What it means
Most financial arrangements involve a stream of level payments. A mortgage is repaid in equal monthly instalments.
A lease is paid in equal quarterly rents. A bond pays the same coupon every six months.
A pension pays the same amount every month. Valuing any of these means adding up the payments after discounting each for the time until it arrives, and because the payments are equal the sum has a closed formula: the present value of an annuity is the payment multiplied by an annuity factor that depends only on the interest rate and the number of payments.
The same mathematics runs forward: the future value of a series of equal savings deposits is the deposit multiplied by a future value annuity factor. Two conventions matter.
An ordinary annuity pays at the end of each period, which is how loans and most bonds work. An annuity due pays at the start of each period, which is how rent and many leases work; each payment arrives one period earlier and the value is higher by one period's interest.
A perpetuity is an annuity that never ends, and its present value is simply the payment divided by the rate. As a retirement product, an annuity is insurance against living longer than your money.
The buyer pays a lump sum, usually from a pension fund, and the insurer pays a guaranteed income. A lifetime annuity pays until death; a fixed-term annuity pays for a set number of years; variations add inflation linking, a guarantee period, a continuing payment to a surviving spouse, or an income that starts only at an advanced age.
The insurer prices the annuity on interest rates, the buyer's age and health, and the options chosen. The trade-off is certainty against flexibility: the income cannot run out, but the capital is gone and cannot usually be recovered if circumstances change.
Understanding annuity mathematics is what allows a borrower to check a loan quote, a lessee to compare buying with leasing, an investor to price a bond and a retiree to judge whether an insurer's offer is fair.
In practice
Real-world examples.
Example
A car lease of $450 a month for 36 months with payments at the start of each month is an annuity due; at 0.5% a month its present value is $450 x 32.87 x 1.005 = $14,866.
Example
A company valuing a ten-year lease commitment at $100,000 a year discounts it as an annuity at its borrowing rate of 7%, giving a liability of $702,400.
Example
A retiree uses half her pension fund to buy an annuity covering her essential expenses and keeps the rest invested for flexibility.
Think of it
“An annuity is a stream of equal payments over time-like monthly loan payments or pension checks.
Formula
Calculation
Present Value of an ordinary annuity = Payment x [ (1 minus (1 + r) to the power minus n) / r ]
Future Value of an ordinary annuity = Payment x [ ((1 + r) to the power n minus 1) / r ]
Payment for a given present value (loan instalment) = PV x [ r / (1 minus (1 + r) to the power minus n) ]
Annuity due values = ordinary annuity values x (1 + r)
where r is the interest rate per period and n the number of payments.
Worked example 1, a loan. A business borrows $50,000 for four years at 6% a year with annual payments in arrears.
- Annuity factor = (1 minus 1.06 to the power minus 4) / 0.06 = (1 minus 0.7921) / 0.06 = 3.465
- Annual payment = $50,000 / 3.465 = $14,430
- Total paid = $57,720, of which $7,720 is interest
Worked example 2, saving. A saver puts $500 a month into an account earning 6% a year (0.5% a month) for ten years (120 payments).
- Future value factor = (1.005 to the power 120 minus 1) / 0.005 = (1.8194 minus 1) / 0.005 = 163.88
- Future value = $500 x 163.88 = $81,940, of which $60,000 is contributions and $21,940 is interest
Worked example 3, a retirement annuity. A 65-year-old has $300,000 and is offered a level lifetime annuity of $18,000 a year. If she lives to 88 (23 years), the payments total $414,000. At a 4% discount rate the present value of 23 annual payments of $18,000 is $18,000 x 14.857 = $267,400, less than the $300,000 paid; at 3% it is $18,000 x 16.444 = $296,000, close to the price. The insurer's margin covers its costs and the risk that she lives past 88, which is the point of the product.Case study
Seen in the real world.
A small manufacturer was offered a choice by an equipment supplier: pay $180,000 now, or $42,000 a year for five years with the first payment on delivery. The owner's instinct was that $210,000 over five years was obviously worse than $180,000 now. His accountant treated the five payments as an annuity due and calculated their present value at the company's 9% borrowing rate: $42,000 x 3.890 x 1.09 = $178,100, marginally less than the cash price.
Because the company's bank overdraft cost 11%, the supplier's implied interest rate of about 9.3% was cheaper than the company's own borrowing, and the instalment option also preserved $138,000 of cash for working capital in the year the new machine came on stream. The company took the instalments. The owner's takeaway was that a stream of payments has a price, and that price is often not the total of the payments.
Watch out
Common mistakes.
- Comparing a lump sum with the undiscounted total of instalments. Money later is worth less than money now; discount the stream.
- Using the annual rate with monthly payments, or the reverse. The rate and the period must match.
- Confusing an ordinary annuity with an annuity due. Payments at the start of each period are worth more than payments at the end.
Questions
People also ask.
What is the difference between an annuity and a perpetuity?
An annuity has a fixed number of payments; a perpetuity continues indefinitely, and its present value is the payment divided by the rate.
Is a retirement annuity a good idea?
It suits people who value guaranteed income for life over flexibility and inheritance. The right answer depends on health, other income, attitude to risk and the rates on offer.
How do I calculate a loan repayment?
Divide the amount borrowed by the present value annuity factor for the loan's rate per period and number of periods, or use a spreadsheet payment function.
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