Unformatted text preview: WEB APPENDIX 5A
CONTINUOUS COMPOUNDING AND DISCOUNTING
In Chapter 5, we dealt only with situations where interest is added at discrete
intervals—annually, semiannually, monthly, and so forth. In some instances,
though, it is possible to have instantaneous, or continuous, growth. In this web
appendix, we discuss present va lue and future value calculations when the interest rate is compounded
continuously. Continuous Compounding
The relationship between discrete and continuous compounding is illustrated in
Figure 5A-1. Panel a shows the annual compounding case, where interest is added
once 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 amount
because interest is earned on interest more often.
Equation 5-1 in the chapter can be applied to any number of compounding
periods per year as follows:
More frequent compounding : FVN ¼ PV 1 þ
A situation in which
interest is added continuously rather than at
discrete points in time. 5A-1 Here INOM is the stated annual rate, M is the number of periods per year, and N is
the number of years. To illustrate, let PV ¼ $100, I ¼ 10%, and N ¼ 5. At various
compounding periods per year, we obtain the following future values at the end of
Annual : FV5 ¼ $100 1 þ
¼ $100ð1:10Þ5 ¼ $161:05
Semiannual : FV5 ¼ $100 1 þ
¼ $100ð1:05Þ10 ¼ $162:89
Monthly : FV5 ¼ $100 1 þ
¼ $100ð1:0083Þ60 ¼ $164:53
Daily : FV5 ¼ $100 1 þ
365 We could keep going, compounding every hour, every minute, every second, and
so forth. At the limit, we could compound every instant, or continuously. The
equation for continuous compounding is as follows:
FVN ¼ PVðeIN Þ 5A-2 Here e is the value 2.7183…. If $100 is invested for five years at 10% compounded
continuously, FV5 is calculated as follows:1
Continuous : FV5 ¼ $100½e0:10ð5Þ ¼ $100ð2:7183:::Þ0:5
¼ $164:87 1 Calculators 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. 5A-1 5A-2 Web Appendix 5A Annual, Semiannual, and Continuous Compounding: Future Value with I ¼ 25% FIGURE 5A-1
a. Annual Compounding b. Semiannual Compounding c. Continuous Compounding Dollars Dollars Dollars 4
3 4 4
Earned $3.0518 $3.4903 $3.2473 3 Interest
Earned 3 2 2 2 1 1 1 0 1 2 3 4 5
Years 0 1 2 3 4 5
Years 1 0 2 3 4 5
Years Continuous Discounting
Equation 5A-2 can be transformed into Equation 5A-3 and used to determine
present values under continuous discounting.
PV ¼ 5A-3 FVN
¼ FVN ðe−IN Þ
eIN 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 ¼ $1, 649
ð2:7183:::Þ0:5 ¼ $1,649
1:649 ¼ $1,000 PROBLEMS
5A-2 5A-3 5A-4 5A-5 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?
PV CONTINUOUS COMPOUNDING In 7 years, you are scheduled to receive money from a
trust 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 in
the account today?
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 annual
rate. If you deposit $1,000 with each bank, what will be the difference in the two bank
account balances after two years?
CONTINUOUS COMPOUNDED INTEREST RATE To purchase your first home 6 years from
today, you need a down payment of $40,000. You currently have $20,000 to invest. To
achieve your goal, what nominal interest rate, compounded continuously, must you earn on
CONTINUOUS COMPOUNDING You have the choice of placing your savings in an
account paying 10.25% compounded annually, an account paying 10.0% compounded
semiannually, or an account paying 9.5% compounded continuously. To maximize
your return, which account would you choose? Web Appendix 5A 5A-6
5A-8 CONTINUOUS COMPOUNDING You have $11,572.28 in an account that has been paying an
annual rate of 9% compounded continuously. If you deposited funds 15 years ago,
how much was your original deposit?
CONTINUOUS COMPOUNDING For a 10-year deposit, what annual rate payable semiannually will produce the same effective rate as 3% compounded continuously?
PAYMENT AND CONTINUOUS COMPOUNDING You place $2,000 in an account that pays
8% 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% compounded
semiannually. If the accounts are to have the same amount at the end of the 3 years, how
much of an initial deposit do you need to make now in the account that pays 9% interest
compounded semiannually? 5A-3 ...
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