The points made about semiannual compounding can be generalized as follows. When compounding periods are more frequent than once a year, use a modified version of Equation A.2:

Here M is the number of times per year compounding occurs. When banks compute daily interest, the value of M is set at 365, and Equation A.2a is applied.

The interest tables can be used when compounding occurs more than once a year. Simply divide the nominal, or stated, interest rate by the number of times compounding occurs, and multiply the years by the number of compounding periods per year. For example, to find the amount to which $1,000 will grow after 6 years with semiannual compounding and a stated 8% interest rate, divide 8% by 2 and multiply the 6 years by 2. Then look in Table A.2 under the 4% column and in the row for Period 12. You will find an interest factor of 1.6010. Multiplying this by the initial $1,000 gives a value of $1,601, the amount to which $1,000 will grow in 6 years at 8% compounded semiannually. This compares with $1,586.90 for annual compounding.

The same procedure applies in all of the cases covered—compounding, discounting, single payments, and annuities. To illustrate semiannual discounting in finding the present value of an annuity, consider the case described in the section "Present Value of an Annuity"—$1,000 a year for 3 years, discounted at 4%. With annual discounting, the interest factor is 2.7751, and the present value of the annuity is $2,775.10. For semiannual discounting, look under the 2% column and in the Period 6 row of Table A.5 to find an interest factor of 5.6014. This is now multiplied by half of $1,000, or the $500 received each 6 months, to get the present value of the annuity, $2,800.70. The payments come a little more rapidly—the first $500 is paid after only 6 months (similarly with other payments)—so the annuity is a little more valuable if payments are received semiannually rather than annually.

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