Annuity Formulas for Present Value and Future Value
Calculating the present and future values of a one time investment is a matter of simple mathematics. Suppose an investor buys an asset that is expected to be worth $10,000 in one year. Assuming a 10% discount rate, what is the present value of the investment?
$9,090.91 = 1,000 / (1 + 0.10)1
However, more complicated investment opportunities require more sophisticated calculation methods to arrive at accurate figures for both present and future values of investments. The standard present and future value formulas assume a one time investment or a one time payout. Some investments are not so simple.
An annuity is a continuous payment of the same amount of money over the course of an investment’s life. Examples might be dividends paid out to common stock holders, a lawsuit settlement, or payments made by the state to a lottery winner. Annuities can also be payments made for a car loan or mortgage. These payments can be viewed as equal cash flows (positive or negative) made over regular intervals such as annually, quarterly, or monthly.
Since the payments of annuity are the same amount made at regular intervals, the calculation of their present and future values are simpler then if the amount or intervals varied. However, they are significantly more tedious than straight present and future value calculations.
Suppose a lottery winner is to be paid $50,000 a year for 20 years (a million-dollar lottery win). The recipient of this annuity wants to know what the present value is of this annuity assuming a 10% discount rate. The present value of an annuity formula is given as:
PVA = CF * [((1 + r)n – 1) / (r * (1 + r)n]
where CF is the regular annuity cash flow, n is the number of periods, and r is the interest rate per period. Using this formula, the present value of the annuity is:
425,678.19 = 50,000 * [((1 + 0.10)20 – 1 / (0.10 * (1 + 0.10)20]
So at a discount rate of 10%, the recipient would be indifferent to receiving the $50,000 each year for the next 20 years or being paid about $425,678 in cash now.
Suppose a car dealership is expected to receive $2,000 a year for 5 years from the sale of a used car. What is the future value of this cash flow assuming a discount rate of 7%? The future value of an annuity formula is given as:
FVA = CF * [((1 + r)n) / r]
$11,501.48 = 2,000 * [((1 + 0.07)5) / 0.07)
So, the future value of the cash flows from the sale of the car is about $11,501. The discount premium is the expected value above the payments as a result of the interest rate. In this case, the premium is about $1,501 (11,501 – (2,000 * 5)).
An annuity is simply a string of equal payments made at defined intervals. The value of an annuity can be calculated if the cash flow, number of payment periods, and interest rate are known. Unequal payments or payments at non-regular intervals represent much more complicated situations in which the effects of interest on cash flows for each time period must be calculated separately and summed to find either the present or future value.