What it means
An annuity is a stream of equal payments made at regular intervals, such as monthly rent, yearly lease payments or loan instalments. To compare a stream like this with a lump sum today, you need to translate every future payment into today's value.
You do that by discounting, which means reducing each payment by an interest rate that reflects the cost of waiting and risk. The present value is lower than the simple total of the payments because later payments are discounted more.
Five payments of $1,000 add up to $5,000, but their present value at 5% is only about $4,329. The higher the discount rate and the longer the stream, the bigger the gap.
This idea underlies many everyday business decisions. It is used to value leases, price bonds, compare a lump-sum settlement with instalments, decide whether to buy or lease equipment, and work out how much a borrower can afford.
Anyone who has taken a loan has been the subject of a present value calculation. There are two main variants.
An ordinary annuity pays at the end of each period, while an annuity due pays at the start, which makes the present value slightly higher because every payment is received one period sooner. The difference is a factor of one plus the discount rate per period.
The choice of discount rate is the most sensitive input. A rate that is too low overstates value and one that is too high understates it, so many companies use their cost of capital or the rate on a similar-risk loan.
Always check that the payment frequency and the rate match, for instance using a monthly rate for monthly payments. Spreadsheets and calculators make the arithmetic simple, but the thinking still needs care.
Write down the payment, the number of periods, the rate per period and the timing before you start, and check that the answer is smaller than the total of the payments. That quick sense check catches most errors in practice.
In practice
Real-world examples.
Example
A company is offered either $120,000 today or $30,000 a year for five years from a legal settlement. At a 6% discount rate, the five payments are worth about $126,371 today. The company accepts the instalments if it trusts the payer.
Example
A retailer signs a lease requiring $5,000 a month for 36 months. The accountant discounts the payments at the retailer's borrowing rate to record the lease liability. The result is smaller than the $180,000 total of payments.
Example
A retiree compares a pension paying $2,000 a month for 20 years with a lump sum offered today. The adviser calculates the present value of the monthly stream at a realistic rate. The client then sees whether the lump sum is fair. If the stream is worth much more than the offer, the adviser suggests negotiating.
Formula
Calculation
PV of ordinary annuity = payment x [1 - (1 + r)^-n] / r, where r is the rate per period and n is the number of periods.
A buyer will receive $10,000 at the end of each year for 3 years and uses a discount rate of 10%. The factor is [1 - (1.10)^-3] / 0.10 = [1 - 0.751315] / 0.10 = 2.48685. PV = $10,000 x 2.48685 = $24,868.50, or about $24,869 when rounded. Checking each payment separately: $10,000 / 1.10 = $9,090.91, $10,000 / 1.21 = $8,264.46 and $10,000 / 1.331 = $7,513.15, which total about $24,869.Case study
Seen in the real world.
Redstone Packaging is a fictional manufacturer used here as an illustration. It had to choose between buying a machine for $90,000 and leasing it for $2,200 a month for four years.
The finance manager calculated the present value of the lease payments using a monthly rate of 0.5%. With 48 payments, the factor was [1 - (1.005)^-48] / 0.005 = 42.58, so the present value was $2,200 x 42.58 = $93,676.
Because $93,676 exceeded the $90,000 purchase price, the illustrative company chose to buy. The lease looked cheaper per month but cost more in today's money.
Watch out
Common mistakes.
- Adding up the payments and calling it the value. That ignores the time value of money and overstates what the stream is worth.
- Using an annual rate with monthly payments. The rate and the period length must match.
- Mixing up ordinary annuity and annuity due. Payments at the start of each period produce a higher present value.
Questions
People also ask.
What discount rate should I use?
Use the rate that reflects the risk and cost of funds, such as your borrowing cost or required return.
Does a higher discount rate raise or lower present value?
It lowers it, because future payments are discounted more heavily.
Can I do this in a spreadsheet?
Yes, the PV function takes the rate, number of periods and payment and returns the present value, shown as a negative number by convention.
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