The present value of an annuity is the amount that a series of equal payments, made at regular intervals for a fixed number of periods, is worth today, given a particular interest or discount rate. An annuity in this sense is any stream of level payments: monthly loan installments, annual bond coupons, lease payments, pension benefits or lottery prize payouts. Discounting each payment back to today and adding them gives the present value.
The calculation answers questions such as how much a lender should advance in exchange for a set of repayments, what a lump sum would be equivalent to a pension stream, or whether a stream of payments is worth more than an upfront offer. It is one of the core applications of the time value of money.
The formula
When payments are made at the end of each period, the formula is:
- is the present value of the annuity.
- is the amount of each payment.
- is the discount rate per period.
- is the number of payments.
The fraction is the present value annuity factor, which can also be read from published tables. When payments are made at the start of each period, as with an annuity due, the result is multiplied by , because each payment arrives one period sooner.
Worked examples
A lottery winner can choose $50,000 a year for 20 years, paid at the end of each year, or a lump sum today. At a discount rate of 6 percent, the annuity factor is (1 minus 1.06 to the power of negative 20) divided by 0.06, which is about 11.4699. The present value is $50,000 times 11.4699, or about $573,496. A lump sum above that figure is the better choice at 6 percent; below it, the annual payments are worth more. If the payments were made at the start of each year, the present value would be $573,496 times 1.06, or about $607,906.
A car buyer can afford $400 a month for five years, and the lender charges 6 percent a year, or 0.5 percent a month. With 60 payments, the annuity factor is about 51.7256, so the buyer can borrow about $20,690. This is how lenders size loans, and the same calculation in reverse gives the monthly payment on a known loan amount, which is then split into interest and principal using an amortization schedule.
Uses in accounting and finance
- Bonds: the price of a bond is the present value of its coupon annuity plus the present value of its face value, discounted at the market rate.
- Leases: lessees measure lease liabilities as the present value of remaining lease payments.
- Pensions and annuity products: insurers price a retirement annuity by discounting expected payments, adjusted for life expectancy.
- Capital budgeting: projects with level annual cash flows are evaluated using annuity factors in net present value (NPV) analysis.
- Legal settlements and structured payments are compared with lump sums on a present value basis.
Common mistakes
The most common error is mismatching the rate and the period: monthly payments need a monthly rate and the number of months. Another is treating payments at the start of each period as if they came at the end, which understates their value. A third is applying the level-payment formula to payments that grow over time, such as inflation-linked pensions, which need a growing annuity formula. Finally, the discount rate should reflect the risk of the payments: a guaranteed stream from a government justifies a lower rate than payments promised by a weak borrower. The related calculation of what level payments will accumulate to is covered in the entry on the future value of an ordinary annuity, and the corresponding loan structures are described in the entry on the loan.