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W E B A P P E N D I X 5 A

CONTINUOUS COMPOUNDING AND DISCOUNTING

In Chapter 5, we dealt only with situations where interest is added at discreteintervalsannually, semiannually, monthly, and so forth. In some instances,though, it is possible to have instantaneous, or continuous, growth. In this webappendix, we discuss present value and future value calculations when the interest rate is compoundedcontinuously.

Continuous Compounding

The relationship between discrete and continuous compounding is illustrated inFigure 5A-1. Panel a shows the annual compounding case, where interest is addedonce a year; Panel b shows the situation when compounding occurs twice a year;and Panel c shows interest being earned continuously. As the graphs show, the

more frequent the compounding period, the larger the final compounded amountbecause interest is earned on interest more often.

Equation 5-1 in the chapter can be applied to any number of compoundingperiods per year as follows:

More frequent compounding: FVN PV 1 INOM

M

MN5A-1

Here INOM is the stated annual rate, M is the number of periods per year, and N isthe number of years. To illustrate, let PV $100, I 10%, and N 5. At variouscompounding periods per year, we obtain the following future values at the end offive years:

Annual: FV5 $100 1 0:10

1

51

$1001:105 $161:05

Semiannual: FV5 $100 1 0:10

2

52 $1001:0510 $162:89

Monthly: FV5 $100 1 0:10

12

512 $1001:008360 $164:53

Daily: FV5 $100 1 0:10

365

5365 $164:86

We could keep going, compounding every hour, every minute, every second, andso forth. At the limit, we could compound every instant, or continuously. Theequation for continuous compounding is as follows:

FVN PVeIN 5A-2

Here e is the value 2.7183. If $100 is invested for five years at 10% compoundedcontinuously, FV5 is calculated as follows:

1

Continuous : FV5 $100e0:105 $1002:7183:::0:5

$164:87

1Calculators with exponential functions can be used to evaluate Equation 5A-2. For example, with an HP-10BII, you

would type .5, press the ex key to get 1.6487, and then multiply by $100 to get $164.87.

Continuous

Compounding

A situation in which

interest is added con-

tinuously rather than at

discrete points in time.

5A-

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Continuous Discounting

Equation 5A-2 can be transformed into Equation 5A-3 and used to determinepresent values under continuous discounting.

PV FVN

eIN FVNe

IN5A-3

Thus, if $1,649 is due in 10 years and if the appropriate continuous discount rate, I,is 5%, the present value of this future payment will be $1,000.

PV

$1,649

2:

7183:::

0:5

$1,649

1:649 $1,000

PROBLEMS

5A-1 FV CONTINUOUS COMPOUNDING If you receive $15,000 today and can invest it at a 6%annual rate compounded continuously, what will be your ending value after 15 years?

5A-2 PV CONTINUOUS COMPOUNDING In 7 years, you are scheduled to receive money from atrust established for you by your grandparents. When the trust matures, there will be$200,000 in the account. If the account earns 9% compounded continuously, how much is inthe account today?

5A-3 FV CONTINUOUS COMPOUNDING Bank A offers a nominal annual interest rate of 7%

compounded daily, while Bank B offers continuous compounding at a 6.9% nominal annualrate. If you deposit $1,000 with each bank, what will be the difference in the two bankaccount balances after two years?

5A-4 CONTINUOUS COMPOUNDED INTEREST RATE To purchase your first home 6 years fromtoday, you need a down payment of $40,000. You currently have $20,000 to invest. Toachieve your goal, what nominal interest rate, compounded continuously, must you earn onthis investment?

5A-5 CONTINUOUS COMPOUNDING You have the choice of placing your savings in anaccount paying 10.25% compounded annually, an account paying 10.0% compoundedsemiannually, or an account paying 9.5% compounded continuously. To maximizeyour return, which account would you choose?

Annual, Semiannual, and Continuous Compounding: Future Value with I 25%FIGURE 5A-1

Dollars

InterestEarned

a. Annual Compounding

4

3

2

1

0 1 2 3 4 5

$3.0518

Years

Dollars

InterestEarned

b. Semiannual Compounding

4

3

2

1

210 3 4 5

$3.2473

Years

Dollars

InterestEarned

c. Continuous Compounding

4

3

2

1

0 1 2 3 4 5

$3.4903

Years

5A-2 Web Appendix 5A

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5A-6 CONTINUOUS COMPOUNDING You have $11,572.28 in an account that has been paying anannual rate of 9% compounded continuously. If you deposited funds 15 years ago,how much was your original deposit?

5A-7 CONTINUOUS COMPOUNDING For a 10-year deposit, what annual rate payable semian-nually will produce the same effective rate as 3% compounded continuously?

5A-8 PAYMENT AND CONTINUOUS COMPOUNDING You place $2,000 in an account that pays8% interest compounded continuously. You plan to hold the account for exactly 3 years.

At the same time in another account, you deposit money that earns 9% compoundedsemiannually. If the accounts are to have the same amount at the end of the 3 years, howmuch of an initial deposit do you need to make now in the account that pays 9% interestcompounded semiannually?

Web Appendix 5A 5A