## 3

P (1 + 2i)

Pi

P (1+ 2i) + Pi = P (1 + 3i)

n

P (1 + (n-1)i)

Pi

P (1+ (n-1)i) + Pi = P (1 + ni)

earns P*i dollars (\$8) of interest since under simple compounding, interest is paid only on the original principal amount P (\$100). Thus at the end of year 2, the balance in the account is obtained by adding P dollars (the original principal) plus P*i (the interest from year 1) plus P*i (the interest from year 2) to obtain P+P*i+P*i (\$100+\$8+\$8=\$116). After some algebraic manipulation, this can be written conveniently mathematically as P*(1+2*i) dollars (\$100*1.16=\$116).

Table 4.4 extends the above logic to year 3 and then generalizes the approach for year n. If we return our attention to our original goal of developing a formula for Fn which is expressed only in terms of the present amount P, the annual interest rate i, and the number of years n, the above development and Table 4.4 results can be summarized as follows:

Example 3

Determine the balance which will accumulate at the end of year 4 in an account which pays 10%/yr simple interest if a deposit of \$500 is made today.

Fn = P * (1 + n*i) F4 = 500 * (1 + 4*0.10) F4 = 500 * (1 + 0.40) F4 = 500 * (1.40) F4 = \$700

### 4.6.4 Compound Interest

For compound interest, interest is earned (charged) on the original principal amount plus any accumulated interest from previous years at the rate of i% per year (i%/ yr). Table 4.5 illustrates the annual calculation of compound interest. In the Table 4.5 and the formulas which follow, i is expressed as a decimal amount (i.e., 8% interest is expressed as 0.08).

At the beginning of year 1 (end of year 0), P dollars (e.g., \$100) are deposited in an account earning i%/yr (e.g., 8%/yr or 0.08) compound interest. Under compound interest, during year 1 the P dollars (\$100) earn P*i dollars (\$100*0.08 = \$8) of interest. Notice that this the same as the amount earned under simple compounding. This result is expected since the interest earned in previous years is zero for year 1. At the end of the year 1 the balance in the account is obtain by adding P dollars (the original principal, \$100) plus P*i (the interest earned during year 1, \$8) to obtain P+P*i (\$100+\$8=\$108). Through algebraic manipulation, the end of year 1 balance can be expressed mathematically as P*(1+i) dollars (\$100*1.08=\$108).

During year 2 and subsequent years, we begin to see the power (if you are a lender) or penalty (if you are a borrower) of compound interest over simple interest. The beginning of year 2 is the same point in time as the end of year 1 so the balance in the account is P*(1+i) dollars (\$108). During year 2 the account earns i% interest on the original principal, P dollars (\$100), and it earns i% interest on the accumulated interest from year 1, P*i dollars (\$8). Thus the interest earned in year 2 is [P+P*i]*i dollars ([\$100+\$8]*0.08=\$8.64). The balance at the end of year 2 is obtained by adding P dollars (the original principal) plus P*i (the interest from year 1) plus [P+P*i]*i (the interest from year 2) to obtain P+P*i+[P+P*i]*i dollars (\$100+\$8+\$8.64=\$116.64). After some algebraic ma

Table 4.5 The Mathematics of Compound Interest

Year (t)

Amount At Beginning Of Year

Interest Earned During Year

Amount At End Of Year

(Ft)

0 0