Today's value of future annuity payments.

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, given a required rate of return. This is the mathematical backbone of pension valuations, mortgage amortisation, lease accounting, personal injury settlements, and many insurance products. If you have ever compared a lump-sum pension transfer value against a promised income stream, you were computing an annuity present value — whether you realised it or not.

The formula that does the work

PV = PMT × [1 − (1 + r)^−n] ÷ r

Where PMT is the payment per period, r is the discount rate per period, and n is the number of periods. The expression in square brackets is called the annuity factor. It compresses a stream of identical future payments into a single present-day number by accounting for how each payment is worth less the further in the future it sits.

Worked example: 1,000 a year for 20 years discounted at 5 per cent. Annuity factor = (1 − 1.05^−20) ÷ 0.05 = 12.462. Present value = 1,000 × 12.462 = 12,462. So a promise of 1,000 a year for 20 years is worth approximately 12,462 today to someone who requires a 5 per cent return.

The three levers that change the answer

The discount rate dominates everything. At 3 per cent the same 1,000 × 20-year stream is worth 14,877. At 7 per cent it is worth 10,594. At 10 per cent it drops to 8,514. A 1 per cent change in discount rate moves present value by roughly 8 to 15 per cent depending on the term.

Term length compounds the rate effect. A 10-year stream at 5 per cent is worth roughly 7.7 times the annual payment. A 30-year stream at 5 per cent is worth roughly 15.4 times. Longer streams are worth more in absolute terms but less per year because later payments are discounted more heavily.

Payment timing (ordinary vs annuity due). This tool assumes ordinary annuity — payments at the end of each period. If payments are made at the start of each period (annuity due, common for rent), multiply the result by (1 + r) to get the correct present value.

Applications where this formula surfaces

Pension transfer values. When a defined-benefit pension offers a Cash Equivalent Transfer Value (CETV), the scheme is essentially computing the present value of the promised lifetime income stream (more accurately, a life-contingent annuity, which is a variant of this calculation). The CETV quote reflects their discount rate assumptions — typically tied to gilt yields plus a prudence margin. When gilt yields rise, CETVs fall, sometimes by 30 to 50 per cent in a few months.

Mortgages. A 200,000 mortgage at 5 per cent over 25 years has a monthly payment of roughly 1,169. The mortgage balance at any point is the present value of the remaining monthly payments discounted at the mortgage rate. This is why amortisation schedules show principal and interest components.

Lease accounting (IFRS 16). Companies booking operating leases must compute the present value of lease payments to record the right-of-use asset and lease liability on the balance sheet. The discount rate used is the implicit rate in the lease or the incremental borrowing rate.

Personal injury and divorce settlements. Courts frequently convert a stream of future income or support payments into a lump sum using present value math. The discount rate used in courts (the Ogden rate, currently reviewed periodically) has varied from 2.5 per cent to minus 0.75 per cent, driving lump sum sizes up and down by enormous amounts.

What the calculator does not include

Inflation. The discount rate and payment amount can be nominal or real (inflation-adjusted), but they must be consistent. If you are valuing a nominal 1,000 a year stream, use a nominal discount rate. If you are valuing an inflation-indexed stream (like a CPI-linked annuity or pension), use a real discount rate. Mixing them produces nonsense.

Mortality and life contingency. A true life annuity pays until death, not for a fixed number of periods. Valuing a life annuity requires mortality tables in addition to this formula. The basic annuity present value here assumes guaranteed payments for the specified term regardless of who lives or dies.

Taxes. Pension income, annuity income, and investment returns have different tax treatments. Compare gross streams to gross streams or net streams to net streams — do not mix.

Choosing the right discount rate

The discount rate should reflect the return you could earn on the same money elsewhere at similar risk. For a pension transfer decision, comparing against expected investment returns (typically 4 to 6 per cent real for balanced portfolios) is common. For a contract valuation, corporate cost of capital may be relevant. For a low-risk comparison, use gilt yields at the relevant duration.

The mistake to avoid: using your savings rate (0.5 per cent) as a discount rate will inflate the present value dramatically and bias you toward taking the income stream. Using your maximum growth ambition (10 per cent) will deflate the present value and bias you toward taking the lump sum. Use a rate that honestly reflects likely achievable return on the alternative.