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8-6 Compound Interest CH 8]

their principal depending on their market conditions. With semiannual compounding the
interest period is for six months and the interest is computed twice each year. Interest rates
are generally stated on an annual basis such as 8% annual and the semiannual rate is half
of the annual rate or (8% ÷ 2 =) 4%. Let’s illustrate this with the simple interest equation of:
I = P R T

where P is $1,500, R is 8% and T is one year; then the calculation is:

P R T
I = $1,500 x 0.08 x 1yr = $120.00

When calculating interest semiannually using the simple interest equation, the same $1,500
principal is used. The Time variable is changed for semiannual as noted below.

P R T
I = $1,500 x 0.08 x (6 mo/12 mo per yr) = $60.00,

The interest is one-half ($60) the interest for a full year when interest is calculated annually.

In compounding, this $60 interest is added to the principal increasing it to ($1,500 + 60
=) $1,560. When interest due is calculated at the next semiannual payment, 6 months later,
the calculation becomes:

P R T
I = $1,560 x 0.08 x (6 mo/12 per yr) = $62.40.

Note that the $62.40 interest due with this second calculation is larger than the first
calculation because the $60 interest earned in the first calculation has been added to the
beginning principal of $1,500; Thus the interest earned is now earning interest, the
advantage of compound interest and the joy of capitalism—having your money work for you.
In the simple interest equation, you are calculating only the interest earned on the
principal at a given interest rate for the interest period of time; This is Chapter 7. With
compound interest, the interest calculation is the same only now the interest earned is
added to the principal, increasing the principal, and the calculation is made again using the
same terms and yielding a higher interest return.
If in our example, the investment was allowed to accumulate for 3 years, with
semiannual interest calculations and payments, the interest is calculated (3 yr x 2 times/
year =) 6 times. As the interest is calculated six times there are six interest periods.

Example B: Willie Jones deposited $1,500 in a fund that pays 8% interest compounded
semiannually. Calculate the value of his account at the end of 3 years.

Solution (1): Original principal ............................................................. $1,500.00
Interest for first period ($1,500 x 0.08 x 6/12) ........................................ 60.00
Principal for second period ..................................................................... $1,560.00
Interest for second period ($1,560.00 x 0.08 x 6/12) ............................... 62.40
Principal for third period ........................................................................ $1,622.40
Interest for third period ($1,622.40 x 0.08 x 6/12) .................................. 64.90
Principal for fourth period ...................................................................... $1,687.30
Interest for fourth period ($1,687.30 x 0.08 x 6/12) ................................ 67.49
Principal for fifth period .......................................................................... $1,754.79
Interest for fifth period ($1,754.79 x 0.08 x 6/12) .................................... 70.19
Principal for sixth period ........................................................................ $1,824.98
Interest for sixth period ($1,824.98 x 0.08 x 6/12) ................................... 73.00
Amount in fund at the end of 3 years ...................................................... $1,897.98

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