Introduction to Corporate Finance

Ross et al.: Fundamentals

of Corporate Finance, Sixth

Edition, Alternate Edition

III. Valuation of Future

Cash Flows

6. Discounted Cash Flow
   Valuation

(^210) © The McGraw−Hill
Companies, 2002
Taking It to the Limit: A Note on
Continuous Compounding
If you made a deposit in a savings account, how often could your money be com-
pounded during the year? If you think about it, there isn’t really any upper limit. We’ve
seen that daily compounding, for example, isn’t a problem. There is no reason to stop
here, however. We could compound every hour or minute or second. How high would
the EAR get in this case? Table 6.3 illustrates the EARs that result as 10 percent is com-
pounded at shorter and shorter intervals. Notice that the EARs do keep getting larger,
but the differences get very small.
As the numbers in Table 6.3 seem to suggest, there is an upper limit to the EAR. If
we let qstand for the quoted rate, then, as the number of times the interest is com-
pounded gets extremely large, the EAR approaches:
EAR eq
1 [6.6]
where eis the number 2.71828 (look for a key labeled “ex” on your calculator). For ex-
ample, with our 10 percent rate, the highest possible EAR is:
EAR eq
1
2.71828.10
1
1.1051709
1
10.51709%
In this case, we say that the money is continuously, or instantaneously, compounded.
What is happening is that interest is being credited the instant it is earned, so the amount
of interest grows continuously.
180 PART THREE Valuation of Future Cash Flows

TABLE 6.3

Compounding

Frequency and Effective

Annual Rates

Compounding Number of Times Effective

Period Compounded Annual Rate

Year 1 10.00000%

Quarter 4 10.38129

Month 12 10.47131

Week 52 10.50648

Day 365 10.51558

Hour 8,760 10.51703

Minute 525,600 10.51709

What’s the Law?

At one time, commercial banks and savings and loan associations (S&Ls) were restricted in

the interest rates they could offer on savings accounts. Under what was known as Regulation

Q, S&Ls were allowed to pay at most 5.5 percent and banks were not allowed to pay more

than 5.25 percent (the idea was to give the S&Ls a competitive advantage; it didn’t work). The

law did not say how often these rates could be compounded, however. Under Regulation Q,

then, what were the maximum allowed interest rates?

The maximum allowed rates occurred with continuous, or instantaneous, compounding.

For the commercial banks, 5.25 percent compounded continuously would be:

EXAMPLE 6.11