What it means
An annuity, in finance, is simply a series of equal payments made at regular intervals. Loan repayments, lease payments and pension income are common examples.
Each payment is worth less the further away it is, so valuing the stream means discounting each payment back to today. The PVIFA does that in a single step.
Instead of discounting each payment separately and adding them up, you take the factor for the chosen rate and number of periods and multiply by the payment. For example, a PVIFA of 2.4869 means that three annual payments of $1 are together worth $2.4869 today.
This is how loan amounts are worked out. If you know how much you can afford to pay each month, multiplying by the PVIFA tells you how much you can borrow.
Going the other way, dividing a loan amount by the PVIFA gives the regular payment, which is how repayment schedules are built. Businesses use the factor for lease-versus-buy decisions, bond valuations and project appraisals.
Whenever a decision involves a steady stream of equal cash flows, the factor speeds up the maths. It is also built into spreadsheet functions that calculate payments and present values.
The factor assumes the payments arrive at the end of each period, which is called an ordinary annuity. If payments arrive at the start of each period, as with many leases, the annuity is called an annuity due and the result must be multiplied by (1 + r).
Missing this adjustment makes the value too low. A nuance is that the factor only works if the payments are equal and the rate stays the same.
If payments grow each year or the rate changes, a different approach is needed. In those cases, analysts discount each cash flow separately or use a growing annuity formula.
In practice
Real-world examples.
Example
A company is considering a machine that will save $15,000 a year for five years. At a discount rate of 8%, the PVIFA is 3.9927. The present value of the savings is 15,000 x 3.9927 = about $59,891, which the company compares with the machine's price.
Example
A bank approves a borrower to repay $2,000 a month for 24 months at a monthly rate of 1%. The PVIFA is [1 - 1.01^-24] / 0.01 = 21.2434. The most the bank would lend is 2,000 x 21.2434 = about $42,487.
Example
A landlord receives rent of $30,000 a year on a ten-year lease. With a discount rate of 6%, the PVIFA is 7.3601. The present value of the lease income is 30,000 x 7.3601 = about $220,803.
Formula
Calculation
PVIFA = [1 - (1 + r) ^ -n] / r
Present value of an annuity = payment x PVIFA
Suppose a company receives $10,000 at the end of each year for three years and the discount rate is 10%. PVIFA = [1 - 1/1.331] / 0.10 = (1 - 0.7513) / 0.10 = 2.4869. Present value = 10,000 x 2.4869 = $24,869. Checking by discounting each payment: 9,091 + 8,264 + 7,513 = $24,868, which matches apart from rounding. The same factor works in reverse for a loan: a borrower who can pay $10,000 a year for three years at 10% can borrow about $24,869 today.Case study
Seen in the real world.
Fernhill Printing is an illustrative, fictional business choosing between buying a press for $130,000 and leasing it for $30,000 a year for five years, paid at the end of each year. The finance manager discounted the lease payments at 8%, the company's borrowing rate.
The PVIFA for five years at 8% is 3.9927. The present value of the lease is 30,000 x 3.9927 = about $119,781, which is below the purchase price of $130,000. On this simple comparison, leasing was cheaper by roughly $10,219 in present value terms.
She then noted that buying would give the company a machine with resale value and tax benefits, which the lease would not. The illustrative lesson was that the factor makes the cash comparison easy, but other factors still need to be weighed. Fernhill's owners also asked her to repeat the comparison at 6% and 10%, since a small change in the discount rate moves the answer, and she found that the lease stayed cheaper at both rates.
Watch out
Common mistakes.
- Using the annuity factor when payments are uneven, as the factor needs equal payments each period.
- Ignoring whether payments are made at the start or end of each period, which changes the answer by a factor of (1 + r).
- Mixing an annual rate with monthly payments, so the rate and the number of periods do not match.
Questions
People also ask.
What is the difference between PVIF and PVIFA?
PVIF discounts a single future sum, while PVIFA adds up the discount factors for a whole series of equal payments.
How is PVIFA used to find a loan payment?
Divide the loan amount by the PVIFA for the loan's rate and term, and the result is the regular repayment.
Does a higher interest rate raise or lower the PVIFA?
It lowers it, because each payment is discounted more heavily, so the stream is worth less today.
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