21.5 21.4 A Limit to Compounding
The rows in Table 21.2 (page 873) show a trend: More frequent compounding yields more interest. But as the frequency of compounding increases, the interest tends to a limiting amount, shown in the far-right column.
Why is this so? Basically, because the extra interest added to the principal from more frequent compounding is not on deposit for the entire year, interest is credited to the account (and begins earning interest) at the end of each quarter. For example, in the case of quarterly compounding, the interest earned in the first quarter is then on deposit for only three of the four quarters of the year. In the first row of Table 21.1, in the case of quarterly compounding, $25 in interest is posted at the end of the first quarter and is part of the principal for only the remaining three quarters of the year, earning
As compounding is done more and more often, smaller and smaller amounts of interest on interest are added.
Let’s see what happens with the crazy interest rate of 100% per year compounded times per year. For an initial balance of $1, the amount at the end of one year-from the compound interest formula, with and —is
As increases, this amount gets closer and closer to a special number called (see Spotlight 21.2). This is illustrated in Table 21.4, where the dots (ellipses) indicate that more decimal places follow. Try those values of m on your calculator.
| 1 | 2.0000000… |
| 10 | 2.5937424… |
| 100 | 2.7048138… |
| 1,000 | 2.7169239… |
| 1,000,000 | 2.7182804… |
881
The Number Spotlight 21.2
The number e is similar to the familiar number in several respects. Both arise naturally, in finding the area and circumference of circles, and in compounding interest continuously ( is also the base for the system of “natural” logarithms). In addition, neither number is rational (expressible as the ratio of two integers, such as 7/2) or even algebraic (the solution of a polynomial equation with integer coefficients, such as ); we say that they are transcendental numbers. Finally, no pattern has ever been found in the digits of the decimal expansion of either number.
In addition to its fundamental importance in banking and growth of populations, the number occurs naturally in several other common contexts as well.
A custom in some families is for each member to buy a holiday gift for just one other member (colloquially, “guy”). This practice allows everyone to take part in giving and receiving without having to buy a gift for every family member. In advance of the holiday, all the members’ names are put into a hat and each member draws out a name at random. If anyone draws his or her own name, the drawing is annulled and is redone. What is the probability that the first drawing is successful? (This problem is often called the hat-check problem, after a whimsical imaginary 19th-century situation in which men who check their hats at a theater checkroom get hats back at random.)
The answer tends toward as the family size increases. In other words, we can expect such a drawing to be successful only about 37% of the time. For families of sizes 2, 3, and 4, the chances are 50%, 33%, and 38%. So 1/e is a good approximation even for small families.
The situation is usually complicated further by the additional restriction that the drawing is also annulled if any husband or wife draws the other’s name (you’re supposed to give a gift to your spouse, regardless!). In this case, for a large family in which all members are paired off, the probability of a successful drawing turns out to be approximately , with the chances for 2, 3, and 4 couples being 17%, 11%, and 14%.
The same results hold if, instead of the husband- wife restriction, no one can have the same “guy” as the previous year. But if that rule is imposed in addition to the husband-wife rule, then the chance of a successful drawing goes down to —too small for the drawing ceremony to be fun anymore!
For a general interest rate , as becomes larger and larger, the limiting amount is , and the interest method is called continuous compounding. The APY is (). (You can calculate powers of using the or button on your calculator. On some calculators, this button is the function of the button marked or . For example, to calculate , press , press and enter 0.10. You get 1.105170918.)
Continuous Compounding DEFINITION
Continuous compounding is the method of calculating interest that yields what compound interest tends toward with more and more frequent compounding per period.
EXAMPLE 7 Continuous Compounding
For $1000 at an annual rate of 10%, compounded times in the course of a single year, what is the balance at the end of the year?
This quantity gets closer and closer to as the number of compoundings increases. No matter how frequently interest is compounded—daily, hourly, every second, infinitely often (“continuously”)—the original $1000 at the end of one year cannot grow beyond $1105.17. The values after 5 and 10 years are shown in the lower rows of Table 21.2 (page 873).
882
Self Check 7
Suppose that $1,000,000 earns 2% annual interest compounded one million times per year. How much interest is earned at the end of the year?
- $20,201.34
Continuous Interest Formula RULE
For a principal deposited in an account at a nominal annual rate , compounded continuously, the balance after years is
We illustrate with $1000 at 10%. For one year, we have and
To find the amount in the account after 5 years, we have :
exactly as shown in the rightmost column of Table 21.2 (page 873).
It makes virtually no difference whether compounding is done daily or continuously over the course of a year. Most banks apply a daily periodic rate (based on compounding continuously) to the balance in the account each day and post interest daily (rounded to the nearest cent). The daily nominal rate (for a non-leap year) is , so each day the balance of the account is multiplied by , the daily effective rate. Except for the rounding in posting interest, the effect is the same as continuous compounding throughout the year, because the compound interest formula gives , which simplifies to the formula from the continuous interest formula.
For example, for a principal of $1000 and an interest rate of 5%, interest compounded daily over a year yields an amount
while continuous compounding yields . Both round to the same $1051.27.
