Actuarial Mathematics for Modelling · Annuity and accumulation functions

Perpetuities, Deferred and m-thly Annuities: Formulas and Valuation

Updated 11 October 2026 · Fact-checked

A perpetuity pays forever, so its present value is 1/i for payments at year end and 1/d for payments at the start. A deferred annuity is valued as v^m × a_n, discounting the annuity back over the deferment. An m-thly annuity of 1 per year uses i^(m) or d^(m) in the denominator instead of i or d.

Understand Perpetuities, Deferred and m-thly Annuities

Start with the basic annuity-immediate, a_n = (1 − v^n) ÷ i. It values 1 paid at the end of each year for n years. Every topic on this page is a small change to that idea.

A perpetuity has no end date. Let n grow large and v^n goes to zero (for i > 0). So a_∞ = 1/i. Think of it this way: if you hold 1/i in a bank earning i per year, you can withdraw the interest of 1 each year and the capital stays intact. For payments at the start of each year, ä_∞ = 1/d. This equals 1/i + 1, because the first payment is paid now and the rest is a perpetuity-immediate.

A deferred annuity starts later. If an annuity of n payments has its first payment at time m + 1, it is deferred m years. Value the annuity as if it started at time m, then discount that value back m years. That gives m| a_n = v^m a_n. You can also write it as a_(m+n) − a_m, because you take all payments up to time m + n and remove the first m.

An m-thly annuity pays more often than once a year, for example monthly (m = 12). The convention is that the total paid per year is 1, so each payment is 1/m. The effective rate per year is still i. We use the nominal rate i^(m) = m[(1+i)^(1/m) − 1] and the nominal discount rate d^(m) = m[1 − v^(1/m)]. Then a_n^(m) = (1 − v^n) ÷ i^(m) for payments at the end of each m-th of a year, and ä_n^(m) = (1 − v^n) ÷ d^(m) for payments at the start.

A continuous annuity is the limit as m goes to infinity. Payments flow at rate 1 per year, and i^(m) and d^(m) both tend to δ, the force of interest. So ā_n = (1 − v^n) ÷ δ. Always keep the numerator 1 − v^n the same and change only the denominator to match the payment timing.

Key rules to remember

Perpetuity-immediate: a_∞ = 1/i; 1 paid at the end of each year forever. Needs i > 0.
Perpetuity-due: ä_∞ = 1/d = 1/i + 1; First payment at time 0. d = i ÷ (1 + i) = iv.
Deferred annuity-immediate: m| a_n = v^m a_n = a_(m+n) − a_m; Deferred m years. First payment at time m + 1, last at time m + n.
Deferred annuity-due: m| ä_n = v^m ä_n = ä_(m+n) − ä_m; First payment at time m. Last at time m + n − 1.
m-thly annuity-immediate: a_n^(m) = (1 − v^n) ÷ i^(m) = (i ÷ i^(m)) a_n; Payments of 1/m at the end of each m-th of a year. Total 1 per year.
m-thly annuity-due: ä_n^(m) = (1 − v^n) ÷ d^(m) = (d ÷ d^(m)) ä_n; Payments of 1/m at the start of each m-th of a year.
Nominal rates: i^(m) = m[(1+i)^(1/m) − 1] ; d^(m) = m[1 − v^(1/m)]; Both are convertible m-thly. 1 + i = (1 + i^(m)/m)^m = (1 − d^(m)/m)^(−m).
Link between m-thly due and immediate: ä_n^(m) = a_n^(m) + (1 − v^n)/m; The due annuity has an extra payment of 1/m at time 0 and one fewer at time n.
m-thly perpetuities: a_∞^(m) = 1/i^(m) ; ä_∞^(m) = 1/d^(m); Set v^n = 0 in the m-thly formulas.
Continuous annuity: ā_n = (1 − v^n) ÷ δ = (i ÷ δ) a_n ; ā_∞ = 1/δ; Payment rate 1 per year. δ = ln(1 + i).
Continuous accumulation: s̄_n = ((1+i)^n − 1) ÷ δ = (1+i)^n ā_n; Accumulated value at time n of a continuous payment stream.

How to solve Perpetuities, Deferred and m-thly Annuities questions

Use the same routine for any question on perpetuities, deferred annuities or m-thly annuities. Draw the timeline first. It stops most errors.

1Draw a timeline. Mark the valuation date, the time of the first payment and the time of the last payment (or write 'forever').
2Find the payment frequency and the size of each payment. If a question gives a payment per period, convert to total per year to use the m-thly factors, or work directly with the per-period rate.
3Decide whether payments are in advance (due) or in arrears (immediate), and whether they are continuous.
4Identify the rate you were given: effective i, nominal i^(m), nominal d^(m) or δ. Convert so you have i, then find i^(m), d^(m) or δ as needed.
5Value the annuity at the time just before its first payment (immediate) or at the time of its first payment (due). This is the valuation point of the standard formula.
6If the annuity is deferred, multiply by v^m, where m is the number of years from the valuation date to that point. Check the count with your timeline.
7Multiply by the annual payment amount and compute. Keep at least six decimal places in factors before the final rounding.
8Sense-check. A perpetuity must exceed any finite annuity with the same payments. A due annuity must exceed the matching immediate annuity.

Quickest way: Subtract and rescale

When to use it: Use when a question gives a standard annuity value (a_n) and asks for an m-thly, continuous or deferred value, or when you want to avoid recomputing v^n repeatedly.

Compute 1 − v^n once. It is the numerator of every non-deferred formula above.
Divide by the right denominator: i, d, i^(m), d^(m) or δ.
For a deferred annuity, use a_(m+n) − a_m if both factors are already available; otherwise use v^m a_n.
For a perpetuity, skip the numerator: use 1/i, 1/d, 1/i^(m) or 1/d^(m).
For a due m-thly annuity, if you have the immediate value, add (1 − v^n)/m instead of recomputing.

Common mistakes in Perpetuities, Deferred and m-thly Annuities

Using v^(m) instead of v^(m−1) or the wrong number of years for the deferment.
Students read 'deferred 5 years' and 'first payment at time 5' as the same thing.
Fix: Deferred m years for an annuity-immediate means the first payment is at time m + 1. Check with your timeline. Then discount by v^m.
Using 1/i for a perpetuity whose first payment is now.
The formula 1/i is memorised without its timing condition.
Fix: 1/i is for payments at the end of each year. If the first payment is at time 0, use 1/d, or 1/i + 1.
Using i instead of i^(m) in an m-thly annuity formula.
The numerator 1 − v^n looks the same, so the denominator is copied from the annual case.
Fix: For payments m times a year, divide 1 − v^n by i^(m) (immediate) or d^(m) (due). Compute i^(m) = m[(1+i)^(1/m) − 1] from the effective rate.
Treating a 'nominal rate of 12% pa convertible monthly' as an effective annual rate of 12%.
The word 'nominal' is skipped.
Fix: Divide by m to get the monthly rate (1%), then find the annual effective rate (1.01^12 − 1), or use i^(12) = 0.12 directly in the denominator.
Using the continuous formula with i instead of δ.
Students remember ā_n = (1 − v^n)/δ but then plug in i by habit.
Fix: Compute δ = ln(1 + i) first. Check that δ < i for i > 0, which is a quick sanity test.
Paying 1 per m-th of a year instead of 1/m.
The question says 'monthly payment of ₹1,000' and the student uses the factor for 1 per month without adjusting.
Fix: The standard factor assumes total 1 per year. If each payment is ₹1,000 monthly, total per year is ₹12,000, so PV = 12,000 × a^(12). Or use the monthly effective rate with a standard annuity of 1,000 per month.

Worked examples

Example 1

A pension pays ₹10,000 at the end of each year for 15 years. The first payment is at the end of year 6. Find the present value today at 8% effective per year.

Show the solution

Timeline: first payment at time 6, last at time 20. This is an annuity-immediate of 15 payments, deferred 5 years.
PV = 10,000 × v^5 × a_15 at i = 8%.
v = 1/1.08. 1.08^15 = 3.172169, so v^15 = 0.315242.
a_15 = (1 − 0.315242) ÷ 0.08 = 8.559479.
1.08^5 = 1.469328, so v^5 = 0.680583.
v^5 a_15 = 0.680583 × 8.559479 = 5.825437.
Check: a_20 − a_5 = 9.818147 − 3.992710 = 5.825437. This matches.
PV = 10,000 × 5.825437 = 58,254.37.

Answer: About ₹58,254.

Example 2

An annuity pays ₹1,000 at the start of each quarter for 10 years. Find its present value at 10% effective per year.

Show the solution

Total paid per year is 4 × 1,000 = ₹4,000. So PV = 4,000 × ä_10^(4).
The formula is ä_10^(4) = (1 − v^10) ÷ d^(4).
v^10 = 1.1^(−10) = 0.385543, so 1 − v^10 = 0.614457.
1.1^(1/4) = 1.024114, so v^(1/4) = 1/1.024114 = 0.976454.
d^(4) = 4 × (1 − 0.976454) = 0.094184.
ä_10^(4) = 0.614457 ÷ 0.094184 = 6.52403.
Check using i^(4) = 4 × 0.024114 = 0.096455. a_10^(4) = 0.614457 ÷ 0.096455 = 6.37042. Add (1 − v^10)/4 = 0.153614. Total 6.52403. This matches.
PV = 4,000 × 6.52403 = 26,096.1.

Answer: About ₹26,096.

Exam tips

In multiple-choice questions, check the timing of the first payment before calculating. Wrong options are often built from the right numbers with the wrong deferment or the wrong 1/i versus 1/d.
For written questions, define your notation. Write 'm| a_n = v^m a_n' or the equivalent and state the rate you use. Marks are given for method, even if arithmetic slips.
In the computer-based paper, build i^(m), d^(m) and δ from i with formulas in cells or R, rather than typing rounded values. Test your function on a known case, such as a perpetuity limit.
When you get a value, test it. Check that a due annuity exceeds the immediate annuity, that the m-thly value lies between the annual and continuous values, and that a deferred value is smaller than the non-deferred value.
Keep the final answer to the precision the question asks, but carry at least six decimals in intermediate factors.

Practice questions from Annuity and accumulation functions

Perpetuities, Deferred and m-thly Annuities in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Perpetuities, Deferred and m-thly Annuities: frequently asked questions

What is the present value formula for a perpetuity in actuarial notation?

For 1 paid at the end of each year forever, a_∞ = 1/i. For payments at the start of each year, ä_∞ = 1/d, where d = i/(1+i). Both need i > 0. For a payment of ₹5,000 at 8%, the immediate perpetuity is worth 5,000 ÷ 0.08 = ₹62,500, and the due perpetuity is worth ₹67,500.

How do I value a deferred annuity?

Value the annuity at the time just before its first payment (immediate) or at its first payment (due), then discount back to today by v^m, where m is the deferment. For an annuity-immediate this gives m| a_n = v^m a_n. Always check the timeline to count m correctly.

What does a_n^(m) mean and how is it different from a_n?

a_n^(m) is the present value of an annuity paying 1/m at the end of each m-th of a year for n years, so 1 per year in total. It equals (1 − v^n) ÷ i^(m). It is slightly larger than a_n because payments are received earlier on average.

What is the formula for the continuous annuity ā_n?

ā_n = (1 − v^n) ÷ δ, where δ = ln(1 + i). It equals (i/δ) a_n. It values a payment stream at rate 1 per year flowing continuously for n years.

Can I use the perpetuity formula if the interest rate is zero?

No. The formula 1/i needs i > 0. At i = 0 the perpetuity has infinite value because the payments never stop and nothing is discounted.