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What is the 'Present Value Of An Annuity'

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

BREAKING DOWN 'Present Value Of An Annuity'

Because of the financial concept known as the time value of money, 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.

Example: Calculation of an Ordinary Annuity's Present Value 

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 an opportunity to receive an annuity that pays $50,000 per year for the next 25 years, with discount rate of 6% or a lump sum payment of $650,000, 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 and 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 the year, 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 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.

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