# Present Value of an Annuity

As originally defined, an annuity was a level stream of payments to an individual, lasting for the duration of the individual’s lifetime. The term “present value” denotes the technique for calculating the present worth of values received in the past or future (or both). Since an annuity contract guarantees payments to the holder commencing at a specific future time and continuing indefinitely, the present value technique is needed to express the contract’s current value.

The logic behind present value theory is that value received in the future is not equivalent to an equal nominal value received today. This is most conveniently explained using a monetary example. Offered a choice between $10 received today and $10 received in one year’s time, anybody would prefer immediate receipt of the money. If the annual interest rate on liquid funds is (say) 3%, immediate receipt would provide the individual with $10.30 in one year’s time, an improvement of $0.30 over delayed receipt of the money.

Although the time value of money is usually cited as the justification for present value analysis, it is not. Even in a non-monetary economy where no explicit interest is paid, present receipt of value would still be preferred to future receipt, given nominal equivalence. Consumption is the end-in-view behind all economic activity and immediate receipt allows for quicker (hence more) consumption. The use of money not only makes our day-to-day life easier and more productive, it also simplifies present-value analysis and calculations, which is why money always figures in present-value analysis.

There is a formula for calculating present value in situations similar to those of an annuity. If we assume annual compounding for simplicity, it is:

PV = F/i (1 - (1/1+i)ⁿ), where PV = present value, F = a fixed or level sum paid at regular intervals, i = the relevant interest rate or discount rate and n = the number of years over which payments are made. i is assumed to be an annual interest rate. (More frequent compounding would produce a different formula, in which the number of annual compounding periods is multiplied by the number of payment years to produce a larger exponent.)

A special kind of bond, called a consol, provides a fixed coupon payment indefinitely, with no maturity date. (It was occasionally sold by governments in the past.) It is therefore a perpetuity; n is equal to infinity. As n approaches infinity, (1+i)ⁿ approaches infinity and 1/ (1+i)ⁿ approaches zero, making PV = F/i.

In the case of an annuity, n is equal to the number of years remaining in the individual’s average life expectancy at the time annuity payments begin. Insurance companies are legally required to hold reserves against the necessity to pay holders of insurance contracts, and calculations such as this help them to calculate how much in reserves to hold. On average, the number of people who surprise them by outliving their average life expectancy (therefore receiving more annuity payments than the company expected to pay out) is counterbalanced by the number of people who die unexpectedly before reaching their average life expectancy (therefore relieving the insurance company of making expected payments).

It is worthwhile to understand how the parameters of the formula are calculated by the company – or anybody else who might want to make a similar calculation. F, the regular level payment, is fixed by the terms of the annuity contract and known to all participants in the annuity agreement. From the insurance company’s perspective, the interest rate (i) is the company’s “cost of capital” – the rate of return it would have to earn to compensate its owner/investors for investing their money in the company’s stock.

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Speak with an advisor over the phone about annuities for FREE.In principle, that cost of capital should reflect two key determinants of real value – risk and inflation. Investors must be compensated commensurate with the level of risk they run by investing their money. Specifically, they must receive a rate of return at least equal to that available on the best alternative investment of equal risk. If they don’t get it, they will withdraw their money in favor of that investment alternative. The more risky is the company’s investment, the higher i should be. Similarly, the higher future inflation is expected to be, the higher i should be in order to compensate investors for the loss of purchasing power their investment returns will suffer.

The higher i is, the lower is the present value – the less valuable is the loss of the F paid out to annuity holders, relative to the amount of value that would just compensate investors for that loss. The higher inflation is, the less valuable is the fixed F paid out to annuity holders, in terms of purchasing power.

Once PV is calculated, the insurance company knows the sum of money that will have to be raised during the accumulation period of a deferred annuity in order to be able to make the promised annuity payments. At that point, the company could then make another calculation – reversing the procedure of the present value formula – to determine the annual contributions it would have to receive from the annuity holder (and invest) in order to accumulate the necessary sum.

An annuity consumer could use the same procedure in order to calculate how much to save in order to generate his or her own “annuity” or – alternatively – how much to pay in contributions for an annuity that would generate a known amount of level annual income in retirement. In practice, the consumer need not utilize formulas; instead, plug-in websites allow the consumer to supply parameters such as contribution amounts, desired income in retirement and interest rates. As long as all but one of the parameters are supplied, the website program can calculate the missing value.

Whereas the insurance company will use its cost of capital as the discount rate or interest rate, the consumer will supply his or her own “cost of capital” – that is, the rate of return the consumer could earn on the best alternative investment of equal risk to that of the annuity.

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