Suppose you make a periodic payment, of a fixed amount, into an investment fund, and the fund earns a given interest rate for a period of time. How much will it amount to at the end of the given time period? Your payments form an annuity certain and the value they will amount to are referred to as the future value of the annuity, or the amount of the annuity. When you finish this chapter, you should understand what the future value of an annuity means, how to compute its value, and where it might be used in practical work.
uses of the future value of an annuity
Here are some examples of the future value of an annuity.
1. You are in your employer's 401(k) program and contribute a certain amount every payday to your 401(k) fund. Your employer matches part of your contribution. Assuming a given rate of return, how much will you have when you are ready to retire?
2. You are giving each of your children $12,000 annually, the maximum amount allowed without gift and estate tax considerations. How much will they have when you retire, assuming a given earnings rate in the funds?
3. You have bought an annuity from an insurance company and contribute a certain amount annually to the annuity. How much will you have at a given future time?
4. You contribute $2,000 annually to your IRA. How much will you have at retirement?
5. You contribute $500 monthly to a fund for the down payment on a house. When will you be able to buy a house?
6. You contribute $400 monthly to a fund for your children's college education. How much will you have when they go off to college?
Many readers will have such funds, especially as defined contribution retirement plans become more common and replace the old defined benefit plans, both as a corporate benefit and as an individual retirement effort. In early 2001, some people began suggesting that the Social Security system also have some self-managed plans as part of the Social Security program.
the equation for the future value of an annuity
From Chapter 5, we have the equation for the present value of an annuity:
To obtain the value t periods in the future, we simply multiply by (1 + i)t, compounding the value t periods into the future. We obtain
Future value =
i Equation 7.2
This is the equation for the future value of an annuity of 1, after t periods, at interest rate i per period.
The extensions to continuous functions follow the outline in Chapter 3 and will not be repeated here.
the tables for amount of annuity
Look at pages 135 to 142 for the amount (future value) of an annuity. You can see that as the interest rate i increases, the future value increases, because the amount deposited earns at a higher rate. As the number of periods increases, the amount increases, for two reasons. First, more payments have been made into the fund; second, the fund has been earning for a longer period.
some investment policy implications
You can see that as time passes, the fund increases. However, the contribution to the fund remains at 1. After a while, depending on the fund's earnings rate, the amount earned by the fund each period exceeds the amount contributed each period.
For example, after 13 years, at 6% interest, the value is 16.8699. The interest earnings during the next year will be 1.01, larger than the 1.00 contributed. At 19 years, the interest earnings are about double the annual contributions. You can see that after a relatively few years, at a reasonable interest rate, the management of the fund becomes more important than continuing the contributions, because the fund earns more than the contribution. From the viewpoint of individual financial management, this shows the importance of early and continued saving.
The future value of an annuity is the amount accumulated, using both the periodic contributions and an assumed interest rate earned per period. The amount equals the present value of an annuity multiplied by the amount at compound interest for the assumed interest rate and number of periods.
The equation for the future value of an annuity is
- for interest rate i and time t.
Practical applications include figuring accumulations in retirement and savings accounts.
You earn $30,000 annually and can contribute up to 10% into your 401(k) program, with your employer matching up to 4% of your salary. You contribute 5% of your salary annually. You wish to retire on $20,000 annually. Assume that you can earn 6% on your 401(k). When will you be able to retire? How will this change as the earnings rate changes? Try rates from 3% to 10% annually. Now change the contribution rate to 10% of earnings. How does this change your projected retirement date? Try earnings rates from 3% to 10% annually. Which is more important, increasing your contribution or increasing the earnings rate?
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