. I can deposit it in my bank account to earn interest in
the interim. In our ideal, perfect capital market (without taxes, in¬‚ation, transaction costs, or
other issues), you can of course do better: you can always shift money between time periods
at an “exchange rate” that re¬‚ects the time value of money.
The shifting-at-will is worth illustrating. Assume that you have $150 cash on hand and that you
An “eager” consumer
will take the project. have exclusive access to a project that costs $100, and returns $200 next year. The appropriate
interest rate (cost of capital) is 10%”but you really want to live it up today. So, how much can
you consume? And, would you take the project? Here is the NPV prescription in a perfect
An “eager beaver”
consumer still takes any
positive NPV project.
• Sell the project in the competitive market for its NPV:

’$100 + = $81.82 .
1 + 10%

• Spend the $150 + ($181.82 ’ $100) = $231.82 today. You will be better o¬ taking the
project than consuming just your $150 cash at hand.

Now, assume that you are Austin Powers, the frozen detective, who cannot consume this year.
A “sleeper” consumer
also takes the project. So, how much will you be able to consume next year? And, would you take the project? The
NPV answer is

• Sell the project in the competitive market for

’$100 + = $81.82 .
1 + 10%

• Put the $81.82 into the bank for 10% today. Get $90 next year.

• Also put your $150 into the bank at 10% interest.

• Next year, consume $90 + $165 = $255.20.

Of course, an equally simple solution would be to take the project, and just put your remaining
$50 into a bank account.
¬le=perpetuities.tex: RP
Section 3·1. Separating Investment Decisions and Present Values From Other Considerations.

The point of this argument is simple: regardless of when you need cash, you are better o¬ The moral of the story:
consumption and
taking all positive NPV projects, and then using the capital markets to shift consumption into
investment decisions can
your preferred time for consumption. It makes no sense to let consumption decisions in¬‚uence be separated in a perfect
investment decisions. This is called the separation of decisions: you can make investment environment.
decisions without concern for your consumption preferences. (This separation of investment
and consumption decisions does not always hold in imperfect markets, in which you can face
di¬erent borrowing and lending interest rates. Chapter 6 will discuss this.)

Side Note: Of course, after they have lost their clients™ money, many brokers like to confuse this basic truth
by claiming that they invested their clients™ money for the long-term, and not for the short-term. This presumes
that long-term investments do worse in the short-run than short-term investment. This makes little sense,
because if this were the case, your broker should purchase the short-term investment, and sell it when it is
relatively worth more than the long-term investment in order to purchase relatively more of the (then relatively
cheaper) long-term investment. So, the fact is that no matter whether an investor needs money sooner or later,
the broker should always purchase the highest NPV investments. This gives clients the most wealth today”if
you care about future consumption, you can always save the extra cash from ¬nding the highest NPV investments

Solve Now!
Q 3.1 What is the main assumption that allows you to independently consider investment (pro-
ject) choices without regard to when you need wealth (or how much money you currently have
at hand)?

Q 3.2 You have $500 and a terminal illness. You cannot wait until the following project com-
pletes: The project costs $400 and o¬ers a rate of return of 15%, although equivalent interest
rates are only 10%. What should you do?

3·1.B. Corporate Valuation: Growth as Investment Criteria?

A similar question to that posed in the previous subsection is “Would it make more sense to The price should
incorporate the
invest in companies that grow more quickly rather than slowly”? If you wish, you can think of
attributes of the ¬rms.
this question as asking whether you should buy stocks in a fast-growing company like Microsoft
or in a slow-growing company like Exxon. The answer is that it does not matter in our perfect
world. Whether a company is growing fast or growing slow is already incorporated in the ¬rm™s
price today, which is just the discounted net present value of the ¬rm™s cash ¬‚ows that will
accrue to the owners. Therefore, neither is the better deal.
For example, consider company “Grow” (G) that will produce Should we invest in a
fast grower or a slow
Gt=1 = $100 G2 = $150 G3 = $250 ,

and company “Shrink” (S) that will produce

St=1 = $100 S2 = $90 S3 = $80 .

Is G not a better company to buy than S?
There is no uncertainty involved, and both ¬rms face the same cost of capital of 10% per annum. Let™s ¬nd out: Compute
the values.
Then the price of G today is

$100 $150 $250
PVt=0 ( G ) = + + ≈ $402.71 , (3.5)
(1 + 10%) (1 + 10%) (1 + 10%)3
1 2

and the price of S today is

$100 $90 $80
PVt=0 ( S ) = + + ≈ $225.39 . (3.6)
(1 + 10%) (1 + 10%) (1 + 10%)3
1 2
¬le=perpetuities.tex: LP
36 Chapter 3. More Time Value of Money.

If you invest in G, then next year you will have $100 cash, and own a company with $150 and
Your investment dollar
grows at the same rate, $250 cash ¬‚ows coming up. G™s value at time 1 (so PV now has subscript 1) will thus be
disconnected from the
cash ¬‚ow rate.
$150 $250
PVt=1 ( G ) = $100 + + ≈ $442.98 . (3.7)
(1 + 10%)1 (1 + 10%)2

Your investment will have earned a rate of return of $442.98/$402.71’1 = 10%. If you instead
invest in S, then next year you will receive $100 cash, and own a company with “only” $90 and
$80 cash ¬‚ows coming up. S™s value will thus be

$90 $80
PVt=1 ( S ) = $100 + + ≈ $247.93 . (3.8)
(1 + 10%) (1 + 10%)2

Your investment will have earned a rate of return of $247.39/$225.39 ’ 1 = 10%. In either
case, you will earn the fair rate of return of 10%. So”whether cash ¬‚ows are growing at a rate
of +50%, ’10%, +237.5%, or ’92% is irrelevant: the ¬rms™ market prices today already re¬‚ect
their future growth rates. There is no necessary connection between the growth rate of the
underlying project cash ¬‚ows or earnings, and the growth rate of your investment money (i.e.,
your expected rate of return). Make sure you understand the thought experiment here: This
statement that higher-growth ¬rms do not necessarily earn a higher rate of return does not
mean that a ¬rm in which managers succeed in increasing the future cash ¬‚ows at no extra
investment cost will not be worth more. Such ¬rms will indeed be worth more, and the current
owners will bene¬t from the rise in future cash ¬‚ows, but this will also be re¬‚ected immediately
in the price at which you can purchase this ¬rm.

3·1.C. The Value Today is just “All In¬‚ows” or just “All Out¬‚ows”

Now, the same argument applies to dividends: in the end, all earnings must be paid out (i.e., as
Dividend Payout Timing
can shift around, too. dividends). This does not need to occur at the same time: your earnings can grow today, and
your dividends can be zero or be shrinking today. In our earlier example, ¬rm G could be a slow
dividend payer or a fast dividend payer. It could pay $100 now, $150 next year and $250 in two
years. Or, it could reinvest the money, e¬ectively on your behalf, (at the same 10%, of course),
and then pay one big lump sum dividend of $100·(1+10%)2 +$150·(1+10%)+$250 = $536 at
the end of period 2. The dividend payout policy does not a¬ect G™s value today. The important
point is that the net present value of your total earnings and your total dividends must both
be equal to the price of the ¬rm in our perfect world”or you would get something for nothing
or lose something for nothing.
¬le=perpetuities.tex: RP
Section 3·1. Separating Investment Decisions and Present Values From Other Considerations.

Important: In a perfect market, the price and value of the ¬rm are determined
by the net present value of the ¬rm™s underlying projects. In total, the cash ¬‚ows
from the ¬rm™s projects belong to the ¬rm™s claim holders. Therefore, the net
present value of the ¬rm™s projects also must be the same as the net present value
of the ¬rm™s payouts to all claimants.

Firm Value = PV( “All Project Payouts” ) = PV( “All Project Cash Flows” ) . (3.9)
All Future All Future

The same logic applies to stock and debt. Debt receives some cash ¬‚ows generated
by the projects, which are then paid out as principal or interest. Similarly, stock
receives some cash ¬‚ows generated by the projects (sometimes casually called
earnings), which are then paid out as dividends.

Stock Value = =
PV( Dividends ) PV( Earnings to Stock ) .
All Future All Future
Debt Value = PV( Principal + Interest ) = PV( Cash Flows to Debt ) .
All Future
All Future

The time patterns of in¬‚ows or out¬‚ows only matters in determining net present
values. Beyond this in¬‚uence, it does not matter whether the ¬rm is a fast-earnings
grower, a slow-earnings grower, a fast-dividend payer, or a slow-dividend payer”
each ¬rm should be a fair investment. There is no value created by shifting earn-
ings or dividends across periods.

This simple insight is the basis of the “Modigliani-Miller” (M&M) theorems, which won two Nobel This is sometimes called
the M&M theorem, but
prizes in economics. (We will explain them in more detail in Chapter 21.) Remember, though,
holds in perfect markets
that the “perfect market” assumption is important”the value of the ¬rm is only the discounted only.
value of all future dividends or all future earnings if markets are not too far from perfect. This
is reasonable enough an assumption for large company stocks traded in the United States,
but not necessarily the case for small, privately held ¬rms. You should also realize that over
any limited time horizon, neither dividends nor earnings may represent value well”dividends
can be zero for a while, earnings can be negative, and the ¬rm can still have tremendous and
positive value.
There is an important corollary. If General Electric is about to win or has just had some great Any wealth gains accrue
to existing shareholders,
luck, having won a large defense contract (like the equivalent of a lottery), shouldn™t you pur-
not to new investors.
chase GE stock to participate in the windfall? Or, if Wal-Mart managers do a great job and have
put together a great ¬rm, shouldn™t you purchase Wal-Mart stock to participate in this wind-
fall? The answer is that you cannot. The old shareholders of Wal-Mart are no dummies. They
know the capabilities of Wal-Mart and how it will translate into cash ¬‚ows. Why should they
give you, a potential new shareholder, a special bargain for something that you contributed
nothing to? Just providing more investment funds is not a big contribution”after all, there are
millions of other investors equally willing to provide funds at the appropriately higher price.
It is competition”among investors for providing funds and among ¬rms for obtaining funds”
that determines the expected rate of return that investors receive and the cost of capital that
¬rms pay. There is actually a more general lesson here. Economics tells us that you must
have a scarce resource if you want to earn above-normal pro¬ts. Whatever is abundant and/or
provided by many will not be tremendously pro¬table.
¬le=perpetuities.tex: LP
38 Chapter 3. More Time Value of Money.

Solve Now!
Q 3.3 Presume that company G pays no interim dividends, so you receive $536 at the end of the
project. What is the G™s market value at time 1, 2, and 3? What is your rate of return in each
year? Assume that the cost of capital is still 10%.

Q 3.4 Presume that company G pays out the full cash ¬‚ows in earnings each period. What is G™s
market value at time 1, 2, and 3? What is your rate of return in each year?

Q 3.5 Which dividend stream increases the value of the ¬rm? Do you prefer a ¬rm paying a lot
of dividends, or a ¬rm paying no dividends until the very end?

Q 3.6 The discount rate is 15%/annum over all periods. Firm F ™s cash ¬‚ows start with $500 and
grow at 20% per annum for 3 years. Firm S™s cash ¬‚ows also start with $500 but shrink at 20%
per annum for 3 years. What are the prices of these two ¬rms, and what is the expected growth
rate of your money that you would invest into these two companies?

3·2. Perpetuities

We now proceed to our second subject of this chapter”the shortcut formulas to compute the
“Perpetuities” and
“Annuities” are projects present values of certain cash streams. A perpetuity is a project with a cash ¬‚ow that repeats
with special kinds of
forever. If the cost of capital (the appropriate discount rate) is constant and the amount of
cash ¬‚ows, which permit
money remains the same or grows at a constant rate, perpetuities lend themselves to quick
the use of short-cut
present value solutions”very useful when you need to come up with quick rule of thumb
estimates. Though the formulas may seem a bit intimidating at ¬rst, using them will quickly
become second nature to you.

3·2.A. The Simple Perpetuity Formula

Table 3.1. Perpetuity Stream of $2 With Interest Rate r = 10%

Cash Present
Time Flow Factor Value Cumulative
0 Nothing! You have no cash ¬‚ow here!
1 $2 $1.82 $1.82
(1 + 10%)1
2 $2 $1.65 $3.47
(1 + 10%)2
3 $2 $1.50 $4.97
(1 + 10%)3
. . . . .
. . . . .
. . . . .
50 $2 $0.02 $19.83
(1 + 10%)50
. . . . .
. . . . .
. . . . .
1/ $2/
t $2 (1 + 10%)t (1 + 10%)t
. . . . .
. . . . .
. . . . .
= $20.00
Net Present Value (Sum):
¬le=perpetuities.tex: RP
Section 3·2. Perpetuities.

At a constant interest rate of 10%, how much money do you need to invest today to receive the An Example Perpetuity
that pays $2 forever.
same dollar amount of interest of $2 each year, starting next year, forever? Such a payment
pattern is called a simple perpetuity. It is a stream of cash ¬‚ows that are the same for each
period and continue forever. Table 3.1 shows a perpetuity paying $2 forever if the interest rate
is 10% per annum.
To con¬rm the table™s last row, which gives the perpetuity™s net present value as $20, you can The Shortcut Perpetuity
spend from here to eternity to add up the in¬nite number of terms. But if you use a spreadsheet
to compute and add up the ¬rst 50 terms, you will get a PV of $19.83. If you add up the ¬rst
100 terms, you will get a PV of $19.9986. Trust me that the sum will converge to $20. This
is because there is a nice shortcut to computing the net present value of the perpetuity if the
cost of capital is constant.
Perpetuity PV = = $20
= .
The “t+1” in the formula is to remind you that the ¬rst cash ¬‚ow begins the following period,
not this period”the cash ¬‚ows are the same in 1 period, in 2 periods, etc.

Important: A stream of constant cash ¬‚ows, CF dollars each period and forever,
beginning next period, and is discounted at the same annual cost of capital r
forever is worth
PVt = .

The easiest way for you to get comfortable with perpetuities is to solve some problems.
Solve Now!
Q 3.7 From memory, write down the perpetuity formula. Be explicit on when the ¬rst cash ¬‚ow

Q 3.8 What is the PV of a perpetuity paying $5 each month, beginning next month, if the monthly
interest rate is a constant 0.5%/month (6.2%/year)?

Q 3.9 What is the PV of a perpetuity paying $15 each month, beginning next month, if the annual
interest rate is a constant 12.68% per year?

Q 3.10 Under what interest rates would you prefer a perpetuity that pays $2 million a year to a
one-time payment of $40 million?

Anecdote: The Oldest Institutions and Perpetuities
Perpetuities assume that projects last forever. But nothing really lasts forever. The oldest Western institution
today may well be the Roman Catholic Church, which is about 2,000 years old. The oldest existing corporation in
the United States is The Collegiate Reformed Protestant Dutch Church of the City of New York, formed in 1628
and granted a corporate charter by King William in 1696. The Canadian Hudson™s Bay Company was founded
in 1670, and claims to be the oldest continuously incorporated company in the world.
Guantanamo Naval Base was leased from Cuba in 1903 as a perpetuity by the United States in exchange for 2,000
pesos per annum in U.S. gold, equivalent to $4,085. In a speech, Fidel Castro has rede¬ned time as “whatever is
inde¬nite lasts 100 years.” In any case, the Cuban government no longer recognizes the agreement, and does
not accept the annual payments”but has also wisely not yet tried to expel the Americans.
Perpetuity bonds, called Consols, are fairly common in Britain, but not in the United States, because the American
IRS does not permit corporations deducting interest payments on Consols.
¬le=perpetuities.tex: LP
40 Chapter 3. More Time Value of Money.

Q 3.11 In Britain, there are Consol bonds that are perpetuity bonds. (In the United States, the
IRS does not allow companies to deduct the interest payments on perpetual bonds, so U.S. cor-
porations do not issue Consol bonds.) What is the value of a Consol bond that promises to pay
$2,000 per year if the prevailing interest rate is 4%?

3·2.B. The Growing Perpetuity Formula

Table 3.2. Perpetuity Stream With CF+1 = $2, Growth Rate g = 5%, and Interest Rate r = 10%

Discount Present
Time Cash Flow Rate Value Cumulative
0 Nothing. You have no cash ¬‚ows here.

(1 + 5%)0 ·$2 = (1 + 10%)1
1 $2.000 $1.818 $1.82
1 2
(1 + 5%) ·$2 = (1 + 10%)
2 $2.100 $1.736 $3.56
2 3
(1 + 5%) ·$2 = (1 + 10%)
3 $2.205 $1.657 $5.22
. . . . .
. . . . .
·$2 =
. . . . .
(1 + 10%)30
(1 + 5%) ·$2 =
30 $8.232 $0.236 $30.09
. . . . . .
. . . . . .
·$2 =
. . . . . .
. . .
. . .
(1 + 5%)t’1 ·$2 = (1 + 10%)t
t . . .
. . . . . .
. . . . . .
·$2 =
. . . . . .

= $40.00
Net Present Value (Sum):

What if the cash ¬‚ows are larger every period? A generalization of the perpetuity formula is
A growing perpetuity
assumes that cash ¬‚ows the growing perpetuity formula, in which the cash ¬‚ows grow by a constant rate g each period.
grow by a constant rate
The cash ¬‚ows of a sample growing perpetuity”which pays $2 next year, grows at a rate of 5%,
and faces a cost of capital of 10%”are shown in Table 3.2. The present value of the ¬rst 50
terms adds up to $36.28. The ¬rst 100 terms add up to $39.64. The ¬rst 200 terms add up to
$39.98. Eventually, the sum approaches the formula
PV of Growing Perpetuity0 = = $40
10% ’ 5%
= .
r ’g

As before, the “t+1” indicates that cash ¬‚ows begin next period, not this period, and r is the
interest rate minus g, the growth rate of your cash ¬‚ows. Note that the growth timing occurs
one period after the discount factor timing. For example, the cash ¬‚ow at time 30 is discounted
by (1 + r )30 , but its cash ¬‚ow is C0 multiplied by a growth factor of (1 + g)29 . We shall see later
that the growing perpetuity formula is most commonly used when nominal project cash ¬‚ows
are assumed to grow by the rate of in¬‚ation. We will use this formula extensively to obtain
“terminal values” in our ¬nal chapter on pro formas.
¬le=perpetuities.tex: RP
Section 3·2. Perpetuities.

Important: A stream of cash ¬‚ows, growing at a rate of g each period and
discounted at a constant interest rate r (which must be higher than g) is worth
CFt+1 (3.14)
PVt = .
r ’g

The ¬rst cash ¬‚ow, CFt+1 occurs next period, the second cash ¬‚ow of
CFt+2 = CFt+1 · (1 + g) occurs in two periods, and so forth, forever.

The growing annuity formula is worth memorizing.

What would happen if the cash ¬‚ows grew faster than the interest rate (g ≥ r )? Wouldn™t the Non-sensible answers.
formula indicate a negative PV? Yes, but this is because the entire scenario would be non-sense.
The PV in the perpetuities formulas is only less than in¬nity, because in today™s dollars, each
term in the sum is a little less than the term in the previous period. If g were greater than
r , however, the cash ¬‚ow one period later would be worth more even in today™s dollars”and
taking a sum over an in¬nite number of increasing terms would yield in¬nity as the value. A
value of in¬nity is clearly not sensible, as nothing in this world is worth an in¬nite amount of
money today. And, therefore, the growing perpetuity formula yields a non-sensical negative
value if g ≥ r ”as it should!
Solve Now!
Q 3.12 From memory, write down the growing perpetuity formula.

Q 3.13 What is the PV of a perpetuity paying $5 each month, beginning this month (in 1 second),
if the monthly interest rate is a constant 0.5%/month (6.2%/year), and the cash ¬‚ows will grow
at a rate of 0.1%/month (1.2%/year)?

3·2.C. A Growing Perpetuity Application: Individual Stock Valuation with Gordon Growth

With their ¬xed interest and growth rates and eternal payment requirements, perpetuities are Perpetuities are
rarely exactly correct. But they can be very helpful for quick back-of-the-envelope estimates.
approximations, but
For example, consider a stable business with pro¬ts of $1 million next year. Because it is stable, often give a useful upper
its pro¬ts are likely to grow at the in¬‚ation rate of, say, 2% per annum. This means it will earn bound.
$1,020,000 in two years, $1,040,400 in three years, etc. The ¬rm faces a cost of capital of 8%.
The growing perpetuity formula indicates that this ¬rm should probably be worth no more than

$1, 000, 000
Business Value = ≈ $16, 666, 667
8% ’ 2%
Business Value0 = ,
r ’g
because in reality, the ¬rm will almost surely not exist forever. Of course, in real life, there are
often even more signi¬cant uncertainties: next year™s pro¬t may be di¬erent, the ¬rm may grow
at a di¬erent rate (or may grow at a di¬erent rate for a while) or face a di¬erent cost of capital
for one-year loans than it does for thirty-year loans. Thus, $16.7 million should be considered
a quick-and-dirty useful approximation, perhaps for an upper limit, and not an exact number.
The growing perpetuity model is sometimes directly applied to the stock market. For example, The “Gordon Growth
Model”: constant eternal
if you believe that a stock™s dividends will grow by g = 5% forever, and if you believe that the
dividend growth.
appropriate rate of return is r = 10%, and you expect the stock to earn and/or pay dividends
of D = $10 next year, then you would feel that a stock price of

D1 $10
P0 = = = $200 (3.16)
r ’g 10% ’ 5%
¬le=perpetuities.tex: LP
42 Chapter 3. More Time Value of Money.

would be appropriate. In this context, the growing perpetuity model is often called the Gordon
growth model, after its inventor Myron Gordon.
Let us explore the Gordon growth model a bit more. In October 2004, Yahoo!Finance listed
Estimating the cost of
capital for GE. General Electric (GE) with a dividend yield of 2.43%. This is dividends divided by the stock
price, D/P , although it may be that dividends are from this year and not forward-looking.
(Fixing this would change our numbers only very little, so we shall not bother.) Rearrange our
formula 3.16:
= r ’g = 2.43% .
Therefore, we know that the market believes that the appropriate cost of capital (r ) for General
Electric exceeds its growth rate of dividends (g) by about 2.4%. Yahoo!Finance further links to a
summary of GE™s cash ¬‚ow statement, which indicates that GE paid $7.643 billion in dividends
in 2003, and $6.358 billion in 2001. Over these two years, the growth rate of dividends was
about 9.6% per annum ($6.358 · (1 + 9.6%)2 ≈ $7.643). Therefore, if we believe 9.6%/year is
a fair representation of the eternal growth rate of GE™s dividends, then the ¬nancial markets
valued GE as if it had a per-annum cost of capital of about

r= + g ≈ 2.4% + 9.6% ≈ 12% .

It is also not uncommon to repeat the same exercise with earnings”that is, presume that stock
You can do the same
with earnings. market values are capitalized as if corporate earnings were eternal cash ¬‚ows growing at a
constant rate g. Again, Yahoo!Finance gives us all the information we need. GE™s “trailing P/E”
ratio”calculated as the current stock price divided by historical earnings”was 21, its “forward
P/E” ratio”calculated as the price divided by analysts™ expectations of next year™s dividends”
was 18.5. The latter is P0 /E1 , and thus closer to what we want. Yahoo!Finance further tells us
that GE™s earnings growth was 6.3%”the g in our formula. Therefore,

E1 E1 1 1
P0 = ’ r= +g = +g ≈ + 6.3% ≈ 11.7% . (3.19)
r ’g P0 P0 /E1 18.5

It is important that you recognize that these are just models”approximations”that you cannot
Keep perspective!
take too seriously (in terms of accuracy). GE will not last forever, earnings are not the cash ¬‚ows
we need (more in Chapter 9), the discount rate is not eternally constant, earnings will not grow
forever at 6.3%, etc. However, the numbers are not uninteresting and probably not too far o¬,
either. GE is a very stable company that is likely to be around for a long time, and you could
do a lot worse than assuming that the cost of capital (for investing of projects that are similar
to GE stock ownership) is somewhere around 12% per annum”say, somewhere between 10%
to 14% per annum.
Solve Now!
Q 3.14 An eternal patent swap contract states that the patentee will pay the patenter $1.5 million
next year. The contract terms state growth with the in¬‚ation rate, which runs at 2% per annum.
The appropriate cost of capital is 14%. What is the value of this patenting contract?

Q 3.15 How would the patent swap contract value change if the ¬rst payment did not occur next
year, but tonight?

Q 3.16 A stock is paying a quarterly dividend of $5 in one month. The dividend is expected to
increase every quarter by the in¬‚ation rate of 0.5% per quarter”so it will be $5.025 in the next
quarter (i.e., paid out in four months). The prevailing cost of capital for this kind of stock is 9%
per annum. What should this stock be worth?

Q 3.17 If a $100 stock has earnings that are $5 per year, and the appropriate cost of capital
for this stock is 12% per year, what does the market expect the ¬rm™s “as-if-eternal dividends” to
grow at?
¬le=perpetuities.tex: RP
Section 3·3. The Annuity Formula.

3·3. The Annuity Formula

The second type of cash ¬‚ow stream that lends itself to a quick formula is an annuity, which is An Annuity pays the
same amount for T
a stream of cash ¬‚ows for a given number of periods. Unlike a perpetuity, payments stop after
T periods. For example, if the interest rate is 10% per period, what is the value of an annuity
that pays $5 per period for 3 periods?
Let us ¬rst do this the slow way. We can hand-compute the net present value to be
$5 $5 $5
PV0 = + + ≈ $12.4343
1 + 10% (1 + 10%)2 (1 + 10%)3
= + + .
(1 + r0,1 ) (1 + r0,2 ) (1 + r0,3 )

So, what is the shortcut to compute the net present value of an annuity? It is the annuity
formula, which is
1 ’ [1/(1 + 10%)]3
PV = $5 · ≈ $12.4343 ,
1 ’ [1/(1 + r )]T
PV = CFt+1 · = .
Is this really a short-cut? Maybe not for 3 periods, but try a 360-period annuity, and let me
know which method you prefer. Either works.

Important: A stream of constant cash ¬‚ows, beginning next period and lasting
for T periods, and discounted at a constant interest rate r , is worth
CFt+1 1 (3.22)
PVt = · 1’ .
(1 + r )T

3·3.A. An Annuity Application: Fixed-Rate Mortgage Payments

Most mortgages are ¬xed rate mortgage loans, and they are basically annuities. They promise Mortgages are annuities,
so the annuity formula
a speci¬ed stream of equal cash payments each month to a lender. A 30-year mortgage with
is quite useful.
monthly payments is really a 360-payments annuity. (The “annu-ity” formula should really be
called a “month-ity” formula in this case.) So, what would be your monthly payment if you took
out a 30-year mortgage loan for $500,000 at an interest rate of 7.5% per annum?
Before you can proceed further, you need to know one more bit of institutional knowledge here: Lenders quote interest
rates using the same
Mortgage providers”like banks”quote interest by just dividing the mortgage quote by 12, so
convention that banks
the true monthly interest rate is 7.5%/12 = 0.625%. (They do not compound; if they did, the use.
monthly interest rate would be (1 + 7.5%)1/12 ’ 1 = 0.605%.)
So our 30-year mortgage is an annuity with 360 equal payments with a discount rate of 0.625% The mortgage payment
can be determined by
per month. Its PV of $500,000 is the amount that you are borrowing. We want to determine
solving the Annuity
the ¬xed monthly cash ¬‚ow that gives the annuity this value: formula.

CFt+1 1
$500, 000 = · 1’ ≈ CFt+1 · 143.02
(1 + 0.625%)360
CFt+1 1
= · 1’ .
(1 + r )T

Solving this for the cash ¬‚ow tells you that the monthly payment on your $500,000 mortgage
will be $3,496.07 for 360 months, beginning next month.
¬le=perpetuities.tex: LP
44 Chapter 3. More Time Value of Money.

Uncle Sam allows mortgage borrowers to deduct the interest, but not the principal, from
Side Note:
their tax bills. The IRS imputes interest on the above mortgage as follows: In the ¬rst month, Uncle Sam
proclaims 0.625%·$500, 000 = $3, 125 to be the tax-deductible mortgage interest payment. Therefore, the
principal repayment is $3, 496.07 ’ $3, 125 = $371.07 and remaining principal is $499,628.93. The follow-
ing month, Uncle Sam proclaims 0.625%·$499, 628.93 = $3, 122.68 to be the tax-deductible interest payment,
$3, 496.07 ’ $3, 122.68 = $373.39 as the principal repayment, and $499,255.54 as the remaining principal. And
so on.

3·3.B. An Annuity Example: A Level-Coupon Bond

Let us exercise our new found knowledge in a more elaborate example”this time with bonds.
Coupon bonds pay not
only at the ¬nal time. Bonds come in many di¬erent varieties, but one useful classi¬cation is into coupon bonds
and zero bonds (short for zero coupon bonds). A coupon bond pays its holder cash at many
di¬erent points in time, whereas a zero bond pays only a single lump sum at the maturity of
the bond. Many coupon bonds promise to pay a regular coupon similar to the interest rate
prevailing at the time of the bond™s original sale, and then return a “principal amount” plus a
¬nal coupon at the end of the bond.
For example, think of a coupon bond that will pay $1,500 each half-year (semi-annual payment
Bonds are speci¬ed by
their promised payout is very common) for ¬ve years, plus an additional $100,000 in 5 years. This payment pattern
is so common that it has specially named features: A bond with coupon payments that remain
the same for the life of the bond is called a level-coupon bond. The $100,000 here would be
called the principal, in contrast to the $1,500 semi-annual coupon. Level bonds are commonly
named by just adding up all the coupon payments over one year (here, $3,000), and dividing
this sum of annual coupon payments by the principal. So this particular bond would be called a
“3% semi-annual coupon bond” ($3,000 coupon per year, divided by the principal of $100,000).
Now, the “3% coupon bond” is just a naming convention for the bond with these speci¬c cash
¬‚ow patterns”it is not the interest rate that you would expect if you bought this bond. In
Section 2·3.C, we called such name designations interest quotes, as distinct from interest rates.
Of course, even if the bond were to cost $100,000 today (and we shall see below that it usually
does not), the interest rate would not be 3% per annum, but (1 + 1.5%)2 ’ 1 ≈ 3.02% per annum.

Side Note: Par value is a vacuous concept, sometimes used to compute coupon payout schedules. Principal
and par value, and/or interest and coupon payment need not be identical, not even at the time of issue, much
less later. For the most part, par value is best ignored.

We now solve for the value of our coupon bond. Incidentally, you may or may not ¬nd the
annuity formula helpful”you can use it, but you do not need it. Our task is to ¬nd the value of
a “3% coupon bond” today. First, we write down the payment structure for our 3% semi-annual
coupon bond. This comes from its de¬ned promised patterns,

A Typical Coupon Bond
Due Bond Due Bond
Year Date Payment Year Date Payment
0.5 Nov 2002 $1,500 3.0 May 2005 $1,500
1.0 May 2003 $1,500 3.5 Nov 2005 $1,500
1.5 Nov 2003 $1,500 4.0 May 2006 $1,500
2.0 May 2004 $1,500 4.5 Nov 2006 $1,500
2.5 Nov 2004 $1,500 5.0 May 2007 $101,500

Second, we need to determine the appropriate expected rates of return to use for discounting.
Step 2: ¬nd the
appropriate costs of We shall assume that the prevailing interest rate is 5% per annum, which translates into 2.47%
for 6 months, 10.25% for two years, etc.

Maturity Yield Maturity Yield
6 Months 2.47% 36 Months 15.76%
12 Months 5.00% 42 Months 18.62%
18 Months 7.59% 48 Months 21.55%
24 Months 10.25% 54 Months 24.55%
30 Months 12.97% 60 Months 27.63%
¬le=perpetuities.tex: RP
Section 3·3. The Annuity Formula.

Our third step is to compute the discount factors, which are just 1/(1 + r0,t ), and to multiply Step 3: Compute the
discount factor is
each future payment by its discount factor. This will give us the present value (PV) of each
1/(1 + r0,t ).
bond payment, and therefore the bond overall value:

Due Bond Rate of Discount Present
Year Date Payment Return Factor Value
0.5 Nov 2002 $1,500 2.47% 0.976 $1,463.85
1.0 May 2003 $1,500 5.00% 0.952 $1,428.57
1.5 Nov 2003 $1,500 7.59% 0.929 $1,349.14
2.0 May 2004 $1,500 10.25% 0.907 $1,360.54
2.5 Nov 2004 $1,500 12.97% 0.885 $1,327.76
3.0 May 2005 $1,500 15.76% 0.864 $1,295.76
3.5 Nov 2005 $1,500 18.62% 0.843 $1,264.53
4.0 May 2006 $1,500 21.55% 0.823 $1,234.05
4.5 Nov 2006 $1,500 24.55% 0.803 $1,204.31
5.0 May 2007 $101,500 27.63% 0.784 $79,527.91
Sum $91,501.42

We now know that, in our perfect world, we would expect this 3% level-coupon bond to be Common naming
conventions for this
trading for $91,501.42 today. Because the current price of the bond is below the so-named
type of bond: coupon
¬nal principal payment of $100,000, our bond would be said to trade at a discount. (The rate is not interest rate!
opposite would be a bond trading at a premium.)
The above computation is a bit tedious. Can we translate it into an annuity? Yes! We will work Using the annuity to
make this faster.
in half-year periods. We thus have 10 coupon cash ¬‚ows, each $1,500, at a per-period interest
rate of 2.47%. So, according to our formula, the coupon payments are worth
1 ’ [1/(1 + r )]T
PV = CFt+1 ·
1 ’ [1/(1 + 2.47%)]10 (3.24)
= $1, 500 ·

≈ .
$13, 148.81

In addition, we have our $100,000 repayment of principal, which is worth
$100, 000
PV = ≈ $78, 352.62
1 + 27.63%
PV = ≈ $78, 352.62 .
(1 + r0,5 )

Together, these present values of our bond™s cash ¬‚ows add up to $91, 501.42.
Prevailing Interest Rates and Bond Values: We already know that the value of one ¬xed future The effect of a change in
interest rates.
payment and the interest rate move in opposite directions. Given that we now have many
payments, what would happen if the economy-wide interest rates were to suddenly move from
5% per annum to 6% per annum? The semi-annual interest rate would now increase from 2.47%

r= 1 + 6% ’ 1 ≈ 2.96% ⇐ (1 + 2.96%) · (1 + 2.96%) ≈ (1 + 6%) .

To get the bond™s new present value, reuse our formula
1 ’ [1/(1 + r )]T CFT
PV = CFt+1 · +
1 + r0,T
1 ’ [1/(1 + 2.96%)]10 $100, 000 (3.27)
= $1, 500 · +
(1 + 2.96%)10

≈ + ≈ $87, 549.70 .
$12, 823.89 $74, 725.82

So, our bond would have lost $3,951.72, or 4.3% of our original investment”which is the same
inverse relation between bond values and prevailing economy-wide interest rates that we ¬rst
saw on Page 24.
¬le=perpetuities.tex: LP
46 Chapter 3. More Time Value of Money.

Important Repeat of Quotes vs. Returns: Never confuse a bond designation with the interest
Interest Rates vs.
Coupon Rates. it pays. The “3%-coupon bond” is just a designation for the bond™s payout pattern. Our bond
will not give you coupon payments equal to 1.5% of your $91,502.42 investment (which would
be $1,372.52). The prevailing interest rate (cost of capital) has nothing to do with the quoted
interest rate on the coupon bond. We could just as well determine the value of a 0%-coupon
bond, or a 10% coupon bond, given our prevailing 5% economy-wide interest rate. Having said
all this, in the real world, many corporations choose coupon rates similar to the prevailing
interest rate, so that at the moment of inception, the bond will be trading at neither premium
nor discount. So, at least for this one brief at-issue instant, the coupon rate and the economy-
wide interest rate may actually be fairly close. However, soon after issuance, market interest
rates will move around, while the bond™s payments remain ¬xed, as designated by the bond™s
coupon name.
Solve Now!
Q 3.18 If you can recall it, write down the annuity formula.

Q 3.19 What is the PV of a 360 month annuity paying $5 per month, beginning at $5 next month,
if the monthly interest rate is a constant 0.5%/month (6.2%/year)?

Q 3.20 Mortgages are not much di¬erent from rental agreements. For example, what would
your rate of return be if you rented your $500,000 warehouse for 10 years at a monthly lease
payment of $5,000? If you can earn 5% elsewhere, would you rent out your warehouse?

Q 3.21 What is the monthly payment on a 15-year mortgage for every $1,000 of mortgage at
an e¬ective interest rate of 6.168% per year (here, 0.5% per month)?

Q 3.22 Solve Fibonacci™s annuity problem from Page 23: Compare the PV of a stream of quar-
terly cash ¬‚ows of 75 bezants vs. the PV of a stream of annual cash ¬‚ows of 300 bezants. Pay-
ments are always at period-end. The interest rate is 2 bezants per month. What is the relative
value of the two streams? Compute the di¬erence for a 1-year investment ¬rst.

Q 3.23 In L™Arithmetique, written in 1558, Jean Trenchant posed the following question: “In the
year 1555, King Henry, to conduct the war, took money from bankers at the rate of 4% per fair
[quarter]. That is better terms for them than 16% per year. In this same year before the fair
of Toussaints, he received by the hands of certain bankers the sum of 3,945,941 ecus and more,
which they called ˜Le Grand Party™ on the condition that he will pay interest at 5% per fair for
41 fairs after which he will be ¬nished. Which of these conditions is better for the bankers?”
Translated, the question is whether a perpetuity at 4% per quarter is better or worse than a
41-month annuity at 5%.

Q 3.24 Assume that a 3% level-coupon bond has not just 5 years with 10 payments, but 20 years
with 40 payments. Also, assume that the interest rate is not 5% per annum, but 10.25% per
annum. What are the bond payment patterns and the bond™s value?

Q 3.25 Check that the rates of return in the coupon bond valuation example on Page 45 are

Q 3.26 In many a de¬ned contribution pension plan, the employer provides a ¬xed percentage
contribution to the employee™s retirement. Let us assume that you must contribute $4,000 per
annum beginning next year, growing annually with the in¬‚ation rate of 2%/year. What is this
individual™s pension cost to you of hiring a 25-year old, who will stay with the company for 35
years? Assume a discount rate of 8% per year. NOTE: You need the growing annuity formula 3.28,
which you should look up.
¬le=perpetuities.tex: RP
Section 3·3. The Annuity Formula.

3·3.C. The Special Cash Flow Streams Summarized

I am not a fan of memorization, but you must remember the growing perpetuity formula. (It The growing annuity
formula ” use is rare.
would likely be useful if you could also remember the annuity formula.) These formulas are
used in many di¬erent contexts. There is also a growing annuity formula, which nobody
remembers, but which you should know to look up if you need it. It is

(1 + g)T
PVt = · 1’ . (3.28)
(1 + r )T
r ’g

It is sometimes used in the context of pension cash ¬‚ows, which tend to grow for a ¬xed number
of time periods and then stop. However, even then it is not a necessary device. It is often more
convenient and ¬‚exible to just work with the cash ¬‚ows themselves within a spreadsheet.
Figure 3.1 summarizes the four special cash ¬‚ow formulas. The present value of a growing A summary
perpetuity must decline (r > g), but if g > 0 declines at a rate that is slower than that of
the simple perpetuity. The annuity stops after a ¬xed number of periods, here T = 7, which
truncates both the cash ¬‚ow stream and its present values.
¬le=perpetuities.tex: LP
48 Chapter 3. More Time Value of Money.

Figure 3.1. The Four Payo¬ Streams and Their Present Values

Simple Perpetuity Growing Perpetuity

Future Cash Flows Future Cash Flows

Forever Forever
NOW 1 2 3 4 5 6 7 Time NOW 1 2 3 4 5 6 7 Time

Their Present Values Their Present Values

Forever Forever
NOW 1 2 3 4 5 6 7 Time NOW 1 2 3 4 5 6 7 Time

Formula: PV = Formula: PV =
r ’g

Simple Annuity (T = 7) Growing Annuity (T = 7)

Future Cash Flows Future Cash Flows
No More

No More


NOW 1 2 3 4 5 6 7 Time NOW 1 2 3 4 5 6 7 Time

Their Present Values Their Present Values
No More

No More


NOW 1 2 3 4 5 6 7 Time NOW 1 2 3 4 5 6 7 Time

CF 1 CF1
Formula: PV = · 1’ Formula: PV = · 1’
1+r r ’g 1+r
¬le=perpetuities.tex: RP
Section 3·4. Summary.

3·4. Summary

The chapter covered the following major points:

• In a perfect market, consumption and investment decisions can be made independently.
You should always take the highest NPV projects, and use the capital markets to shift
cash into periods in which you want it.

• In a perfect market, ¬rms are worth the present value of their assets. Whether they grow
fast or slow is irrelevant except to the extent that this determines their PV. Indeed, ¬rms
can shift the time patterns of cash ¬‚ows and dividends without changing the underlying
¬rm value.

• In a perfect market, the gains from sudden surprises accrue to old owners, not new capital
provides, because old owners have no reason to want to share the spoils.

• The PV of a growing perpetuity”with constant-growth (g) cash ¬‚ows CF beginning next
year and constant per-period interest rate r ”is

PVt = (3.30)
r ’g

• The application of the growing perpetuity formula to stocks is called the Gordon dividend
growth model.

• The PV of an annuity”T periods of constant CF cash ¬‚ows (beginning next year) and
constant per-period interest rate r ”is

1 ’ [1/(1 + r )]T
PVt = CFt+1 · . (3.31)

• Fixed-rate mortgages are annuities, and therefore can be valued with the annuity formula.
¬le=perpetuities.tex: LP
50 Chapter 3. More Time Value of Money.

Solutions and Exercises

1. The fact that you can use capital markets to shift money forth and back without costs.
2. Take the project. If you invest $400, the project will give $400 · (1 + 15%) = $460 next period. The capital
markets will value the project at $418.18. Sell it at this amount. Thereby, you will end up being able to
consume $500 ’ $400 + $418.18 = $518.18.
3. For easier naming, we call year 0 to be 2000. The ¬rm™s present value in 2000 is $536/1.103 ≈ $402.71”but
we already knew this. If you purchase this company, its value in 2001 depends on a cash ¬‚ow stream that is
$0 in 2001, $0 in year 2002 and $536 in year 2003. So, it will be worth $536/1.102 ≈ $442.98 in 2001. In
2002, your ¬rm will be worth $536/1.10 = $487.27. Finally, in 2003, it will be worth $536. Each year, you
expect to earn 10%, which you can compute from the four ¬rm values.
4. Again, we call year 0 2000. The ¬rm™s present value in 2000 is based on dividends of $100, $150, and $250
in the next three years. So, the ¬rm value in 2000 is the $402.71 in Formula 3.5. The ¬rm value in 2001 is in
Formula 3.7, but you immediately receive $100 in cash, so the ¬rm is worth only $442.98 ’ $100 = $342.98.
As an investor, you would have earned a rate of return of $442.98/$402.71 ’ 1 = 10%. The ¬rm value in 2002
PVt=2 ( G ) = ≈ $227.27 . (3.32)
(1 + 10%)
but you will also receive $150 in cash, for a total ¬rm-related wealth of $377.27. In addition, you will have the
$100 from 2001, which would have grown to $110”for a total wealth of $487.27. Thus, starting with wealth of
$442.98 and ending up with wealth of $487.27, you would have earned a rate of return of $487.27/$442.98 ’
1 = 10%. A similar computation shows that you will earn 10% from 2002 ($487.27) to 2003 ($536.00).
5. It makes no di¬erence!
6. F ™s cash ¬‚ows are $500, $600, and $720. Its value is therefore $1,361.88. Firm S™s cash ¬‚ows are $500, $400,
and $320. Its value is therefore $947.65. Both ¬rms o¬er your investment dollar a 15% rate of return.

7. CF1 /r . The ¬rst cash ¬‚ow occurs next period, not this period.
8. PV = CF1 /r = $5/0.005 = $1, 000.
9. PV = CF1 /r = $15/.01 = $1, 500.
10. You would prefer the perpetuity if the interest rate/cost of capital was less than 5%.
11. PV = $2, 000/4% = $50, 000.


. 3
( 39)