ńņš. 3 |

translated into the same unitsā”dollars todayā”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 ļ¬nd out in later chapters,

despite its conceptual simplicity, the application of NPV in the real world is often surprisingly

diļ¬cult, because you must estimate cash ļ¬‚ows and discount factors.

ļ¬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 ļ¬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

capital.

(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 ā’ 1 = 0; for money in one year, it is

r0,1 is 1.051 ā’ 1 = 5%; for money in two years, it is r0,2 is 1.052 ā’ 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%) ā 0.952 dollars today. A dollar in two years is worth 1/(1 + 5%)2 ā 0.907.

You must now multiply the promised payoļ¬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 ļ¬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 ā“$31.85,

A negative version of the

same project. and you would be better oļ¬ 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

1

t r r0,t PV( CFt )

CFt

1 + r0,t

Today ā“$900 5.00% 0.00% 1.000 ā“$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 ā“$100 5.00% 27.63% 0.784 ā“$78.35

Net Present Value (Sum): $68.15

As a manager, you must estimate your project cash ļ¬‚ows. The appropriate interest rate (also called cost of capital

in this context) is provided to you by the opportunity cost of your investorsā”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 ļ¬‚ow in two

years??

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?

ļ¬le=constantinterest.tex: RP

29

Section 2Ā·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 ļ¬nal payoļ¬ 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 bondā™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 bondā™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 ļ¬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 oļ¬ering a ļ¬rm promise of $100 in 1 year is $95. What is the implied

interest rate? If the bondā™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?

ļ¬le=constantinterest.tex: LP

30 Chapter 2. The Time Value of Money.

2Ā·5. Summary

The chapter covered the following major points:

ā¢ Returns must not be averaged, but compounded over time.

ā¢ The time-value of money means that one dollar today is worth more than one dollar

tomorrow. Put diļ¬erently, the future value of one dollar is less than the present value of

one dollar.

ā¢ 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 ļ¬‚owā”the investment costā”today.

ā¢ The NPV formula can be written as

CF1 CF2

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.

ā¢ The Net Present Value Capital Budgeting Rule states that you should accept projects with

a positive NPV and reject projects with a negative NPV.

ā¢ 100 basis points are equal to 1%.

ā¢ Interest rate quotes are not interest ratesā”misunderstandings often reigns supreme. For

example, stated annual rates may not be the eļ¬ective rates that you will earn if you put

money into the bank. Or, a 3% coupon bond does not oļ¬er its investors a 3% rate of return.

If in doubt, ask!

ā¢ 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 bondsā™ future payouts, and therefore decreases todayā™s price of the bond.

ļ¬le=constantinterest.tex: RP

31

Section 2Ā·5. Summary.

Solutions and Exercises

1. Deļ¬nitely yes. Foregone salary is a cost that you are bearing. This can be reasonably estimated, and many

economic consulting ļ¬rms regularly do so. As to non-monetary beneļ¬ts, there is the reputation that the

degree oļ¬ers you, the education that betters you, and the beer consumption pleasure, if applicable.

2. Inļ¬‚ows: Value of Implicit Rent. Capital Gain if house appreciates. Outļ¬‚ows: Maintenance Costs. Transaction

Costs. Mortgage Costs. Real Estate Tax. Theft. Capital Loss if house depreciates. And so on.

$1, 050 ā’ $1, 000

3. r = = 5%.

$1, 000

$25

4. r = = 2.5%.

$1, 000

5. 200.

6. 9.8%.

x ā’ $250

7. r = 30% = ā’ x = (1 + 30%) Ā· $250 = $325.

$250

8. (1 + 20%)5 ā’ 1 ā 149%.

9. (1 + 100%)1/5 ā’ 1 ā 14.87%.

10. $2, 000 Ā· (1 + 25%)15 = $56, 843.42.

11. 1.0520 ā’ 1 ā 165.3%, so you would end up with $200 Ā· (1 + 165.3%) ā $530.66.

12.

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

4

r0,0.25 =

13. r0,2 = (1 + r0,1 ) Ā· (1 + r1,2 ) ā’ 1 = (1 + 5%) Ā· (1 + 5%) ā’ 1 = 10.25%.

14. r0,2 = (1 + r0,1 )10 ā’ 1 = 62.89%.

r0,2 = (1 + r0,1 )100 ā’ 1 == 130.5 = 13, 050% In words, this is about 130 times the initial investment, and

15.

substantially more than 500% (5 times the initial investment).

16. About 22.5 years.

17. [1 + (ā’1/3)]5 ā’ 1 ā ā’86.83%. So about $2,633.74 is left.

First, solve for the interest rate: 1d Ā·(1 + r )5 = 2d ā’ r = 14.87%. Therefore, in 100 years, he will have

18.

(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 ā 1.0021758. So, your $100,000 will grow into $100,217.58. You can compute

this diļ¬erently: (1 + 12%)( 1/52.15).

22. 12%/365 = 3.288bp/day.

23. (1 + 0.03288%)7 ā 1.003116. So, your $100,000 will grow into $100,230.36.

24. (1 + 0.12/365)365 ā $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%) Ā· (1 + 5%)] = 0.9070.

30. It is todayā™s value in dollars for one future dollar, i.e., at a speciļ¬c point in time in the future.

31. The ļ¬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).

ļ¬le=constantinterest.tex: LP

32 Chapter 2. The Time Value of Money.

32. ā’$900 + $200/(1 + 8%)1 + $200/(1 + 8%)2 + $400/(1 + 8%)3 + $400/(1 + 8%)4 ā’ $100/(1 + 8%)5 ā $0.14. The

NPV is positive, therefore this is a worthwhile project.

$250 ā’ x

33. r = 30% = ā’ x = $250/(1 + 30%) ā $192.31.

x

34. $150/(1 + 5%) ā $142.86.

35. $150/(1 + 5%/365)365 ā $142.68.

The ļ¬rst tuition payment is worth $30, 000/(1 + 6%)1/2 ā $29, 139. The second tuition payment is worth

36.

$30, 000/(1 + 6%)3/2 ā $27, 489. Thus, the total present value is $56,628.

37. Good. Your future payments would be worth less in todayā™s money.

38. The original interest rate is $100/$95 ā’ 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 ā’1.4%.

(All answers should be treated as suspect. They have only been sketched, and not been checked.)

CHAPTER 3

More Time Value of Money

Quick Formulas for Perpetuities and Annuities

last ļ¬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 diļ¬erent

questions that remain:

1. What should inļ¬‚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 ļ¬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 ļ¬rm should make

any diļ¬erence

The answer to the second question will be that there are indeed valuation formulas for projects

that have peculiar cash ļ¬‚ow patternsā”perpetuities and annuities. Their values are easy to

compute when interest rates are constant. This often makes them useful āquick-and-dirtyā

tools for approximations. They are not only in wide use, but also are necessary to compute

cash ļ¬‚ows for common bonds (like mortgages) and to help you understand the economics of

corporate growth.

33

ļ¬le=perpetuities.tex: LP

34 Chapter 3. More Time Value of Money.

3Ā·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 ļ¬rms

better investments than slow-growing ļ¬rms?

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

The answer is noā”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

market.

not agreeā”that is, if you value money tomorrow more than you value money todayā”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, inļ¬‚ation, transaction costs, or

other issues), you can of course do better: you can always shift money between time periods

at an āexchange rateā that reļ¬‚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 āeagerā 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%ā”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

market:

An āeager beaverā

consumer still takes any

positive NPV project.

ā¢ Sell the project in the competitive market for its NPV:

$200

(3.1)

ā’$100 + = $81.82 .

1 + 10%

ā¢ Spend the $150 + ($181.82 ā’ $100) = $231.82 today. You will be better oļ¬ 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 āsleeperā 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

ā¢ Sell the project in the competitive market for

$200

(3.2)

ā’$100 + = $81.82 .

1 + 10%

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

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

ā¢ 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.

ļ¬le=perpetuities.tex: RP

35

Section 3Ā·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 oļ¬ 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 inļ¬‚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

diļ¬erent borrowing and lending interest rates. Chapter 6 will discuss this.)

Side Note: Of course, after they have lost their clientsā™ money, many brokers like to confuse this basic truth

by claiming that they invested their clientsā™ 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 todayā”if

you care about future consumption, you can always save the extra cash from ļ¬nding the highest NPV investments

today.

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 oļ¬ers a rate of return of 15%, although equivalent interest

rates are only 10%. What should you do?

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

A similar question to that posed in the previous subsection is āWould it make more sense to The price should

incorporate the

invest in companies that grow more quickly rather than slowlyā? If you wish, you can think of

attributes of the ļ¬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 ļ¬rmā™s

price today, which is just the discounted net present value of the ļ¬rmā™s cash ļ¬‚ows that will

accrue to the owners. Therefore, neither is the better deal.

For example, consider company āGrowā (G) that will produce Should we invest in a

fast grower or a slow

grower?

(3.3)

Gt=1 = $100 G2 = $150 G3 = $250 ,

and company āShrinkā (S) that will produce

(3.4)

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

Is G not a better company to buy than S?

There is no uncertainty involved, and both ļ¬rms face the same cost of capital of 10% per annum. Letā™s ļ¬nd out: Compute

the values.

Then the price of G today is

$100 $150 $250

PVt=0 ( G ) = + + ā $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 ) = + + ā $225.39 . (3.6)

(1 + 10%) (1 + 10%) (1 + 10%)3

1 2

ļ¬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 ļ¬‚ows coming up. Gā™s value at time 1 (so PV now has subscript 1) will thus be

disconnected from the

cash ļ¬‚ow rate.

$150 $250

PVt=1 ( G ) = $100 + + ā $442.98 . (3.7)

(1 + 10%)1 (1 + 10%)2

Your investment will have earned a rate of return of $442.98/$402.71ā’1 = 10%. If you instead

invest in S, then next year you will receive $100 cash, and own a company with āonlyā $90 and

$80 cash ļ¬‚ows coming up. Sā™s value will thus be

$90 $80

PVt=1 ( S ) = $100 + + ā $247.93 . (3.8)

(1 + 10%) (1 + 10%)2

1

Your investment will have earned a rate of return of $247.39/$225.39 ā’ 1 = 10%. In either

case, you will earn the fair rate of return of 10%. Soā”whether cash ļ¬‚ows are growing at a rate

of +50%, ā’10%, +237.5%, or ā’92% is irrelevant: the ļ¬rmsā™ market prices today already reļ¬‚ect

their future growth rates. There is no necessary connection between the growth rate of the

underlying project cash ļ¬‚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 ļ¬rms do not necessarily earn a higher rate of return does not

mean that a ļ¬rm in which managers succeed in increasing the future cash ļ¬‚ows at no extra

investment cost will not be worth more. Such ļ¬rms will indeed be worth more, and the current

owners will beneļ¬t from the rise in future cash ļ¬‚ows, but this will also be reļ¬‚ected immediately

in the price at which you can purchase this ļ¬rm.

3Ā·1.C. The Value Today is just āAll Inļ¬‚owsā or just āAll Outļ¬‚owsā

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, ļ¬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, eļ¬ectively on your behalf, (at the same 10%, of course),

and then pay one big lump sum dividend of $100Ā·(1+10%)2 +$150Ā·(1+10%)+$250 = $536 at

the end of period 2. The dividend payout policy does not aļ¬ect Gā™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 ļ¬rm in our perfect worldā”or you would get something for nothing

or lose something for nothing.

ļ¬le=perpetuities.tex: RP

37

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

Important: In a perfect market, the price and value of the ļ¬rm are determined

by the net present value of the ļ¬rmā™s underlying projects. In total, the cash ļ¬‚ows

from the ļ¬rmā™s projects belong to the ļ¬rmā™s claim holders. Therefore, the net

present value of the ļ¬rmā™s projects also must be the same as the net present value

of the ļ¬rmā™s payouts to all claimants.

Firm Value = PV( āAll Project Payoutsā ) = PV( āAll Project Cash Flowsā ) . (3.9)

All Future All Future

The same logic applies to stock and debt. Debt receives some cash ļ¬‚ows generated

by the projects, which are then paid out as principal or interest. Similarly, stock

receives some cash ļ¬‚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

(3.10)

Debt Value = PV( Principal + Interest ) = PV( Cash Flows to Debt ) .

All Future

All Future

The time patterns of inļ¬‚ows or outļ¬‚ows only matters in determining net present

values. Beyond this inļ¬‚uence, it does not matter whether the ļ¬rm is a fast-earnings

grower, a slow-earnings grower, a fast-dividend payer, or a slow-dividend payerā”

each ļ¬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 āModigliani-Millerā (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 āperfect marketā assumption is importantā”the value of the ļ¬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 ļ¬rms. You should also realize that over

any limited time horizon, neither dividends nor earnings may represent value wellā”dividends

can be zero for a while, earnings can be negative, and the ļ¬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), shouldnā™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 ļ¬rm, shouldnā™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 ļ¬‚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 contributionā”after all, there are

millions of other investors equally willing to provide funds at the appropriately higher price.

It is competitionā”among investors for providing funds and among ļ¬rms for obtaining fundsā”

that determines the expected rate of return that investors receive and the cost of capital that

ļ¬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 proļ¬ts. Whatever is abundant and/or

provided by many will not be tremendously proļ¬table.

ļ¬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 Gā™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 ļ¬‚ows in earnings each period. What is Gā™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 ļ¬rm? Do you prefer a ļ¬rm paying a lot

of dividends, or a ļ¬rm paying no dividends until the very end?

Q 3.6 The discount rate is 15%/annum over all periods. Firm F ā™s cash ļ¬‚ows start with $500 and

grow at 20% per annum for 3 years. Firm Sā™s cash ļ¬‚ows also start with $500 but shrink at 20%

per annum for 3 years. What are the prices of these two ļ¬rms, and what is the expected growth

rate of your money that you would invest into these two companies?

3Ā·2. Perpetuities

We now proceed to our second subject of this chapterā”the shortcut formulas to compute the

āPerpetuitiesā and

āAnnuitiesā are projects present values of certain cash streams. A perpetuity is a project with a cash ļ¬‚ow that repeats

with special kinds of

forever. If the cost of capital (the appropriate discount rate) is constant and the amount of

cash ļ¬‚ows, which permit

money remains the same or grows at a constant rate, perpetuities lend themselves to quick

the use of short-cut

formulas.

present value solutionsā”very useful when you need to come up with quick rule of thumb

estimates. Though the formulas may seem a bit intimidating at ļ¬rst, using them will quickly

become second nature to you.

3Ā·2.A. The Simple Perpetuity Formula

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

Cash Present

Discount

Time Flow Factor Value Cumulative

0 Nothing! You have no cash ļ¬‚ow here!

1/

1 $2 $1.82 $1.82

(1 + 10%)1

1/

2 $2 $1.65 $3.47

(1 + 10%)2

1/

3 $2 $1.50 $4.97

(1 + 10%)3

. . . . .

. . . . .

. . . . .

1/

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):

ļ¬le=perpetuities.tex: RP

39

Section 3Ā·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 ļ¬‚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 conļ¬rm the tableā™s last row, which gives the perpetuityā™s net present value as $20, you can The Shortcut Perpetuity

Formula.

spend from here to eternity to add up the inļ¬nite number of terms. But if you use a spreadsheet

to compute and add up the ļ¬rst 50 terms, you will get a PV of $19.83. If you add up the ļ¬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.

$2

Perpetuity PV = = $20

10%

(3.11)

CFt+1

= .

PVt

r

The āt+1ā in the formula is to remind you that the ļ¬rst cash ļ¬‚ow begins the following period,

not this periodā”the cash ļ¬‚ows are the same in 1 period, in 2 periods, etc.

Important: A stream of constant cash ļ¬‚ows, CF dollars each period and forever,

beginning next period, and is discounted at the same annual cost of capital r

forever is worth

CFt+1

(3.12)

PVt = .

r

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 ļ¬rst cash ļ¬‚ow

occurs.

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 Hudsonā™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 redeļ¬ned time as āwhatever is

indeļ¬nite lasts 100 years.ā In any case, the Cuban government no longer recognizes the agreement, and does

not accept the annual paymentsā”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.

ļ¬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%?

3Ā·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 ļ¬‚ows here.

(1 + 5%)0 Ā·$2 = (1 + 10%)1

1 $2.000 $1.818 $1.82

1 2

(1 + 5%) Ā·$2 = (1 + 10%)

2 $2.100 $1.736 $3.56

2 3

(1 + 5%) Ā·$2 = (1 + 10%)

3 $2.205 $1.657 $5.22

. . . . .

. . . . .

Ā·$2 =

. . . . .

29

(1 + 10%)30

(1 + 5%) Ā·$2 =

30 $8.232 $0.236 $30.09

. . . . . .

. . . . . .

Ā·$2 =

. . . . . .

. . .

. . .

(1 + 5%)tā’1 Ā·$2 = (1 + 10%)t

t . . .

. . . . . .

. . . . . .

Ā·$2 =

. . . . . .

= $40.00

Net Present Value (Sum):

What if the cash ļ¬‚ows are larger every period? A generalization of the perpetuity formula is

A growing perpetuity

assumes that cash ļ¬‚ows the growing perpetuity formula, in which the cash ļ¬‚ows grow by a constant rate g each period.

grow by a constant rate

The cash ļ¬‚ows of a sample growing perpetuityā”which pays $2 next year, grows at a rate of 5%,

forever.

and faces a cost of capital of 10%ā”are shown in Table 3.2. The present value of the ļ¬rst 50

terms adds up to $36.28. The ļ¬rst 100 terms add up to $39.64. The ļ¬rst 200 terms add up to

$39.98. Eventually, the sum approaches the formula

$2

PV of Growing Perpetuity0 = = $40

10% ā’ 5%

(3.13)

CFt+1

= .

PVt

r ā’g

As before, the āt+1ā indicates that cash ļ¬‚ows begin next period, not this period, and r is the

interest rate minus g, the growth rate of your cash ļ¬‚ows. Note that the growth timing occurs

one period after the discount factor timing. For example, the cash ļ¬‚ow at time 30 is discounted

by (1 + r )30 , but its cash ļ¬‚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 ļ¬‚ows

are assumed to grow by the rate of inļ¬‚ation. We will use this formula extensively to obtain

āterminal valuesā in our ļ¬nal chapter on pro formas.

ļ¬le=perpetuities.tex: RP

41

Section 3Ā·2. Perpetuities.

Important: A stream of cash ļ¬‚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 ā’g

The ļ¬rst cash ļ¬‚ow, CFt+1 occurs next period, the second cash ļ¬‚ow of

CFt+2 = CFt+1 Ā· (1 + g) occurs in two periods, and so forth, forever.

The growing annuity formula is worth memorizing.

What would happen if the cash ļ¬‚ows grew faster than the interest rate (g ā„ r )? Wouldnā™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 inļ¬nity, because in todayā™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 ļ¬‚ow one period later would be worth more even in todayā™s dollarsā”and

taking a sum over an inļ¬nite number of increasing terms would yield inļ¬nity as the value. A

value of inļ¬nity is clearly not sensible, as nothing in this world is worth an inļ¬nite amount of

money today. And, therefore, the growing perpetuity formula yields a non-sensical negative

value if g ā„ r ā”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 ļ¬‚ows will grow

at a rate of 0.1%/month (1.2%/year)?

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

Models

With their ļ¬xed interest and growth rates and eternal payment requirements, perpetuities are Perpetuities are

imperfect

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 proļ¬ts of $1 million next year. Because it is stable, often give a useful upper

its proļ¬ts are likely to grow at the inļ¬‚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 ļ¬rm faces a cost of capital of 8%.

The growing perpetuity formula indicates that this ļ¬rm should probably be worth no more than

$1, 000, 000

Business Value = ā $16, 666, 667

8% ā’ 2%

(3.15)

CF1

Business Value0 = ,

r ā’g

because in reality, the ļ¬rm will almost surely not exist forever. Of course, in real life, there are

often even more signiļ¬cant uncertainties: next yearā™s proļ¬t may be diļ¬erent, the ļ¬rm may grow

at a diļ¬erent rate (or may grow at a diļ¬erent rate for a while) or face a diļ¬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 āGordon Growth

Modelā: constant eternal

if you believe that a stockā™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 ā’g 10% ā’ 5%

ļ¬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:

D

(3.17)

= r ā’g = 2.43% .

P

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 GEā™s cash ļ¬‚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 Ā· (1 + 9.6%)2 ā $7.643). Therefore, if we believe 9.6%/year is

a fair representation of the eternal growth rate of GEā™s dividends, then the ļ¬nancial markets

valued GE as if it had a per-annum cost of capital of about

D

(3.18)

r= + g ā 2.4% + 9.6% ā 12% .

P

It is also not uncommon to repeat the same exercise with earningsā”that is, presume that stock

You can do the same

with earnings. market values are capitalized as if corporate earnings were eternal cash ļ¬‚ows growing at a

constant rate g. Again, Yahoo!Finance gives us all the information we need. GEā™s ātrailing P/Eā

ratioā”calculated as the current stock price divided by historical earningsā”was 21, its āforward

P/Eā ratioā”calculated as the price divided by analystsā™ expectations of next yearā™s dividendsā”

was 18.5. The latter is P0 /E1 , and thus closer to what we want. Yahoo!Finance further tells us

that GEā™s earnings growth was 6.3%ā”the g in our formula. Therefore,

E1 E1 1 1

P0 = ā’ r= +g = +g ā + 6.3% ā 11.7% . (3.19)

r ā’g P0 P0 /E1 18.5

It is important that you recognize that these are just modelsā”approximationsā”that you cannot

Keep perspective!

take too seriously (in terms of accuracy). GE will not last forever, earnings are not the cash ļ¬‚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 oļ¬,

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 annumā”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 inļ¬‚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 ļ¬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 inļ¬‚ation rate of 0.5% per quarterā”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 ļ¬rmā™s āas-if-eternal dividendsā to

grow at?

ļ¬le=perpetuities.tex: RP

43

Section 3Ā·3. The Annuity Formula.

3Ā·3. The Annuity Formula

The second type of cash ļ¬‚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 ļ¬‚ows for a given number of periods. Unlike a perpetuity, payments stop after

years.

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 ļ¬rst do this the slow way. We can hand-compute the net present value to be

$5 $5 $5

PV0 = + + ā $12.4343

1 + 10% (1 + 10%)2 (1 + 10%)3

(3.20)

CF1 CF2 CF3

= + + .

PV0

(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 ā’ [1/(1 + 10%)]3

PV = $5 Ā· ā $12.4343 ,

10%

(3.21)

1 ā’ [1/(1 + r )]T

PV = CFt+1 Ā· = .

PV

r

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 ļ¬‚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 = Ā· 1ā’ .

(1 + r )T

r

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

Most mortgages are ļ¬xed rate mortgage loans, and they are basically annuities. They promise Mortgages are annuities,

so the annuity formula

a speciļ¬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 āannu-ityā formula should really be

called a āmonth-ityā 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 providersā”like banksā”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 ā’ 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 ļ¬xed monthly cash ļ¬‚ow that gives the annuity this value: formula.

CFt+1 1

$500, 000 = Ā· 1ā’ ā CFt+1 Ā· 143.02

(1 + 0.625%)360

0.625%

(3.23)

CFt+1 1

= Ā· 1ā’ .

PV

(1 + r )T

r

Solving this for the cash ļ¬‚ow tells you that the monthly payment on your $500,000 mortgage

will be $3,496.07 for 360 months, beginning next month.

ļ¬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 ļ¬rst month, Uncle Sam

proclaims 0.625%Ā·$500, 000 = $3, 125 to be the tax-deductible mortgage interest payment. Therefore, the

principal repayment is $3, 496.07 ā’ $3, 125 = $371.07 and remaining principal is $499,628.93. The follow-

ing month, Uncle Sam proclaims 0.625%Ā·$499, 628.93 = $3, 122.68 to be the tax-deductible interest payment,

$3, 496.07 ā’ $3, 122.68 = $373.39 as the principal repayment, and $499,255.54 as the remaining principal. And

so on.

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

Let us exercise our new found knowledge in a more elaborate exampleā”this time with bonds.

Coupon bonds pay not

only at the ļ¬nal time. Bonds come in many diļ¬erent varieties, but one useful classiļ¬cation is into coupon bonds

and zero bonds (short for zero coupon bonds). A coupon bond pays its holder cash at many

diļ¬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 bondā™s original sale, and then return a āprincipal amountā plus a

ļ¬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 speciļ¬ed by

their promised payout is very common) for ļ¬ve years, plus an additional $100,000 in 5 years. This payment pattern

patterns.

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

ā3% semi-annual coupon bondā ($3,000 coupon per year, divided by the principal of $100,000).

Now, the ā3% coupon bondā is just a naming convention for the bond with these speciļ¬c cash

ļ¬‚ow patternsā”it is not the interest rate that you would expect if you bought this bond. In

Section 2Ā·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 ā’ 1 ā 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 ļ¬nd the

annuity formula helpfulā”you can use it, but you do not need it. Our task is to ļ¬nd the value of

a ā3% coupon bondā today. First, we write down the payment structure for our 3% semi-annual

coupon bond. This comes from its deļ¬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: ļ¬nd the

appropriate costs of We shall assume that the prevailing interest rate is 5% per annum, which translates into 2.47%

capital.

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%

ļ¬le=perpetuities.tex: RP

45

Section 3Ā·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

ļ¬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 ļ¬‚ows, each $1,500, at a per-period interest

rate of 2.47%. So, according to our formula, the coupon payments are worth

1 ā’ [1/(1 + r )]T

PV = CFt+1 Ā·

r

1 ā’ [1/(1 + 2.47%)]10 (3.24)

= $1, 500 Ā·

2.47%

ā .

$13, 148.81

In addition, we have our $100,000 repayment of principal, which is worth

$100, 000

PV = ā $78, 352.62

1 + 27.63%

(3.25)

CF

PV = ā $78, 352.62 .

(1 + r0,5 )

Together, these present values of our bondā™s cash ļ¬‚ows add up to $91, 501.42.

Prevailing Interest Rates and Bond Values: We already know that the value of one ļ¬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%

to

ā

2

(3.26)

r= 1 + 6% ā’ 1 ā 2.96% ā (1 + 2.96%) Ā· (1 + 2.96%) ā (1 + 6%) .

To get the bondā™s new present value, reuse our formula

1 ā’ [1/(1 + r )]T CFT

PV = CFt+1 Ā· +

1 + r0,T

r

1 ā’ [1/(1 + 2.96%)]10 $100, 000 (3.27)

= $1, 500 Ā· +

(1 + 2.96%)10

2.96%

ā + ā $87, 549.70 .

$12, 823.89 $74, 725.82

So, our bond would have lost $3,951.72, or 4.3% of our original investmentā”which is the same

inverse relation between bond values and prevailing economy-wide interest rates that we ļ¬rst

saw on Page 24.

ļ¬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 ā3%-coupon bondā is just a designation for the bondā™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 bondā™s payments remain ļ¬xed, as designated by the bondā™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 diļ¬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 eļ¬ective interest rate of 6.168% per year (here, 0.5% per month)?

Q 3.22 Solve Fibonacciā™s annuity problem from Page 23: Compare the PV of a stream of quar-

terly cash ļ¬‚ows of 75 bezants vs. the PV of a stream of annual cash ļ¬‚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 diļ¬erence for a 1-year investment ļ¬rst.

Q 3.23 In Lā™Arithmetique, written in 1558, Jean Trenchant posed the following question: āIn 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 ā˜Le Grand Partyā™ on the condition that he will pay interest at 5% per fair for

41 fairs after which he will be ļ¬nished. Which of these conditions is better for the bankers?ā

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 bondā™s value?

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

correct.

Q 3.26 In many a deļ¬ned contribution pension plan, the employer provides a ļ¬xed percentage

contribution to the employeeā™s retirement. Let us assume that you must contribute $4,000 per

annum beginning next year, growing annually with the inļ¬‚ation rate of 2%/year. What is this

individualā™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.

ļ¬le=perpetuities.tex: RP

47

Section 3Ā·3. The Annuity Formula.

3Ā·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 ā” use is rare.

would likely be useful if you could also remember the annuity formula.) These formulas are

used in many diļ¬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

CFt+1

PVt = Ā· 1ā’ . (3.28)

(1 + r )T

r ā’g

It is sometimes used in the context of pension cash ļ¬‚ows, which tend to grow for a ļ¬xed number

of time periods and then stop. However, even then it is not a necessary device. It is often more

convenient and ļ¬‚exible to just work with the cash ļ¬‚ows themselves within a spreadsheet.

Figure 3.1 summarizes the four special cash ļ¬‚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 ļ¬xed number of periods, here T = 7, which

truncates both the cash ļ¬‚ow stream and its present values.

ļ¬le=perpetuities.tex: LP

48 Chapter 3. More Time Value of Money.

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

Simple Perpetuity Growing Perpetuity

Future Cash Flows Future Cash Flows

E E

Forever Forever

E E

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

Their Present Values Their Present Values

E E

Forever Forever

E E

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

CF CF1

Formula: PV = Formula: PV =

r ā’g

r

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

Future Cash Flows Future Cash Flows

No More

No More

(T=7)

(T=7)

E E

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

(T=7)

(T=7)

E E

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

T T

1+g

CF 1 CF1

Formula: PV = Ā· 1ā’ Formula: PV = Ā· 1ā’

1+r r ā’g 1+r

r

ļ¬le=perpetuities.tex: RP

49

Section 3Ā·4. Summary.

3Ā·4. Summary

The chapter covered the following major points:

ā¢ 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.

ā¢ In a perfect market, ļ¬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, ļ¬rms

can shift the time patterns of cash ļ¬‚ows and dividends without changing the underlying

ļ¬rm value.

ā¢ 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.

ā¢ The PV of a growing perpetuityā”with constant-growth (g) cash ļ¬‚ows CF beginning next

year and constant per-period interest rate r ā”is

CFt+1

PVt = (3.30)

.

r ā’g

ā¢ The application of the growing perpetuity formula to stocks is called the Gordon dividend

growth model.

ā¢ The PV of an annuityā”T periods of constant CF cash ļ¬‚ows (beginning next year) and

constant per-period interest rate r ā”is

1 ā’ [1/(1 + r )]T

PVt = CFt+1 Ā· . (3.31)

r

ā¢ Fixed-rate mortgages are annuities, and therefore can be valued with the annuity formula.

ļ¬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 Ā· (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 ā’ $400 + $418.18 = $518.18.

3. For easier naming, we call year 0 to be 2000. The ļ¬rmā™s present value in 2000 is $536/1.103 ā $402.71ā”but

we already knew this. If you purchase this company, its value in 2001 depends on a cash ļ¬‚ow stream that is

$0 in 2001, $0 in year 2002 and $536 in year 2003. So, it will be worth $536/1.102 ā $442.98 in 2001. In

2002, your ļ¬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 ļ¬rm values.

4. Again, we call year 0 2000. The ļ¬rmā™s present value in 2000 is based on dividends of $100, $150, and $250

in the next three years. So, the ļ¬rm value in 2000 is the $402.71 in Formula 3.5. The ļ¬rm value in 2001 is in

Formula 3.7, but you immediately receive $100 in cash, so the ļ¬rm is worth only $442.98 ā’ $100 = $342.98.

As an investor, you would have earned a rate of return of $442.98/$402.71 ā’ 1 = 10%. The ļ¬rm value in 2002

is

$250

PVt=2 ( G ) = ā $227.27 . (3.32)

(1 + 10%)

but you will also receive $150 in cash, for a total ļ¬rm-related wealth of $377.27. In addition, you will have the

$100 from 2001, which would have grown to $110ā”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 ā’

1 = 10%. A similar computation shows that you will earn 10% from 2002 ($487.27) to 2003 ($536.00).

5. It makes no diļ¬erence!

6. F ā™s cash ļ¬‚ows are $500, $600, and $720. Its value is therefore $1,361.88. Firm Sā™s cash ļ¬‚ows are $500, $400,

and $320. Its value is therefore $947.65. Both ļ¬rms oļ¬er your investment dollar a 15% rate of return.

7. CF1 /r . The ļ¬rst cash ļ¬‚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.

ńņš. 3 |