Compound and Simple Interest

We have seen that if an amount, P, is lent for one interest period at the interest rate, i, the amount that must be repaid at the end of the period is F P(1 i ). But loans may be for several periods. How is the quantity of money that must be repaid computed when the loan is for N interest periods? The usual way is one period at a time. Suppose that the amount P is borrowed for N periods at the interest rate i. The amount that must be repaid at the end of the N periods is P(1 i)^N; that is

F P(1  i)^N (2.1)

This is derived as shown in Table 2.1.

This method of computing interest is called compounding. Compounding assumes that there are N sequential one-period loans. At the end of the first interest period, the borrower owes P(1  i). This is the amount borrowed for the second period. Interest is

The most commonly used interest period is one year. If we say, for example, “6 percent interest” without specifying an interest period, the assumption is that 6 percent interest is paid for a one-year period. However, interest periods can be of any duration. Here are some other common interest periods:

Interest Period Interest Is Calculated:

Semiannually Twice per year, or once every six months Quarterly Four times a year, or once every three months

Monthly 12 times per year

Weekly 52 times per year

Daily 365 times per year

Continuous For infinitesimally small periods CLOSE-UP 2.1 Interest Periods

Table 2.1 Compound Interest Computations

Beginning Amount Owed

of Period Amount Lent Interest Amount at Period End

1 P  Pi  P  Pi  P(1  i)

2 P(1  i)  P(1  i)i  P(1  i)  P(1  i)i  P(1  i)² 3 P(1  i)²  P(1  i)²i  P(1  i)² P(1  i)²i  P(1  i)³

. .

. .

. .

N P(1  i)^N1  [P(1  i)^N1]i  P(1  i)^N

required on this larger amount. At the end of the second period [P(1  i)](1  i) is owed.

This is the amount borrowed for the third period. This continues so that at the end of the (N – 1)th period, P(1  i)^N–1 is owed. The interest on this over the N^th period is [P(1  i)^N–1]i. The total interest on the loan over the N periods is

I_c P(1  i)^N – P (2.2)

I_cis called compound interest. It is the standard method of computing interest where interest accumulated in one interest period is added to the principal amount used to calculate interest in the next period. The interest period used with the compound interest method is called the compounding period.

EXAMPLE 2.2

If you were to lend $100 for three years at 10 percent per year compound interest, how much interest would you get at the end of the three years?

If you lend $100 for three years at 10 percent compound interest per year, you will earn $10 in interest in the first year. That $10 will be lent, along with the original $100, for the second year. Thus, in the second year, the interest earned will be $11 

$110(0.10). The $11 is lent for the third year. This makes the loan for the third year

$121, and $12.10  $121(0.10) in interest will be earned in the third year. At the end of the three years, the amount you are owed will be $133.10. The interest received is then

$33.10. This can also be calculated from Equation (2.2):

I_c F  P  100(1  0.1)³– 100 33.10 ^100*(1 0.1)^3  100 Table 2.2 summarizes the compounding process.

An Excel spreadsheet corresponding to Table 2.2 along with instructions for creat-ing it are provided in the Spreadsheet Savvy box on page 24. _____________________쏋

Table 2.2 Compound Interest Computations for Example 2.2

Beginning Amount Owed

of Year Amount Lent Interest Amount at Year-End

1 100  100 0.1  $110

2 110  110 0.1  $121

3 121  121 0.1  $133.10

If the interest payment for an N-period loan at the interest rate i per period is computed without compounding, the interest amount, I_s, is called simple interest. It is computed as

I_s PiN

Simple interest is a method of computing interest where interest earned during an interest period is not added to the principal amount used to calculate interest in the next period. Simple interest is rarely used in practice, except as a method of calculating approxi-mate interest.

EXAMPLE 2.3

If you were to lend $100 for three years at 10 percent per year simple interest, how much interest would you get at the end of the three years?

The total amount of interest earned on the $100 over the three years would be $30.

This can be calculated by using I_s PiN:

I_s PiN  100(0.10)(3)  30 ____________________________________________쏋

Interest amounts computed with simple interest and compound interest will yield the same results only when the number of interest periods is one. As the number of periods increases, the difference between the accumulated interest amounts for the two methods increases exponentially.

S P R E A D S H E E T S A V V Y

One of the most common uses of spreadsheets is to compute the value of formulas. The formulas written into a cell of a worksheet can be as complicated as necessary to write out the calculation desired. It’s a good idea to use parentheses often, both to indicate the order of execution of the oper-ations within the formula and to make error checking easier. Use  and  for addition and subtrac-tion, * for multiplicasubtrac-tion, / for division, and ^ for exponentiation. Within a formula, you can refer to values that are stored in other cells. For example, if the value for P is in cell B1, the value for i is in cell B2, and the value for N is in cell A8, the expression P(1  i )^N1is entered into a worksheet cell as “B1*(1  B2)^(A8  1).” Note that the  sign is used to indicate that the expression is to be evaluated by Excel.

Worksheets such as the one shown in Table 2.2 are particularly easy to create in a spreadsheet program. It is a good idea to designate one area of the spreadsheet for parameters and to document each. For example, the values for P and i are entered into cells B1 and B2. For documentation, cells A1 and A2 have text indicating what cells B1 and B2 mean. Calculations in other cells of the work-sheet can reference the values of P and i, and these calculations can be easily updated simply by changing the contents of cells B1 or B2 as needed.

In the example worksheet below, Table 2.2 is reproduced for five periods for a principal amount P $100 and an interest rate of i  10 percent. The values are shown in the left-hand table and the formulas used are shown in the right. Start by entering “B1” into cell B4. C4 is computed as the product of cells B4 and B2, and D4 as the sum of B4 and C4. Cell B5, the amount owed at the beginning of period 2, is the amount owed at the end of period 1, D4. Once cells C5 and D5 are entered, rows 6 through 8 can be completed by using the Fill Down function: Highlight cells B5:D5 to B8:D8, and under the Home tab in Excel, select Fill Down. This will copy the formulas from cells B5 to D5 down to rows 6 through 8, making the appropriate row number adjustments auto-matically. Copying and pasting will achieve the same effect.

Important: the “$” entered in cell C4 in reference to B$2 tells Excel not to adjust the row reference to the interest rate when cell B2 is filled down or pasted.

When the number of interest periods is significantly greater than one, the difference between simple interest and compound interest can be very great. In April 1993, a couple in Nevada presented the state government with a $1000 bond issued by the state in 1865.

The bond carried an annual interest rate of 24 percent. The couple claimed the bond was now worth several trillion dollars (Newsweek, August 9, 1993, p. 8). If one takes the length of time from 1865 to the time the couple presented the bond to the state as 128 years, the value of the bond could have been $908 trillion  $1000(1  0.24)¹²⁸. As of 2012, the value would have been $54 quadrillion!

If, instead of compound interest, a simple interest rate given by iN  (24%)(128)  3072 percent were used, the bond would be worth only $31 720  $1000(1  30.72).

Thus, the difference between compound and simple interest can be dramatic, especially when the interest rate is high and the number of periods is large. The graph in Figure 2.2 shows the difference between compound interest and simple interest for the first 20 years of the bond example. As for the couple in Nevada, the $1000 bond was worthless after all—a state judge ruled that the bond had to have been cashed by 1872.

The conventional approach for computing interest is the compound interest method rather than simple interest. Simple interest is rarely used, except perhaps as an intuitive (yet incorrect!) way of thinking of compound interest. We mention simple interest pri-marily to contrast it with compound interest and to indicate that the difference between the two methods can be large.