Find Time Value of Money with One Payment Held for Two Periods or More

Obviously, companies hold most long-term investments for longer than one year. To determine the future value of this investment for longer periods of time, just multiply the interest factor by itself for each year the investment is held. In other words, take the interest factor to the power of the number of years’ held, n:

PV(1 + i)n = FV

Suppose a company invests $400 today for five years, at an interest rate of 12 percent. What’s the future value of this investment?

PV = $400
i = 12%
n = 5
PV(1 + i)n = FV
400(1 + 0.12)5 = FV
400 x 1.76 = FV
$705 = FV

Investing $400 today and holding it for five years at 12 percent will eventually give you $705. You can also use this formula to find the present value required to reach a known future value. If you know that you need to have exactly $900 four years from now (that’s the future value) and that the expected interest rate is 9 percent, you can plug these values into the formula to figure out the present value:

i = 9%
n = 4
FV = $900
PV(1 + i)n = 900
PV x 1.41 = 900
PV = 900 / 1.41
PV = $638

Therefore, if you sock away $638 now at 9-percent annual interest, you’ll have $900 in four years.

As with the one-year version of the formula in the preceding section, treat the unit of (1 + i)n as a single factor to avoid using long formulas to convert between present value and future value.

In these examples, time value of money formulas are applied based on year-long periods, designating the variable n to measure the number of years. For more-precise results, apply time value of money formulas based on shorter periods of time, such as months or even days.

The variable n, then, would measure the number of months or days. That said, the interest rate, or i, always measures the interest rate per period. Therefore, if n equals one year, an annual interest rate of 12 percent would be apropos. However, if n equals one month, you should also express the interest rate by months — say, as 1 percent per month (12 percent divided by 12 months).

Bankers call this monthly compounding. To try daily compounding, where n equals one day, express the interest rate in days. For example, 12 percent divided by 365 days equals 0.0329 percent per day, so that i = 0.000329.

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