What it means
Discounting ten annual payments one by one takes ten calculations and invites ten mistakes. The annuity factor collapses all of them into one number that depends only on the rate and the number of periods.
Once you have the factor, valuation becomes a single multiplication. The method is used heavily in capital investment decisions.
It converts the present value of an asset's whole life cost into an equivalent annual cost, which lets you compare options with different lifespans fairly, such as a machine lasting five years against one lasting eight. Without that conversion, the longer lived option always looks more expensive simply because it covers more years.
The same factor runs in both directions. Going forward, payment multiplied by factor gives present value, which is how lease and annuity pricing works.
Going backward, present value divided by factor gives the annual payment, which is how loan instalments and equivalent annual costs are set. Two cautions apply.
The factor assumes payments are equal and arrive at the end of each period, so uneven cash flows need discounting individually and payments in advance need the extra (1 + r) adjustment. The result is also only as good as the discount rate used, so it is worth testing the answer at a rate a couple of percentage points either side.
Published tables of annuity factors still exist, but a spreadsheet is now the usual tool, and the factor is easy to build from first principles in a single cell. Keeping the rate and the number of periods as separate inputs makes the model far easier to review, because a colleague can change one assumption and see the effect immediately.
It also makes errors visible, since an implausible factor stands out at a glance against the rough rule that a factor can never exceed the number of periods.
In practice
Real-world examples.
Example
A bakery chain compares two ovens, one costing $90,000 over six years and one costing $140,000 over ten years. Converting both to equivalent annual costs shows the larger oven is cheaper per year, which reverses the conclusion the owner drew from the purchase prices.
Example
A lender sets the monthly instalment on a $60,000 five year business loan by dividing the loan by the monthly annuity factor for 60 periods at the monthly rate. The same factor is then used to produce the amortisation schedule.
Example
A property investor values a lease that pays $48,000 a year for twelve years by multiplying the rent by the twelve year annuity factor at her 7% required return, arriving at a present value of about $381,000 before adjusting for the vacancy risk.
Formula
Calculation
Annuity factor = (1 - (1 + r) to the power of -n) / r, and equivalent annual amount = present value / annuity factor. Suppose a packaging line has a total life cycle cost with a present value of $200,000 over five years, and the company uses a 6% discount rate. Calculating the factor: 1.06 to the power of 5 is 1.3382, so 1 / 1.3382 = 0.7473, and (1 - 0.7473) / 0.06 = 0.2527 / 0.06 = 4.2124. The equivalent annual cost is $200,000 / 4.2124 = $47,479 a year. A rival machine with a present value cost of $260,000 over eight years at the same rate has a factor of 6.2098 and an equivalent annual cost of $41,870, so it is the cheaper choice per year of service despite the higher total.Case study
Seen in the real world.
Hollowbrook Dairies is an illustrative, fictional food processor choosing between two chilled storage systems. The first cost $320,000 and would last four years; the second cost $520,000 and would last nine years.
The operations director favoured the cheaper unit on cash grounds. The finance team applied the annuity factor method at the company's 8% discount rate and found equivalent annual costs of about $96,600 for the four year unit and about $83,300 for the nine year unit, a difference of roughly $13,300 a year.
In this illustrative case the board approved the more expensive system and funded it with a term loan, having satisfied itself that the annual cost of service, not the purchase price, was the comparable figure.
Watch out
Common mistakes.
- Using the annuity factor on cash flows that are not equal in every period, which quietly produces a wrong present value.
- Mixing an annual discount rate with monthly periods instead of converting the rate and the number of periods to the same basis.
- Comparing total life cycle costs for assets with different lifespans without converting them to equivalent annual amounts.
Questions
People also ask.
When should I not use the annuity factor?
Whenever the payments vary, or when a large one off cost or residual value sits outside the regular stream and must be discounted on its own.
Does the factor include inflation?
Only if the discount rate used is a nominal rate and the payments are in nominal terms; keep real with real and nominal with nominal.
How do I adjust for payments made in advance?
Multiply the result by one plus the periodic rate, which turns the ordinary annuity factor into an annuity due factor.
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