Annuity Present Value Calculator

Why the present value of an annuity matters

An annuity is a stream of equal periodic payments for a defined number of periods. The present value is what that stream is worth today at a given required rate of return. The same calculation sits underneath pension valuations, mortgage amortisation, lease accounting, personal injury settlements and many insurance products. Comparing a lump-sum offer against a promised income stream is an annuity present value calculation, whether or not it is named as one.

The formula

PV = PMT × [1 − (1 + i)⁻ⁿ] ÷ i, where PMT is the payment per period, i is the discount rate per period as a decimal, and n is the number of periods. The bracketed expression is the annuity factor: it compresses a stream of identical future payments into one present-day number by accounting for each payment being worth less the further out it sits. The calculator reports the factor alongside the value, because it is the part that does not depend on the payment size.

Worked example: 1,000 a period for 20 periods at 5 per cent. The factor is (1 − 1.05⁻²⁰) ÷ 0.05 = 12.4622, so the present value is 1,000 × 12.4622 = 12,462.21.

Matching the rate to the payment frequency

The rate and the period count have to describe the same interval, and this is where the calculation most often goes wrong. For monthly payments the periodic rate is the annual rate divided by twelve and the periods are counted in months: 1,000 a month for 20 years at 6 per cent annual is entered as a rate of 0.5 and 240 periods, not 6 and 20. Entering an annual rate against a monthly count discounts each payment as though a year passed between them.

What moves the answer

The discount rate matters most on long streams and least on short ones. At 3 per cent the same 1,000 a period over 20 periods is worth 14.88 times the payment, at 5 per cent 12.46 times, at 7 per cent 10.59 times, and at 10 per cent 8.51 times. A one-percentage-point rise in the rate reduces the present value by roughly 3 per cent over five periods, about 8 per cent over twenty, and about 12 per cent over forty. The payment is the simplest lever: the present value is exactly proportional to it, so a 10 per cent larger payment is a 10 per cent larger answer at any rate and any term. Term length compounds the rate effect. A 10-period stream at 5 per cent is worth roughly 7.7 times the payment, a 30-period stream roughly 15.4 times. Longer streams are worth more in total but less per period, because later payments are discounted more heavily.

Payment timing also moves the answer, though it is not an input. This tool computes the ordinary annuity, with payments at the end of each period. Where payments fall at the start instead, as rent and most leases do, the stream is worth (1 + i) times as much; that figure is reported as its own row.

Applications where this formula surfaces

Pension transfer values. Where a scheme offers a lump-sum equivalent of a promised income, it is computing the present value of that stream, or more precisely a life-contingent annuity, which is a variant of this calculation. The quote reflects the scheme's discount-rate assumptions, which are usually tied to bond yields plus a prudence margin, so quotes move as those yields move.

Mortgages. A 200,000 mortgage at 5 per cent over 25 years carries a monthly payment of roughly 1,169. The balance at any point is the present value of the remaining payments discounted at the mortgage rate, which is what an amortisation schedule is showing.

Lease accounting. Under IFRS 16 and its equivalents in other reporting regimes, lease payments are discounted to a present value to record the right-of-use asset and the lease liability. The rate used is the rate implicit in the lease, or the incremental borrowing rate where that is not determinable.

Personal injury and divorce settlements. Courts in many jurisdictions convert a stream of future income or support payments into a lump sum using present value math. Some set a statutory discount rate for that conversion and review it periodically. Because the lump sum is the annuity factor times the payment, a one-percentage-point change in that rate moves the award by around 8 per cent on a twenty-period stream and more on longer ones, which is why revisions to it are contested.

What the calculator does not include

Inflation. The discount rate and the payment can both be nominal or both real, but they cannot be mixed. A nominal stream is discounted at a nominal rate; an inflation-indexed stream is discounted at a real one. Mixing the two produces a figure that corresponds to nothing.

Mortality and life contingency. A life annuity pays until death rather than for a fixed count of periods, and valuing one needs mortality tables alongside this formula; the calculation here assumes payments continue for the full term entered.

Taxes. Pension income, annuity income and investment returns are taxed differently, so gross and net streams are not comparable to each other.

Choosing a discount rate

The discount rate stands for the return available on the same money elsewhere at similar risk. A low-risk comparison typically draws on government bond yields at the matching duration; a contract valuation may use a corporate cost of capital; a pension comparison often uses an expected portfolio return. The choice shapes the result more than any other input on long streams. A very low rate inflates the present value, so the income stream looks larger relative to a lump sum; a very high one deflates it and the comparison reverses. A rate that reflects the return actually achievable on the alternative gives a result aligned with that assumption, and the assumption is the thing worth stating alongside the number.