CHAPTER 5

Mathematics of Individual Finance

Future Value of an Annuity

The future value of an annuity refers to the amount that we will accumulate by making regular payments at interest over a period of time. The formula is simply:

FVA n = PMT(1+r)1 + PMT(1+r)2 + ... + PMT(1+r)n

= PMT[(1+r)1 + (1+r)2 + ... + (1+r)n]

where the sum in the bracket is called the future value interest factor of an annuity. These factors can be found in Appendix Table A-4.

Example 5-11

Julie is 23 and has started her first job. She plans to put aside $5,000 per year so that she can make a nice down payment on a house in 6 years. If she makes the payments at the end of each year and earns 8 percent on her money, how much will she have accumulated at the end of 6 years?

Solution 5-11

You need to find the future value of an annuity of $5,000 for 6 years at 8%. Using Appendix Table A-4, the future value interest factor of the annuity is 7.3359. Multiplying this by $5,000 gives us $36,679.50. On a financial calculator, [6 N 8 %i 5000 PMT CPT FV] gives us $36,679.65. Note that some calculators will give the answer as a negative number since they may be set up to regard payments as coming to you. If this bothers you, make your payment a negative number (often by hitting the +/- key) and the answer will be the same, but given as a positive number.

LOTUS has a formula for the future value of an annuity given as @FV(payments , interest, term). In Example 5-11 the appropriate LOTUS formula would be @FV(5000,.08,6).

If payments begin immediately, rather than at the end of the first period, you must calculate the future value of an annuity due. Since the last payment is made at the beginning of the last period, the entire future value of the annuity earns an extra year's interest by the end of the last period. To calculate the future value of an annuity due, merely multiply the future value of the annuity by 1 plus the interest rate.

Example 5-12

In Example 5-11, Julie decides to begin contributing to her house fund immediately. How much will she have at the end of the 6 years?

Solution 5-12

Since this is the future value of an annuity due, multiply the future value of the annuity, which is $36,679.65 by (1 plus the interest rate) or 1.08. This shows that Julie will have $39,614.02 at the end of 6 years. On a financial calculator with a DUE button the entire procedure would be [6 N 8 %i 5000 PMT DUE FV] and the answer will be $39,614.02. On a LOTUS spreadsheet which does not have an annuity due function, you would enter @FV(5000,.08,6)*(1.08).

[Determining Payments]

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