Value Additivity and Present Values of Cash Flow Streams

Present values (or discounted values), henceforth denoted as PV(rather than P), and

0

future values obey the principle of value additivity; that is, the present (future) value

of many cash flows combined is the sum of their individual present (future) values.

This implies that the future value at date tof $14, for example, is the same as the sum

tof 14 $1 payments, each made at date 0, or 14(1 r)^t. Value

of the future values at

additivity also implies that one can generalize equation (9.3) to value a stream of cash

payments as follows:

Result 9.4

Let C,C,... , Cdenote cash flows at dates 1, 2,... , T,respectively. The present value 12T of this cash flow stream

CCC
12r
PV. . .	(9.5)
(1r)12T
(1r)(1r)

if for all horizons the discount rate is r.

Example 9.6:Determining the Present Value of a Cash Flow Stream

Compute the present value of the unlevered cash flows of the hydrogenerator, computed in

Exhibit 9.1.Recall that these cash flows were $96,000 at the end of each of the first five

years and $56,000 at the end of years six to eight.Assume that the discount rate is 10 per-

cent per year.

Answer:The present value of the unlevered cash flows of the project is

$96,000$96,000$96,000$96,000

PV$450,387

1.11.1²³1.1⁴

1.1

$96,000$56,000$56,000$56,000

1.1⁵⁶⁷1.1⁸

1.11.1

Inflation

Chemicals, Inc., has pretax cash flows at $100,000 each year, which do not grow over

time (see Exhibit 9.1). In an inflationary economic environment, however, one typi-

cally expects both revenues and costs, and thus cash flows, to increase over time. The

nominal discount rates—that is, the rates obtained from observed rates of apprecia-

tion directly—apply to nominal cash flows, the observed cash flows, which grow with

inflation. However, it is possible to forecast inflation-adjusted cash flows, often referred

to as real cash flows, which take out the component of growth due to inflation.

When inflation-adjusted cash flow forecasts are employed, they need to be dis-

counted at real discount rates, which are the nominal discount rates adjusted for

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appreciation due to inflation. If iis the rate of inflation per period, and ris the appro-

priate nominal discount rate, the real discount rate per period is:

1r

nominal

r 1

^real1i

IA

To convert the date tnominal cash flow, C, to an inflation-adjusted cash flow, C,use

tt

the formula:

C

IAt

C

^t(1i)^t

The present value of the inflation-adjusted cash flow, using the real rate for discount-

ing back tperiods, is:

IA

C

t

PV

(1r)^t

real

C

t

i)^t

(1C

^t

(1r)^tt

(1r)

realnominal

The present value equality implies the following result.

Result 9.5

Discounting nominal cash flows at nominal discount rates or inflation-adjusted cash flowsat the appropriately computed real interest rates generates the same present value.

Annuities and Perpetuities

There are several special cases of the present value formula, equation (9.5), described

earlier. When CC... CC,the stream of payments is known as a stan-

12T

dard annuity. If Tis infinite, it is a standard perpetuity. Standard annuities and per-

petuities have payments that begin at date 1. The present values of standard annuities

and perpetuities lend themselves to particularly simple equations. Less standard perpe-

tuities and annuities can have any payment frequency, although it must be regular (for

examples, every half period).

Many financial securities have patterns to their cash flows that resemble annuities

and perpetuities. For example, standard fixed-rate residential mortgages are annuities;

straight-coupon bonds are the sum of an annuity and a zero-coupon bond (see Chapter

2). The dividend discount models used to value equity are based on a growing perpe-

tuity formula.

Perpetuities.The algebraic representation of the infinite sum that is the present value

of a perpetuity is

	CCC
PV	...	(9.6a)
	(1(1r)2(1r)3
	r)

Asimple and easily memorized formula for PVis found by first multiplying both sides

of equation (9.6a) by 1/(1 r), implying

PV	CC
	...	(9.6b)
	r)2r)3
1r	(1(1

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Chapter 9

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317

Then, subtract the corresponding sides of equation (9.6b) from equation (9.6a) to obtain

PVC

PV

1r1r

which is equivalent to

rC

PV

1r1r

which simplifies to

C
PV	(9.7)
r

Result 9.6

If ris the discount rate per period, the present value of a perpetuity with payments of Ceach period commencing at date 1 is C/r.

The next example illustrates this result.

Example 9.7:The Value of a Perpetuity

The alumni relations office of the MBA program at Generous Grads University expects that

their new beg-the-students strategy will permanently increase the gift that each graduating

class typically presents to the school at their culmination ceremony.What is the value of the

beg-the-students strategy to the school if each graduating class’s gift is $100,000 larger

beginning next year and the discount rate is 5 percent?

Answer:Using Result 9.6, the value is

PV $2,000,0000 $100,000/.05

Aperpetuity with payments of Ccommencing today is the sum of two types of cash

flows: (1) Astandard perpetuity, paying Cevery period beginning with a payment of

Cat date 1, and (2) a payment of Ctoday. The standard perpetuity in item 1 has a

present value of C/r.The payment of Ctoday in item 2 has a present value of C.Add

them to get the present value of the combined cash flows: CC/r.

Note that the value for the “backward-shifted” perpetuity described in the last para-

graph is also equal to (1 r)C/r.This can be interpreted as the date 1 value of a pay-

ment of C/rat date 0. It should not be surprising that the date 1 value of a perpetuity

that begins at date 1 is the same as the date 0 value of a perpetuity that begins at date

0. If we had begun our analysis with this insight, namely that

PV(1 r) PVC

we could have derived the perpetuity formula in equation (9.7), PVC/r,by solving

the equation immediately above for PV.

Deriving the perpetuity formula in this manner illustrates that value additivity,

along with the ability to combine, separate, and shift cash flow streams, is often use-

ful for valuation insights. As the previous paragraph demonstrates, to obtain a for-

mula for the present value of a complex cash flow, it is useful to first obtain a pres-

ent value for a basic type of cash flow stream and then derive the present value of

the more complex cash flow stream from it. The ability to manipulate and match var-

ious cash flow streams, in whole or in part, is a basic skill that is valuable for finan-

cial analysis.

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318Part IIIValuing Real Assets

Example 9.8 illustrates how to apply this skill in valuing a complex perpetuity.

Example 9.8: Computing the Value of a Complex Perpetuity

What is the value of a perpetuity with payments of $2 every half-year commencing one-half

year from now if r10 percent per year?

Answer:Examine the cash flows of the payoffs, outlined in the following table:

		Cash Flow (in $) at Year
.5	1	1.522.533.5	4	...

...

22222222

This can be viewed as the sum of two perpetuities with annual payments, outlined below:

		Cash Flow (in $) at Year
.5	1	1.522.533.5	4	...

Perpetuity ...

120202020

Perpetuity ...

202020202

Perpetuity 2 is worth $2/r or $20.The first perpetuity is like the second perpetuity except

that each cash flow occurs one-half period earlier.Let us discount the payoffs of perpetuity

1 to year .5 rather than to year 0.At year .5, the value of perpetuity 1 is $2 $2/ror $22.

Discounting $22 back one-half year earlier, we find that its year 0 value is

$22/(1.1)^.5

$20.976

Summing the year 0 values of the two perpetuities generates the date 0 value of the origi-

nal perpetuity with semiannual payments.This is

$22/r

$40.976$2/r$20$20.976

(1r)^.5

The annuity formula derivation in the next subsection also demonstrate how useful it

is to be able to manipulate cash flows in creative ways.

Annuities.Astandard annuity with payments of Cfrom date 1 to date Thas cash

flows outlined in the following table:

Cash Flow at Date

123...	3	...
	TT2T
	1T

CCC...

C000...

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319

Astandard annuity can thus be viewed as the difference between two perpetuities. The

first perpetuity has cash flows outlined in the table below.

Cash Flow at Date

123...	T3	...
	2T
	1T

CCC...

CCC...

The second perpetuity has cash flows that are identical to the first except that they com-

mence at date T1; that is, they are represented by

Cash Flow at Date

123...	3	...
	TT2T
	1T

000...

0CCC...

The first perpetuity has a date 0 value of C/r.The second perpetuity has a date Tvalue

of C/r,implying a date 0 present value of

C/r

(1r)^T

The difference in these two perpetual cash flow streams has the same cash flows as the

annuity. Hence, the date 0 value of the annuity is the difference in the two date 0 values

C/r

PVC/r

(1r)^T

This result is summarized as follows.

Result 9.7

If ris the discount rate per period, the present value of an annuity with payments com-mencing at date 1 and ending at date Tis

C1
PV 1	(9.8)
r)T
r(1

Example 9.9:Computing Annuity Payments

Buffy has just borrowed $100,000 from her rich uncle to pay for her undergraduate educa-

tion at Yale.She has promised to make payments each year for the next 30 years to her

uncle at an interest rate of 10 percent.What are her annual payments?

Answer:The present value of the payments has to equal the amount of the loan,

$100,000.If ris .10, equation (9.8) indicates that the present value of $1 paid annually for

30 years is $9.427.To obtain a present value equal to $100,000, Buffy must pay

$100,000

$10,608

9.427

at the end of each of the next 30 years.

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320Part IIIValuing Real Assets

Growing Perpetuities.Agrowing perpetuityis a perpetual cash flow stream that

grows at a constant rate (denoted here as g) over time, as represented below.

Cash Flow at Date

1234

...

g)	Cg)2	3
(1	(1	C(1 g)	...
CC

If g

r,the present value of this sum is finite and given by the formula

CC(1g)C(1g)2
PV. . .	(9.9a)
(1r)(1r)23
(1r)

Asimpler formula for this present value is found by first multiplying both sides of

1g

equation (9.9a) by yielding

1r

1gCg)Cg)2
(1(1
PV. . .	(9.9b)
1rr)23
(1(1r)

Then, subtract the corresponding sides of equation (9.9b) from equation (9.9a) to obtain

1gC

PV PV

1r1r

When rearranged, this implies

Result 9.8The value of a growing perpetuity with initial payment of C dollars one period from now is

C
PV	(9.10)
r g

One of the most interesting applications of equation (9.10) is the valuation of

stocks. According to the dividend discount model, stocks can be valued as the dis-

counted value of their future dividend stream. Aspecial case of this model arises when

dividends are assumed to be growing at a constant rate, generating a stock price per

share of

div
1
S	(9.11)
0r g

with divnext year’s forecasted dividend per share, rthe discount rate, and gthe div-

1

idend growth rate.

Example 9.10:Valuing a Share of Stock

Assume a stock is going to pay a dividend of $2 per share one year from now and that this

dividend will grow by 4 percent per year, forever.If the discount rate is 8 percent per year,

what is the present value of the share of stock?

Answer:Using equation (9.11), the per share value is

$2

$50

.08 .04

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Chapter 9

Discounting and Valuation

321

Since no stocks pay dividends that grow at a constant rate, the dividend discount for-

mula, as represented in equation (9.11), is not a practical way to value companies. How-

ever, that does not by itself invalidate the premise that the fair value of a stock is the

discounted value of its future dividend stream. After all, dividends are the only cash

flow that the stock produces for its current and future shareholders and thus represent

the only basis on which to establish a fair valuation for the stock. This is true even for

firms that do not currently pay dividends. Such no-dividend companies are not worth-

less—their current no-dividend policy does not imply that the company never will pay

dividends. Moreover, the final cash payout to shareholders, whether it comes in the

form of a liquidating dividend or a cash payout by an acquiring firm or a leveraged

buyout, counts as a dividend in this model. Such terminal payouts are virtually impos-

sible to estimate, and they suggest that there are numerous practical impediments to

valuing firms with the dividend discount model.

Growing Annuities.Agrowing annuityis identical to a growing perpetuity except

that the cash flows terminate at date T.It is possible to derive the present value of

this perpetuity from equation (9.11). Applying the reasoning used to derive the annu-

ity formula, we find that a growing annuity is like the difference between two grow-

ing perpetuities. One commences at date 1 and has a present value given by equa-

tion (9.11). The second perpetuity commences at date T1 and has a present value

equal to

C(1g)^T

(r g)(1r)^T

The numerator C(1 g)^T

is the initial payment of the growing perpetuity and thus

replaces Cin the equation. The second product in the denominator (1 r)^T

would not

appear if we were valuing the perpetuity at date T.However, to find its value at date

0, we discount its date Tvalue an additional Tperiods. The difference between the date

0 values of the first and second perpetuity is given by

C(1g)T
PV 1	(9.12)
r g(1r)T

which is the present value of a T-period growing annuity, with an initial cash flow of

C, commencing one period from now and with a growth rate of g. Unlike perpetuities,

growing annuities (with finite horizons) need not assume that g

r.

Simple Interest

The ability to handle compound interest calculations is essential for most of what we

do throughout this text. Simple interest calculations are less important, but they are

needed to compute accrued interest on bonds and certain financial contracts such as

Eurodollar deposits and savings accounts (see Chapter 2). An investment that pays sim-

ple interest at a rate of rper period earns interest of rtat the end of tperiods for every

dollar invested today.

Time Horizons and Compounding Frequencies

This chapter has developed formulas for present values and future values of cash flows or

cash flow streams based on knowing a compound interest rate (or rate of return or yield)

per period. Different financial securities, however, define the length of this fundamental

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322Part IIIValuing Real Assets

period differently. The fundamental time period for residential mortgages, for example,

is one month because mortgage payments are typically made monthly. The fundamen-

tal period for government bonds and notes and corporate bonds is six months, which

is the length of time between coupon payments.

Annualized Rates.Finance practitioners long ago recognized that it is difficult to

understand the relative profitability of two investments, where one investment states

the amount of interest earned in one month while the other states it over six months.

To facilitate such comparisons, interest rates on all investments tend to be quoted on

an annualized basis. Thus, the rate quoted on a mortgage with monthly payments is 12

times the monthly interest paid per dollar of principal. For the government bond with

semiannual coupons, the rate is twice the semiannual coupon (interest) paid.

The annualization adjustment described is imperfect for making comparisons

between investments since it does not reflect the interest earned on reinvested interest.

As a consequence, annualized interest rates with the same rbut different compound-

ing frequencies mean different things. To make use of the formulas developed in the

previous sections, where ris the interest earned over a single period per dollar invested

at the beginning of the period, one has to translate the rates that are quoted for finan-

cial securities back into rates per period and properly compute the number of periods

over which the future value or present value is taken. Exhibit 9.3 does just this.

Equivalent Rates.Aproper adjustment would convert each rate to the same com-

pounding frequency. If the annualized interest rates for two investments are each trans-

lated into an annually compounded rate, the investment with the higher annually com-

pounded rate is the more profitable investment, other things being equal.

Consider, for example, two 16 percent rates—one compounded annually and the

other semiannually. A16 percent annually compounded rate means $1.00 invested at

the beginning of the year has a value of $1.16 at the end of the year. However, a 16

percent rate compounded semiannually becomes, according to the translation in Exhibit

9.3, an 8 percent rate over a six-month period. At the end of six months, one could

reinvest the $1.08 for another six months. With an 8 percent return over the second six

months, the $1.08 would have grown to $1.1664 by the end of the year. This 16.64

percent rate, the equivalent annually compounded rate,would be a more appropriate

number to use if comparing this investment to one that uses the annually compounded

rate. In general, given the investment with the same interest rate r,but different

EXHIBIT9.3

Translating Annualized Interest Rates with Different Compounding Frequencies into Interest Earned perPeriod

Annualized Interest Rate Quotation Basis

Interest per Period

Length of a Period

Annually compounded	r	1 year
Semiannually compounded	r/2	6 months
Quarterly compounded	r/4	3 months
Monthly compounded	r/12	1 month
Weekly compounded	r/52	1 week
Daily compounded	r/365	1 day
Compounded mtimes a year	r/m	1/myears

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Chapter 9

Discounting and Valuation

323

compounding frequencies, the more frequent the compounding, the faster the growth

rate of the investment.

If the compounding frequency is mtimes a year and the annualized rate is r,the

amount accumulated from an investment of PVafter tyears is:

rmt
PPV 1	(9.13)
tm

Equation (9.13) is another version of equation (9.2), the future value formula, recog-

nizing that the per period interest rate is r/m,and the number of periods in tyears is mt.

If there is continuous compounding, so that mbecomes infinite, this formula has

the limiting value

Pert
PV	(9.14)
t

where eis the base of the natural logarithm, approximately 2.718.

Inverting equations (9.13) and (9.14) yields the corresponding present value

formulas:

		P
		t
PV			(9.15)
		rmt
		1
		m

PV Pe rt	(9.16)
t

To compute the equivalent rate using a different compounding frequency for the same

investment, change m,and find the new rthat generates the same future value Pin

t

equation (9.15). Example 9.11 illustrates the procedure.

Example 9.11:Finding Equivalent Rates with Different Compounding

Frequencies

An investment of $1.00 that grows to $1.10 at the end of one year is said to have a return

of 10 percent, annually compounded.This is known as the “annually compounded rate of

return.”What are the equivalent semiannually and continuously compounded rates of growth

for this investment?

Answer:Its semiannually compounded rate is approximately 9.76 percent and its con-

tinuously compounded rate is approximately 9.53 percent.These are found respectively by

solving the following equations for r

r²

1.10 1

2

and
	1.10 ert

U.S. residential mortgages are annuities based on monthly compounded interest.

The monthly payment per $100,000 of a Y-year mortgage with interest rate ris the

number xthat satisfies the present value equation

12Y

x

$100,000^t

r

t1 1

12

Example 9.12 illustrates how to compute a monthly mortgage payment.

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Example 9.12:Computing Monthly Mortgage Payments

Compute the monthly mortgage payment of a 30-year fixed-rate mortgage of $150,000 at 9

percent.

Answer:The monthly unannualized interest rate is .09/12 or .0075.Hence, the monthly

payment xsolves

360

x

$150,000^t

1.0075

t1

which, after further simplification using the annuity formula from equation (9.8), is solved by

x$1206.93