Annuity calculator

What is an annuity factor?

An annuity is a series of equal payments at equal intervals. Its annuity factor is the present value of 1 paid at the end of each period for n periods: AF = (1 - (1 + r)^-n) / r. Multiply the payment by the factor and you have the present value of the whole series in one step.

At 10% for 5 periods the factor is 3.790787, printed as 3.791 in an annuity table. So $1,000 a year for 5 years, first payment in one year, is worth $3,790.79 today, or $3,791 by the table method. You would receive $5,000 in all; the difference of $1,209.21 is the cost of waiting.

How does the annuity table link to the present value table?

The annuity table is a running total of the present value table. Each row adds one more discount factor. At 10%:

The right-hand column is exactly the 10% column of an annuity table. Adding the rounded factors on the left gives 1.735, 2.486, 3.169 and 3.790, each 0.001 below the column, because each factor was rounded before adding; the table rounds the exact running total instead. That is why the calculator prints both the factor to 3 decimals and the exact factor, and shows the step-by-step table under the result for your own inputs.

Annuity due versus ordinary annuity: what changes?

In an ordinary annuity the first payment comes at the end of period 1. In an annuity due it comes now, at period 0, as with rent or a lease paid in advance. The payment at period 0 is not discounted, so the factor is 1 + the annuity factor for n - 1 periods.

For five payments of $1,000 at 10%: 1 + 3.170 = 4.170, so the annuity due is worth $4,170 by the table method and $4,169.87 exactly. The same figure comes from multiplying the ordinary factor by 1.10: 3.790787 x 1.10 = 4.169865. Pick Now, period 0 under First payment to see the period 0 row in the step table.

How do you value a deferred annuity?

When the first payment comes later than period 1, take the annuity factor to the last payment and subtract the factor for the periods with no payment. Five payments of $1,000 at 10%, the first at the end of year 3, run from year 3 to year 7:

1. Annuity factor years 1 to 7: 4.868.
2. Less annuity factor years 1 to 2: 1.736.
3. Factor for years 3 to 7: 4.868 - 1.736 = 3.132.
4. Present value: $1,000 x 3.132 = $3,132 (exactly $3,132.88).

A second route gives the same answer: value the five payments one period before the first, $1,000 x 3.791 = $3,791 at the end of year 2, then discount two years with the 2-year discount factor, 0.826. That gives $3,131.37 by the table, a little different only because of rounding. The step table under the calculator shows zero cash flows in the deferred years, so you can see where the subtraction comes from.

How do you calculate the future value of an annuity?

Compound forward instead of discounting back: FV = payment x ((1 + r)^n - 1) / r for payments at the end of each period, multiplied by (1 + r) when they are paid at the start. $1,000 a year for 5 years at 10% grows to $6,105.10; paid at the start of each year, to $6,715.61.

Future value and present value are the same stream seen from two dates. The ordinary annuity's present value, $3,790.79, compounded for 5 years at 10% is $3,790.79 x 1.61051 = $6,105.11; the extra cent comes only from rounding the present value to whole cents first. The Future value tab shows the value each payment reaches by the end, so you can see the first payment doing most of the growing.

How do you find the payment, or the rate, from an annuity factor?

Payment. Divide the present value by the factor. $10,000 repaid in 5 equal yearly payments at 10% needs $10,000 / 3.791 = $2,637.83 by the table, or $2,637.97 exactly. The same division turns the present value of a cost into an equal yearly amount, for example to compare assets with different lives: $25,000 spread over 3 years at 10% is $25,000 / 2.487 = about $10,052 a year.

Rate. Work backward through the table. If $10,000 buys 5 payments of $2,500, the factor is 10,000 / 2,500 = 4.000. On the 5-period row, 4.100 sits under 7% and 3.993 under 8%, so the rate is about 7% + (4.100 - 4.000) / (4.100 - 3.993) x 1% = 7.93%. The exact rate is 7.931%. This is the same interpolation used to estimate an internal rate of return.

What is a perpetuity, and how does it relate to an annuity?

A perpetuity is an annuity that never ends. As n grows, (1 + r)^-n shrinks towards zero and the annuity factor closes in on 1 / r. At 10% the factor for 50 payments is 9.915, against a perpetuity factor of 10, which is why the chart under the calculator flattens out.

So a perpetuity of $1,000 a year at 10% is worth $1,000 / 0.10 = $10,000. The first 50 payments carry 99.1% of that value. If the first payment is today, add it: $11,000. If the first payment is at the end of year 3, the formula values the stream at the end of year 2, so discount two years: $10,000 x 0.826 = $8,260 by the table, $8,264.46 exactly.

Worked examples in pounds, dirhams and euros

• Pounds, present value. £5,000 a year for 8 years at 6%, first payment in a year. The 8-year factor at 6% is 6.210, so the value is £31,050 by the table method and £31,048.97 exactly. Paid in advance, the factor is 1 + 5.582 and the exact value is £32,911.91.
• UAE dirhams, payment. AED 500,000 repaid over 12 yearly payments at 7%. The factor is 7.943, so each payment is AED 62,948.51 by the table method and AED 62,950.99 exactly.
• Euros, future value. €3,000 at the start of each year for 20 years at 5% grows to €104,157.76. Paid at the end of each year it reaches €99,197.86.

How do you calculate an annuity in Excel or Google Sheets?

Excel's PV, FV and PMT functions take the rate per period, the number of periods, the payment and an optional type: 0 or omitted for the end of each period, 1 for the start. Money paid out is entered as negative.

• Ordinary annuity: =PV(10%, 5, -1000) returns 3,790.79.
• Annuity due: =PV(10%, 5, -1000, 0, 1) returns 4,169.87.
• Future value: =FV(10%, 5, -1000) returns 6,105.10.
• Payment: =PMT(10%, 5, -10000) returns 2,637.97.
• Deferred annuity: value it at the end of year 2, then discount: =PV(10%, 2, 0, -PV(10%, 5, -1000)) returns 3,132.88.

For monthly payments keep the units matched: the yearly rate divided by 12 and the number of months.

Common mistakes, what is left out, and a glossary

• Using n instead of n - 1 for an annuity due. The factor is 1 + AF(n - 1), not 1 + AF(n).
• Subtracting the wrong factor for a deferral. Payments from year 3 to 7 use AF(7) - AF(2), not AF(7) - AF(3).
• Mixing yearly rates with monthly payments. Rate and periods must use the same interval.
• Expecting the table answer to match the exact one. Factors to 3 decimals can differ by a few units in the last place.

The calculator assumes level payments, one rate for every period and one payment per period. It does not handle growing annuities, tax or changing rates. Glossary: discount factor, the present value of 1 received after t periods; cumulative discount factor, the sum of discount factors over the payment periods; annuity factor, the cumulative discount factor for periods 1 to n; deferred annuity, one whose first payment comes after period 1.

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