Appendix to Chapter 14 Compound Interest © 2004 Thomson Learning/South-Western

Appendix toChapter 14

Compound Interest

© 2004 Thomson Learning/South-Western

2

Interest

Interest is payment for the current use of funds.– For example, an annual rate of 5 percent would

require someone who borrowed $100 to pay $5 in interest.

Throughout this appendix, assume the market has established an annual interest rate which is expressed either as a nominal or real rate (mathematics are the same).

3

Compound Interest

If the funds are invested for more than one time period you will receive compound interest, interest paid on prior interest earned.

For example, $1 invested for one year at interest rate i will provide

.05.1$)05.1(1$)05(.1$1$

have will year youone of end at the percent, 5 If

).1(1$1$1$

i

ii

4

Interest for Two Years

If you leave your money in the bank at the end of the first year, you will earn interest on both the original $1 and on the first year’s interest.

A the end of two years you will have

.)1(1$

)1)(1(1$)1(1$)1(1$2i

iiiii

5


.1$21$1$

)21(1$)1(1$

.21)1(

2

22

22

ii

iii

iii

togrow will$1 years,two of end the At

At the end of two years you will have the sum of three amounts:

– Your original $1.– Two years’ interest on your original $1, ($1·2i)– Interest on your first year’s interest, [($1 ·i) ·i] = $1 ·i2.

6


.1035.1$)1025.1(1$)05.1(1$

have will you years twoof end at the percent, 5 again, If,2

i

This represents the sum of your original $1 plus– Two years’ interest on the $1 ($.10).– The interest on the first year’s interest (5 percent of $.05,

which is $.0025).

7

Interest for Three Years

.157625.1$157625.11$)05.01(1$

toamounts thispercent, 5 If

.)1(1$

)1()1(1$)1(1$)1(1$

3

3

222

i

i

iiiii

If you now leave these funds, which after two years amount to $1·(1 + i)2, in the bank, at the end of three years you will have

8

A General Formula

If you leave $1 in the bank for any number of years, n, your will have at the end of that period,

Value of $1 compounded for n years = $1·(1 + i)n.

With a 5 percent interest rate over a period of 10 years you would have

$1·(1.05)10 = $1·1.62889 = $1.162889.

9

A General Formula

Without compounding, you would have had $1.50 ($1 plus interest of $0.05 per year).

The additional $0.12889 results from compounding.

Table 14A.1 shows the value of $1 compounded for various time periods and interest rates.

10

Table 14A.1: Effects of Compound Interest for Various Interest Rates and Time Periods with an Initial Investment of $1

Interest Rate Years 1 Percent 3 Percent 5 Percent 10 Percent

1 $1.01 $1.03 $1.05 $1.10 2 1.02 1.0609 1.1025 1.2100 3 1.0303 1.0927 1.1576 1.3310 5 1.051 1.159 1.2763 1.6105 10 1.1046 1.344 1.6289 2.5937 25 1.282 2.094 3.3863 10.8347 50 1.645 4.384 11.4674 117.3909 100 2.705 19.219 131.5013 13,780.6123

11

A General Formula

Table 14A.1 demonstrates how compounding becomes very important for long periods.– At 5 percent, $1 grows to $131.50 over 100 years

with $125.50 coming from compound interest.– At 10 percent interest over 100 years, more than

99.9 percent of the $13,780.61 comes from compounding.

12

Compounding with Any Dollar Amount

D dollars invested for n years at an interest rate of i will grow to– Value of $D invested for n years = $D·(1 + i)n.

For example, $1000 invested invested for– 1 year would grows to $1,050 [$1000·(1.05)].– 10 years would grows to $1,629 [$1000·(1.629)].– 100 years would grows to $131,501

[$1000·(131.501)].

13

APPLICATION 14A.1: Compound Interest Gone Berserk

Manhattan Island– The $24 that legend has was paid by Dutch settlers

for Manhattan Island, with a 5 percent interest rate, for the 377 years between 1623 and 2000 would be

.296,560,336,2$)679,356,97(24$)05.1(24$ 377

14


Horse Manure– In 1840 a restriction on the number of horses

allowed into Philadelphia because the horse population was growing at 10 percent per year.

– If the 50,000 population had been allowed to continue to grow at 10 percent the population in the 1990s would have been

.000,000,886,81)10.1(000,50

)1(000,50hourses ofNumber 150

150

i

15


Rabbits– Rabbits were introduced into Australia in the early

1860s, and, with no predators, the population grew at 100 percent per year through 1880.

– Starting with 2 rabbits in 1860, by 1880 the population would have been

.152,097,2)2(2)11(2

)1(2rabbits ofNumber 2020

20

i

16

Present Discounted Value

One dollar received today is more valuable than one dollar received a year from now..

One dollar invested in a bank will grow to more than one dollar in one year.

The present discounted value or present value is the value of future transactions discounted back to the present day to take account of the effect of potential interest payments.

17

Present Discounted Value

Present value reflect the opportunity cost notion.– The present value of a dollar you will not get for one

year is the amount you would have to put in a bank now to have $1 at the end of one year.

– For example, with i = 5 percent, if you invest $.95 today, you will have $1 in one year.

18

An Algebraic Definition

.1$05.19524.0$

and

9524.0$05.1

$

is year one in $1 of (PDV) valuediscontedpresent thepercent, 5 If

.1$)1(1

1$

since )1(

1$ is year one in

$1 of valuediscountedpresent the, is rateinterest theIf

PDV

i

ii

i

i

19


A two year wait involves even greater opportunity costs since you forgo two years of interest.

At i = 5 percent, $0.907 will grow to $1 in two years, so the present value of $1 payable in two years is $0.907.

20


For any interest rate, i, the PDV of $1 payable in two years is

.907.0$1025.1

1$

)05.1(

1$ years twoin payable $1 of PDV

percent 5For

.)1(

1$ years twoin payable $1 of PDV

2

2

i

i

21

General PDV Formulas

With an interest rate of i, the present value of $1 payable after any number of years, n, is simply

This is the reverse of computing compound interest; instead of multiplying by the factor (1 + i)n, you divide by that factor.

.)1(

1$ yearsn in payable $1 of PDV

ni

22

General PDV Formulas

.)1(

$ yearsn in payable $D of PDV

ni

D

The present value of any number of dollars ($D) payable in n years is given by the above formula.

The PDV values for $1 shown in Table 14A.2 are the reciprocals of the values in Table 14A.1.

– Note, a PDV is smaller with higher interest rates or with a longer period.

23

Table 14A.2: Present Discounted Value of $1 for Various Time Periods and Interest Rates

Interest Rate Years until Payment Is

Received 1 Percent 3 Percent 5 Percent 10 Percent 1 $.99010 $.97087 $.95238 $.90909 2 .98030 .94260 .90703 .82645 3 .97059 .91516 .86386 .75131 5 .95147 .86281 .78351 .62093 10 .90531 .74405 .61391 .38555 25 .78003 .47755 .29531 .09230 50 .60790 .22810 .08720 .00852 100 .36969 .05203 .00760 .00007

24

APPLICATION 14A.2: Zero-Coupon Bonds

Federal Treasury bonds sometimes include coupons that promise the owner a certain semiannual interest payments.– For example, a 30 year $1 million bond issued in

2000 at 6 percent would include 60 semiannual coupons in which the government promises to pay $30,000 on January 1 and July 1 of each year.

25

APPLICATION 14A.2: The Invention of Zero-Coupon Bonds

Because coupons can be costly, some financial firms introduced zero-coupon bonds in the 1970s.

A brokerage firm would buy the Treasury bond, “strip” the coupons, and sell these coupons as separate investments.– From the above example, one such coupon would

pay a $30,000 interest payment on July 1, 2020.

26

APPLICATION 14A.2: Applying the PDV Formula

The PDV of $30,000 payable on July 1 in 20 years (2022) at a 5 percent interest rate is

– The coupon would sell for $11,308.

These more convenient investments are now so popular that they are quoted daily in financial newspapers.

.308,11$653.2

000,30$

)05.1(

000,30$

)1(

000,30$2020

i

27

Discounting Payment Streams

Dollars payable at different points of time have different present values.– For example, suppose someone wins a $1 million

state lottery which is payable over 25 years.– The PDV of $40,000 per year for 25 years is

$363,200, certainly less than $1 million.

28

An Algebraic Presentation

Assume a stream of payments that promises $1 per year starting next year and continuing for three years.

Applying the previous formula give us the PDV

.)1(

1$

)1(

1$

1

1$32 iii

PDV

29


At i = 5 percent the value would be

The general formula for 5 years is .7232.2$

8639.0$9070.0$9523.0$)05.1(

1$

)05.1(

1$

05.1

1$32

.)1(

1$

)1(

1$

)1(

1$

)1(

1$

1

1$5432 iiiii

PDV

30


.)1(

1$

)1(

1$

1

1$

is yearsnfor formula general The

2 niiiPDV

Table 14A.3 uses this formula to compute the value of $1 per year for various number of years and interest rates.

– Note, PDV decreases with increasing years and higher interest rates.

31

Table 14A.3: Present Value of $1 per Year for Various Interest Rates

Interest Rate Years of Payment 1 Percent 3 Percent 5 Percent 10 Percent

1 $.99 $.97 $.95 $.91 2 1.97 1.91 1.86 1.74 3 2.94 2.83 2.72 2.49 5 4.85 4.58 4.33 3.79 10 9.47 8.53 7.72 6.14 25 22.02 17.41 14.09 9.08 50 39.20 25.73 18.26 9.91

100 63.02 31.60 19.85 9.99 Forever 100.00 33.33 20.00 10.00

32

Perpetual Payments

How much ($X) would you have to invest at an interest rate of i to yield $1 a year forever?

.1$

$

thatmeansjust But that

,$1$

iX

Xi

33

Perpetual Payments

For example, the present value of $1 per year forever with an interest rate of 5 percent is $20 (=$1/0.05).

A perpetuity is a promise of a certain number of dollars each year, forever.– See the last row of Table 15A.3.

They are illegal in the U.S. but some United Kingdom perpetuities still exist.

34

Varying Payment Streams

The computations, when payments vary from year to year, is shown below where Di represents the amount to be paid in any year i.

.)1()1()1(1 3

32

21n

n

i

D

i

D

i

D

i

DPDV

35

Calculating Yields

The yield is the effective rate of return promised by a payments stream that can be purchased at a certain price.

Using the previous equation with P representing the price becomes:

.)1(

1 where

,)1()1(1

22

1

221

i

DDD

i

D

i

D

i

DPDVP

nn

nn

36

Reading Bond Tables

The bond tables in The Wall Street Journal for August 13, 1999 lists a “5.5% Bond maturing in August 2028” selling for 1,095.

– The bond promises to pay 5.5 percent of its initial face value ($1,000) each year and then repay the $1000 principal when interest payments end in 26 years.

percent. 4.85 (i) yielda in results which

equation, following the using calculated is yieldThe

26262 1000555555095,1

37

Frequency of Compounding

Since the 1960s, banks went from compounding annually to more frequent, usually daily, compounding.

Semiannual Compounding– The bank would pay two times a year.– If you deposit $1 on January 1, by July 1 it will have

grown to be $1·(1 + i/2).

38

Semiannual Compounding

In general, with an interest rate of i, semiannual compounding would yield

).1(1$4/1$)1(1$

)4/1(1$)2/1(1$2

22

iii

iii

:since gcompoundin annual

exceeds This year.one of end at the

)2/1)(2/1(1$ ii

39

A General Treatment

More frequent compounding increases the effective yield than a 5 percent annual interest rate.

Table 14A.4 shows how frequency affects yield.– The gains from monthly are relatively large, but the

gain from monthly to daily or more frequent is small.

40

Table 14A.4: Value of $1 at a 5 Percent Annual Interest Rate Compounded with Different Frequencies and Terms

Frequency Years on Deposit Annual Semiannual Monthly Daily

1 $1.0500 $1.0506 $1.0512 $1.0513 2 1.1025 1.1038 1.1049 1.1052 3 1.1576 1.1596 1.1615 1.1618 5 1.2763 1.2801 1.2834 1.2840 10 1.6289 1.6386 1.6471 1.6487 25 3.3863 3.4371 3.4816 3.4900 50 11.4674 11.8137 12.1218 12.1803 100 131.5013 139.5639 146.9380 148.3607

41

APPLICATION 14A.3: Continuous Compounding

The Amazing Properties of e– e (=2.71828) is the base of the natural logarithms.– This number is used for calculating continuous

compounding interest, in that the one-period result of having $1 compounded instantly at interest rate i is ei.

– Continuous compounding of $D invested for t years, is equal to $D·eit.

42


The Rule of 70– To find the doubling time for an investment we solve

the equation eit = 2.– Taking natural logarithms of both sides gives

– Since ln2 is about 0.70, doubling time is found by dividing 70 by the interest rate, in percent.

.2ln*

it

43


Products and Ratios– Suppose x is growing at rate r1, and y at rate r2.

– Then, if z is the product of x times y, it is growing at

– For example, if price is growing at 3 percent per year and quantity at 7 percent per year, then total revenue is growing at 10 percent per year.

.)( 2121 trrtrtr eeeyxz

44


– If w is the ratio of x to y, then

– If nominal GDP is growing at 6 percent per year and inflation is 4 percent per year, then real GDP is growing at 2 percent per year.

.)( 21

2

1trr

tr

tr

ee

e

y

xw

45


Discounting– The appropriate discount factor for interest that is

compounded continuously is e-it.– The PDV of $10 (treated as received continuously throughout

the year) per year for 20 years at a 5 percent interest rate is

.42.126

)1(200

10

1

20

0

05.0

e

dtePDV t

46

The Present Discounted Value Approach to Investment Decisions

When a firm buys a machine, it is in effect buying a stream of net revenues in future periods.

The present discounted value is the value of this stream to the firm since it takes into account the opportunity costs of purchasing the machine.

47


Suppose the machine is to last n years and provide the marginal value product (Ri) in each year.

Then the PDV of the machine is given by

.)1()1(1 2

21n

n

r

R

r

R

r

RPDV

48


If the price of the machine (P) exceeds the PDV, the firm should always buy, but if the price is less than the PDV, the opportunity cost is too great.

Equilibrium in competitive markets would require the price equal to the PDV.

.)1()1(1 2

21n

n

r

R

r

R

r

RPDVP

49

Present Discounted Value and the Rental Rate

Assume the machine does not depreciate and returns the same marginal value product each year.

This uniform rate will equal the rental rate of the machine (v) since other firms would be willing to pay this for the machine’s use each period.

The PDV is thus

nr

v

r

v

r

vPDV

)1()1(1 2

50

Present Discounted Value and the Rental Rate

Given this perpetuity, and assuming equilibrium so that P = PDV, we obtain the equation below.

Note, this is the same result we obtained in Chapter 14 with no depreciation (d = 0).

.

or

rPv

r

vP

Documents

Appendix to Chapter 14 Compound Interest © 2004 Thomson Learning/South-Western