Copyright © 2008, The McGraw-Hill Companies, Inc.
Here is another easy choice:Example 3.3.2 Which of these two banks is offering the best CD rate?Bank Rate Compounding
Cato National Bank and Trust 4.35% Daily
Meridian Mutual Building and Loan 4.35% QuarterlyBoth rates are the same, but Cato National compounds interest daily. Since more frequent
compounding means more interest overall, we know that Cato National will end up paying
more interest, and so their rate is the better one.Example 3.3.2 illustrates the fact that in making comparisons it is not enough to look at the
rate alone. Compounding also must be taken into account. This makes matters a bit more
complicated. What are we to make of a choice such as this one?Bank Rate Compounding
Bank of Bolivar 4.04% Annual
Richburg Savings Bank 3.98% DailyOn the one hand, Bank of Bolivar’s rate is higher. Yet on the other, Richburg Savings is
offering daily compounding. It is possible that Richburg’s daily compounding will more
than make up for its lower rate, and that despite initial appearances Richburg may in the
end be offering more interest. Yet it is also possible that Richburg’s daily compounding is
not enough to catch up with Bolivar’s higher rate.
Whether or not a higher rate is better than more frequent compounding really depends
on the rates involved. If the annual rate were much higher, we would know which to pick.
We know that even though daily compounding results in more interest, it isn’t that much
more. Given a choice between 8% compounded quarterly and 2% compounded daily and
we can easily tell which is the better rate. Likewise, if the rates were very, very close to
each other we would also know which to choose. Given a choice between 5.51% com-
pounded daily versus 5.52% compounded annually and we know that daily compound-
ing will more than make up for a mere 0.01% difference in the rates. But in cases like
this one, where the rates are far, but not too far, apart, there is no easy way to tell simply
by looking at them which is the winner.
We can resolve the question by actually trying both rates out. We can determine the
future value of the deposit at each bank and see which one is higher. Unfortunately, we
don’t know how much money is involved, nor do we know how long the account will be
open. Fortunately, though, this doesn’t matter. For comparison purposes, we can use any
amount of money as a present value. Interest is calculated as a percent, and so whichever
rate is higher for a $100 deposit will also be higher for a deposit of $100,000, or any other
amount for that matter. Likewise, we can use any period of time we like for comparison
purposes. Whichever rate is faster for 1 year will also be faster for 3^1 ⁄^2 years; if you always
run faster than I do, you will win a race with me regardless of how long the race is.
Since we can choose whatever PV and time we like, we might as well choose some nice
round numbers. Suppose we use PV $100 and a term of 1 year. Then:Bolivar: FV $100(1.0404)^1 $104.04Richburg: FV $100 1 + ___0.0398 365
365
$104.06From this, we can see that in this case daily compounding does indeed make up for the
lower rate. Richburg wins by a small margin. Of course, it would be possible for Bolivar to
raise its rate enough to match, or even beat, Richburg. A worthwhile question to consider
is this: How much would Bolivar have to raise its rate to match Richburg’s? Readers are
encouraged to attempt to answer this question for themselves, though we will hold off
answering this for a bit.
Let’s consider another comparison of different rates and compounding frequencies.3.3 Effective Interest Rates 115cb