9.2Using Discount Rates to Obtain Present Values

Having learned how to obtain cash flows from standard accounting items, and how to

forecast various accounting statements, it is now time to learn how to discount cash

flows to obtain present values. At the beginning of this chapter, we learned that the dis-

count rates used are simply rates of return.

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312Part IIIValuing Real Assets

Single Period Returns and Their Interpretation

Recall from Chapter 4 that over a single period, a return is simply profit over initial

investment. Algebraically

P P
r10	(9.1a)
P
0

where

Pdate 1 investment value (plus any cash distributed) like dividends or

1

coupons

Pdate 0 investment value

0

Hence, an investment of $1 that grows to $1.08 has a rate of return of .08 (or 8 per-

cent) over the period.

There is a close relation between interest on a bank account and the bank account’s

rate of return. Abank account that pays 8 percent interest returns $1 in principal plus

$.08 in interest per dollar invested. In short, the investment value grows from $1 to

$1.08 over the period. Hence, the interest rate paid is another way of expressing the

rate of return on the bank account. However, as Chapter 2 noted, interest is not always

so highly linked to the rate of return. For example, certain types of traded bonds, known

as discount bonds, pay less in legal interest than their promised rate of return, which

is sometimes referred to as their yield to maturity.Interest, in this case, is something

of a misnomer. The appreciation in the price of the bond is a form of implicit interest

that is not counted in the legal calculation of interest.

Equation (9.1a) makes it appear as if prices determine rates of return. Rearranging

equation (9.1a), however, implies

P
1
P	(9.1b)
01r

or

PP(1r)	(9.1c)
10

Equation (9.1b) indicates that the current or present valueof the investment is

determined by the rate of return. Hence, to earn a rate of return of 8 percent over the

period, an investment worth $1.08 at date 1 requires that $1.00 be invested at date 0.

Similarly, an investment worth $1 at date 1 requires that $1.00/1.08 (or approximately

$0.926) be invested at date 0 to earn the 8 percent return. This suggests that a rate of

return is a discount rate that translates future values into their date 0 equivalents. Equa-

tion (9.1c) makes it seem as if the future value of the investment is determined by the

rate of return. Of course, all of these equations and interpretations of rare correct and

state the same thing. Any two of the three variables in equations (9.1) determine the

third. Which equation you use depends on which variable you don’t know.

Rates of Return in a Multiperiod Setting

Exhibit 9.2 illustrates what happens to an investment of Pafter tperiods if it earns a

0

rate of return of rper period and all profit (interest) is reinvested.

Exhibit 9.2 indicates that, when ris the interest rate (or rate of return per period)

and all interest (profit) is reinvested, an investment of Pdollars at date 0 has a future

0

valueat date tof

PP(1 r)t	(9.2)
t0

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Chapter 9

Discounting and Valuation

313

EXHIBIT9.2

The Value of an Investment overMultiple Periods When Interest (Profit) Is Reinvested

Beginning-	End-of-	Initial
of-Period	Period	Principal	Interest (profit)
Date	Date	Balance	Earned over Period	End-of-Period Value

01PPPrP(1 r)

rP

00000

(1 r)P(1 r)(1 r) P(1 r)rP(1 r)²

PrP

12

00000

P(1 r)²²²²³

23P(1 r)(1 r)P(1 r)rP(1 r)

rP

00000

.....

...

.

.

.....

t 1^{t 1t 1}P(1 r)^{t 1}rP(1 r)^t

t 1tP(1 r)P(1 r)(1 r)

rP

00000

The date 0 value of Pdollars paid at date t,also known as the present valueor discounted

t

value,comes from rearranging this formula, so that Pis on the left-hand side; that is

0

P
t
P	(9.3)
0(1r)t

If ris positive, equation (9.3) states that Pis smaller the larger tis, other things being

0

equal. Hence, a dollar in the future is worth less than a dollar today. Cash received

early is better than cash received late because the earlier one receives money, the greater

the interest (profit) that can be earned on it.

Equations (9.2) and (9.3) have added a fourth variable, t, to the present value and

future value equations. It can be an unknown in a finance problem, too, as the fol-

lowing example illustrates.

Example 9.4:Computing the Time to Double YourMoney

How many periods will it take your money to double if the rate of return per period is 4

percent?

Answer:Using equation (9.2), find the tthat solves

P(1 .04)^t

2P

00

which is solved by
	tln(2)/ln(1.04) 17.673 (periods).

The answer to Example 9.4 is approximated as 70 divided by the percentage rate

of interest (4 percent in this case). This rule of 70provides a rough guide to doubling

time without the need to compute logarithms. For example, at a 5 percent rate of return,

doubling would approximately occur every 14 periods, because 14 70/5. The more

precise answer is ln(2)/ln(1.05) 14.207 periods.

Generalizing the Present Value and Future Value Formulas.The present value and

future value formulas generalize to any pair of dates tand tthat are tperiods apart,

12

that is, t tt. Equation (9.2) represents the value of the date tcash flow at date

211

tand equation (9.3) represents the value of the date tcash flow at tif the two dates

221

are tperiods apart. Hence, if t3 and t8, equation (9.2), with t5, would give

12

the value at date 8 of an investment of Pdollars at date 3. In addition, t, t, or tneed

012

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314Part IIIValuing Real Assets

not be whole numbers. In other words, tcould be .5, 3.8, 1/3, or even some irrational

number like . Thus, if t2.6 and t7.1, equation (9.3) with t4.5 ( 7.1 2.6)

12

represents the value at date 2.6 of Pdollars paid at date 7.1.

1

Explicit versus Implicit Interest and Compounding.Exhibit 9.2 and the discussion

of the exhibit reads as though we are examining a bank account that earns compound

interest. Compound interest rates reflect the interest that is earned on interest. Com-

pound interest arises whenever interest earnings are reinvested in the account to increase

the principal balance on which the investor earns future interest. But what about secu-

rities that never explicitly pay interest and thus have no interest or profit to reinvest?

We also can use the compound interest formulas to refer to the yield or rate of return

of these securities.

To see how to apply these formulas when interest (or profit) cannot be reinvested,

consider a zero-coupon bond, which (as Chapter 2 noted) is a bond that promises a sin-

gle payment (known as its face value) at a future date. With a date 0 price of Pand

0

a promise to pay a face value of $100 at date Tand nothing prior to date T,the yield

(or rate of return) on the bond can still be quoted in compound interest terms. This

yield on the bond is the number rthat makes

100 P(1 r)^T

0

implying
	1 100T r 1 P 0	(9.4)

Thus, rmakes $100 equal to what the initial principal on the bond would turn into by

date Tif the bond appreciates at a rate of rper period, and if all profits from appreci-

ation are reinvested in the bond itself.⁵Example 9.5 provides a numerical calculation

of the yield of a zero-coupon bond.

Example 9.5:Determining the Yield on a Zero-Coupon Bond

Compute the per period yield of a zero-coupon bond with a face value of $100 at date 20

and a current price of $45.

Answer:Using the formula presented in equation (9.4):

1

10020

r 1.040733

45

or about 4.07 percent per period.

Since a zero-coupon bond never pays interest and thus provides nothing to reinvest,

the process of quoting a compound interest rate for the bond is merely a convention—

a different way of expressing its price. As in the case of the one-period investment,

quoting a bond’s price in terms of either its yield or its actual price is a matter of per-

sonal preference. Both, in some sense, are different ways of representing the same thing.

Investing versus Borrowing.Investing and borrowing are two sides of the same coin.

For example, when a corporation issues a bond to one of its investors, it is in essence

⁵While such profits are implicit if the bond is merely held to maturity, it is possible to make them

explicit by selling a portion of the bond to realize the profits and then, redundantly, buying back the sold

portion of the bond.

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Chapter 9

Discounting and Valuation

315

borrowing from that investor. The cash flows to the corporation are identical in mag-

nitude but opposite in sign to those of the investor. The rate of return earned by the

investor is viewed as a cost paid by the corporate borrower. Because investing and bor-

rowing are opposite sides of the same transaction, the same techniques used to analyze

investing can be used to analyze borrowing. For instance, in Example 9.5, a $45 loan

paid back in one lump sum after 20 periods at a rate of interest of 4.0733 percent per

period would require payment of $100. Similarly, if you promise to pay back a loan

with a $100 payment 20 periods from now at a loan rate of 4.0733 percent per period,

you are asking to borrow $45 today.