To calculate an annuity factor, divide 1 minus (1 + i)−n by i, where i is the interest rate per payment period and n is the total number of periods. The result is a single multiplier that converts a stream of equal future payments into a present-value lump sum. Multiply the factor by the payment amount, and you have what the whole stream is worth today. That’s the math. The rest is knowing which rate to use, which timing convention applies, and when the IRS requires you to skip the formula and read a factor off a published table instead.
The Three Inputs
Every annuity factor calculation starts with the same three numbers.
The first is the discount rate. For private planning, this is usually an expected rate of return or an inflation assumption. For a tax return, federal law requires the Section 7520 rate, which equals 120 percent of the applicable federal midterm rate, compounded annually, rounded to the nearest two-tenths of a percent.1Office of the Law Revision Counsel. 26 USC 7520 – Valuation Tables The IRS publishes it monthly.2Internal Revenue Service. Section 7520 Interest Rates
The second is the total number of payment periods. Multiply years by payment frequency: ten years of monthly payments is 120 periods; twenty years of quarterly payments is 80.
The third is the periodic interest rate, which you get by dividing the annual rate by the number of payments per year. A 6 percent annual rate with monthly payments becomes 0.005 per period. This is the step most people fumble, and because every later step compounds from it, one small error at the front becomes a large error at the end.
The Ordinary Annuity Factor
An ordinary annuity pays at the end of each period. Structured settlements and most loan schedules work this way. Call the periodic rate i and the number of periods n. The formula runs in three moves:
- Raise (1 + i) to the negative n power. This is the discount factor for a single dollar received at the end of the stream.
- Subtract that number from 1. This captures the total discount applied across all periods.
- Divide the result by i. That quotient is the annuity factor, sometimes called the present value interest factor of an annuity.
Worked Example
Consider a settlement that pays $1,000 per month for five years, discounted at 6 percent annually.
- Periodic rate: 0.06 ÷ 12 = 0.005
- Number of periods: 5 × 12 = 60
- Step 1: (1.005)−60 = 0.74137
- Step 2: 1 − 0.74137 = 0.25863
- Step 3: 0.25863 ÷ 0.005 = 51.7256
The annuity factor is 51.7256. Multiply by $1,000 and the present value is $51,725.60. Carry at least six decimal places in the intermediate steps. Rounding early on a long payment stream can move the final present value by hundreds of dollars.
Annuity Due: Payments at the Start of Each Period
When payments arrive at the beginning of each period rather than the end, the arrangement is called an annuity due. Lease payments, some retirement distributions, and insurance premiums usually work this way. Because each payment lands one period earlier, the present value is slightly higher.
The adjustment is one multiplication. Take the ordinary annuity factor and multiply by (1 + i). Using the example above, 51.7256 × 1.005 = 51.9842, and the present value rises from $51,725.60 to $51,984.20. Miss this step when payments genuinely begin on day one and you undervalue the obligation.
Growing and Deferred Payment Streams
Not every stream is flat. When payments escalate at a fixed rate each period, the growing annuity factor replaces the standard discount with the spread between the discount rate and the growth rate:
Growing annuity factor = [1 − ((1 + g) / (1 + i))n] / (i − g)
Here g is the periodic growth rate. When g equals i exactly, the formula breaks because the denominator is zero, and the factor simplifies to n / (1 + i). In practice the two rates rarely match, but it’s worth knowing the fallback.
A deferred annuity waits some number of periods before payments begin. Compute the ordinary annuity factor as if payments started immediately, then discount that whole factor back over the deferral period. If payments start in 5 years and last 15 years at 5 percent annually, calculate the 15-year factor and multiply by 1/(1.05)5. The deferral discount can cut the present value substantially, which is why a deferred annuity generally costs less upfront than an immediate one.
Doing It in a Spreadsheet
Excel and Google Sheets have a PV function that runs the whole calculation. The syntax is PV(rate, nper, pmt, [fv], [type]).3Microsoft. PV Function
- rate: the periodic rate
- nper: total number of periods
- pmt: payment per period, entered as a negative for cash outflows
- fv: future value, usually left blank or 0
- type: 0 for an ordinary annuity, 1 for an annuity due
To get the annuity factor by itself rather than a dollar value, set the payment to 1. Entering =PV(0.005, 60, -1, 0, 0) returns 51.7256 for the worked example above. That factor multiplies against any payment amount you want, and building comparison tables across different rates and terms is a matter of copying the formula.
When the IRS Requires Its Own Tables
For valuing annuities, life estates, remainders, and reversionary interests on a federal tax return, you don’t run the formula. Federal regulations require the IRS actuarial tables.4eCFR. 26 CFR 20.2031-7 – Valuation of Annuities, Interests for Life or Term of Years, and Remainder or Reversionary Interests The tables currently in force apply to valuation dates on or after June 1, 2023.5Internal Revenue Service. Actuarial Tables
Table B for Fixed-Term Annuities
Table B lists annual annuity factors organized by interest rate and remaining years. Find the row for your Section 7520 rate, the column for your term, and read the factor. Multiply by the annual payment. If someone will receive $12,000 per year for 15 years at a 4.6 percent Section 7520 rate, the Table B factor at that intersection times $12,000 is the present value.5Internal Revenue Service. Actuarial Tables
Table S for Life Annuities
Table S applies when the annuity runs for someone’s lifetime rather than a set term. Factors depend on the annuitant’s age at the valuation date and the applicable Section 7520 rate, and they incorporate mortality assumptions. A 50-year-old’s factor will be materially higher than an 80-year-old’s at the same interest rate, because the expected payout runs longer.5Internal Revenue Service. Actuarial Tables
Table K for Non-Annual Payments
Tables B and S assume annual payments. When payments come monthly, quarterly, or semiannually, multiply the Table B or Table S factor by the matching Table K adjustment for that frequency and interest rate. Skip this step and the present value will be off in one direction or the other, depending on whether payments are more or less frequent than yearly.
The Two-Month Election for Charitable Transfers
For a tax return, use the Section 7520 rate for the month of the transfer. If the transfer qualifies for a charitable deduction, you can elect to use the rate from either of the two months immediately preceding the transfer month instead.6eCFR. 26 CFR 20.7520-2 – Valuation of Charitable Interests A higher Section 7520 rate lowers the value of the annuity interest and raises the remainder. For a charitable remainder trust, where the deduction rides on the remainder, the highest of the three available rates produces the largest deduction. Over a long payout, two-tenths of a percent can shift the deduction by thousands.
What a Wrong Factor Costs
Valuation penalties escalate with the size of the error. A claimed value that reaches 150 percent or more of the correct amount is a substantial valuation misstatement, drawing a 20 percent penalty on the resulting tax underpayment. At 200 percent or more, the penalty doubles to 40 percent.7Office of the Law Revision Counsel. 26 USC 6662 – Imposition of Accuracy-Related Penalty on Underpayments The penalty attaches to the tax underpayment, not the misvalued amount, so a $50,000 valuation error that produces $15,000 of additional tax carries a $3,000 or $6,000 penalty depending on which tier applies.
Using the correct Section 7520 rate for the valuation month and the current IRS tables makes the factor defensible on its face. For anything the standard tables don’t cover, such as unusual payment schedules, multiple lives, or non-standard growth patterns, an actuary using IRS-approved methodology is the safer route.