Present Value Of An Annuity

The present value of an annuity is the current value of future payments from an annuity, given a specified rate of return or discount rate. The annuity's future cash flows are discounted at the discount rate. Thus, the higher the discount rate, the lower the present value of the annuity.

Because of the time value of money concept, receiving money today is worth more than receiving the same amount money in the future because the money today can be invested at a given rate of return. By the same logic, receiving $5,000 today is worth more than getting $1,000 per year for five years. The lump sum invested today is worth more at the end of the five years than the incremental investments of $1,000 each, even if invested at the exact same interest rate.

The formula for the present value of an ordinary annuity, as opposed to an annuity due, is as follows:

P = PMT x ((1 - (1 / (1 + r) ^ n)) / r)

Where:

P = the present value of an annuity stream

PMT = the dollar amount of each annuity payment

r = the interest rate (also known as the discount rate)

n = the number of periods in which payments will be made

Assume an individual has the opportunity to receive an annuity that pays $50,000 per year for the next 25 years with a 6 percent discount rate or a $650,000 lump-sum payment and needs to determine the more rational option. Using the above formula, the present value of this annuity is:

Present value of annuity = $50,000 x ((1 - (1 / (1 + 0.06) ^ 25)) / 0.06) = $639,168

Given this information, the annuity is worth $10,832 less on a time-adjusted basis, so the individual should choose the lump-sum payment over the annuity.

Note: This formula is for an ordinary annuity where payments are made at the end of the period in question. In the above example, each $50,000 payment would occur at the end of each year for 25 years. With an annuity due, the payments are made at the beginning of the period in question. To find the value of an annuity due, simply multiply the above formula by a factor of (1 + r):

P = PMT x ((1 - (1 / (1 + r) ^ n)) / r) x (1 + r)

If the above example of an annuity due, its value would be:

P = $50,000 x ((1 - (1 / (1 + 0.06) ^ 25)) / 0.06) x (1 + 0.06) = $677,518

In this case, the individual should choose the annuity due because it is worth $27,518 more than the lump-sum payment.