Personal finance
Annuity present value calculator
What a series of constant instalments you will receive in future periods is worth today.
When to calculate the present value of an annuity
Present value answers a precise question: what is money I will receive in the future worth today. A sum available in a year is worth less than the same sum available now, because in the meantime the capital could have earned interest. Discounting means bringing each future instalment back to the present by dividing it by the accumulation factor of the periods that separate it from today.
It is the calculation that makes non-homogeneous quantities comparable: a lump-sum settlement against a periodic annuity, a cash purchase against instalment payments, surrendering a policy against keeping the annuity. In all these cases the comparison only means something once the flows are brought to the same instant in time.
The rate to use is the discount rate, that is the return you could obtain by employing the capital elsewhere at comparable risk. The higher the rate, the less future instalments are worth today: this is why rate rises depress the value of bonds already issued.
Numeric example: €1,000 received at year end for 20 years, discounted at 3% a year, is worth 1,000 × (1 − (1.03)^−20) / 0.03 = €14,877.47 today. The nominal total received is €20,000: the €5,122.53 difference is the cost of the deferral.
Common mistakes
- Adding up future instalments to compare them with an amount available today: without discounting, the comparison is meaningless.
- Picking an arbitrary discount rate: it should reflect the alternative return at comparable risk, not an optimistic expectation.
- Applying an annual rate to monthly instalments: as with every financial formula, rate and period must share the same unit.
Frequently asked questions
What is the formula for the present value of an annuity?
For an ordinary annuity A = R × (1 − (1 + i)^−n) / i, where R is the instalment, i the discount rate per period and n the number of periods. For an annuity due the result is multiplied by (1 + i).
Which discount rate should I use?
The return you would get by employing the capital in an alternative of comparable risk. For certain flows the yield of government bonds of matching maturity is often used; for uncertain flows a higher rate that embeds a risk premium.
Why is the present value lower than the sum of the instalments?
Because each future instalment is divided by the accumulation factor of the periods separating it from today. The further away it is, the less it weighs: that is the time value of money.
Can I use it to work out a loan instalment?
Yes, it is the same relationship seen from the other side: the amount lent is the present value of the future instalments. To get a mortgage instalment directly, use the amortization schedule calculator.
How this calculation works
Ordinary annuity: A = R × (1 − (1 + i)^−n) / i. Annuity due: A = R × (1 − (1 + i)^−n) / i × (1 + i). With i = 0 the present value equals R × n. The rate i must be a decimal and refer to the same frequency as the instalments.