Actuarial Mathematics for Modelling · Equation of value

Annuities and Level Payment Cashflows: Formulas and Loan Schedules

Updated 11 October 2026 · Fact-checked

An annuity-certain is a series of payments made at fixed times. You value it by applying the equation of value: set the present value of payments equal to the amount paid or owed today. For level payments, use a(n) = (1 − vⁿ) ÷ i for payments at year end. Adjust for due, deferred, increasing or loan cases.

Understand Annuities and Level Payment Cashflows

An annuity-certain is a set of payments that will be made for sure, at regular intervals, for a fixed term. Rent, loan instalments and fixed coupons are examples. Because the payments are not linked to survival, you only need the interest rate and the timing.

The idea behind every formula is the equation of value: the present value of money in equals the present value of money out, at the same date and the same rate. An annuity formula is just a shortcut for adding up discounted payments. With v = 1 ÷ (1 + i), the payments of 1 at times 1, 2, ..., n have present value v + v² + ... + vⁿ. That is a geometric series, and it sums to (1 − vⁿ) ÷ i.

The timing decides which formula you use. Payments at the end of each period make an annuity-immediate, written a(n) (the symbol is a with a n-angle). Payments at the start of each period make an annuity-due, written ä(n). The due annuity is one period earlier, so every payment is worth (1 + i) times as much. A perpetuity never ends, so vⁿ goes to zero. A deferred annuity starts later, so you discount the whole annuity back by the deferment period.

In an increasing annuity, payments rise by a fixed amount, for example 1, 2, 3, ..., n. A loan is an annuity seen from the lender's side. The loan amount equals the present value of the instalments at the loan rate. Each instalment pays the interest due on the outstanding balance first. The rest repays capital. The balance after any payment equals the present value of the remaining instalments.

Key rules to remember

Discount factor: v = 1 ÷ (1 + i); d = i ÷ (1 + i) = iv; i is the effective rate per payment period. Always match the rate period to the payment period.
Annuity-immediate (payments at end of each period): a(n) = (1 − vⁿ) ÷ i; accumulation s(n) = ((1 + i)ⁿ − 1) ÷ i; a(n) is the value one period before the first payment. s(n) is the value at the time of the last payment.
Annuity-due (payments at start of each period): ä(n) = (1 − vⁿ) ÷ d = (1 + i) × a(n); Value at the time of the first payment. Also ä(n) = 1 + a(n−1).
Perpetuities: a(∞) = 1 ÷ i; ä(∞) = 1 ÷ d; Valid for i > 0. The first is paid at year end, the second at the start.
Deferred annuity: Value now of a(n) deferred m periods = vᵐ × a(n) = a(m+n) − a(m); Payments run from time m+1 to m+n. Check the first payment date carefully.
Increasing annuity-immediate (payments 1, 2, ..., n): (Ia)(n) = (ä(n) − n × vⁿ) ÷ i; Payment of t at time t. Value is at time 0. For (Iä)(n) use (ä(n) − n vⁿ) ÷ d.
Decreasing annuity-immediate (payments n, n−1, ..., 1): (Da)(n) = (n − a(n)) ÷ i; Payment of n at time 1, down to 1 at time n.
Perpetuity increasing by 1 each period: (Ia)(∞) = 1 ÷ (i × d) = (1 + i) ÷ i²; Payments 1, 2, 3, ... at times 1, 2, 3, ...
Loan outstanding balance: Bₜ = X × a(n−t) at rate i, for level instalment X at year ends; Prospective method. Interest in instalment t+1 = i × Bₜ; capital repaid = X − i × Bₜ.
m-thly payments: a⁽ᵐ⁾(n) = (1 − vⁿ) ÷ i⁽ᵐ⁾, where i⁽ᵐ⁾ = m × ((1 + i)^(1/m) − 1); This is for total payment 1 a year paid in m instalments of 1/m each at the end of each m-th period.

How to solve Annuities and Level Payment Cashflows questions

Use this method for any annuity or loan question. It keeps the timing right, which is where most marks are lost.

1Write down the rate and its period. Convert to an effective rate per payment period if needed, for example (1 + i)^(1/12) − 1 for monthly payments.
2Draw a timeline. Mark each payment date, the payment amount and the valuation date.
3Identify the pattern: level, increasing or decreasing; immediate or due; finite, deferred or perpetual.
4Choose a comparison date. Usually pick the date that makes the formula fit, such as one period before the first payment.
5Write the equation of value: PV of inflows = PV of outflows. Put the unknown on one side.
6Use the standard formula, then shift the answer by (1 + i) or v powers if the valuation date is different from the formula date.
7For loans, find the instalment first. Then find the balance prospectively as the PV of remaining payments, or retrospectively. Split each payment into interest and capital.
8Check: the final balance should be zero, and the answer should be sensible in size.

Quickest way: Anchor on the formula date

When to use it: Use when payments are a standard pattern but start or end at an awkward time, as in deferred, due or mixed-date questions under time pressure.

Find the date at which your formula gives the value: one period before the first payment for a(n), the first payment date for ä(n).
Calculate the annuity value there.
Multiply by (1 + i) for each period you must move forward, or by v for each period you must move back.
For a loan, find X = L ÷ a(n). For a balance after t payments, use X × a(n−t) with no need to build the schedule.
For one row of a schedule, interest = i × opening balance and capital = X − interest. Do only the rows asked.

Common mistakes in Annuities and Level Payment Cashflows

Using the annual rate when payments are monthly or half-yearly.
The formula looks the same, so the rate period is overlooked.
Fix: Convert first. Use the effective rate per payment period, and set n as the number of payments.
Treating a deferred annuity as starting m periods after time 0 when it starts m+1 periods after.
Wording like 'deferred 5 years' is confused with the first payment date.
Fix: For payments in arrears, deferred m years means first payment at time m+1. The value is vᵐ a(n). Check with the timeline.
Using a(n) when payments are at the start of each period.
The words immediate and due are mixed up.
Fix: Ask whether the first payment is now or one period away. If now, use ä(n) = (1 + i) a(n).
Using the increasing annuity formula (Ia)(n) with a(n) in the numerator instead of ä(n).
The formula is recalled from memory and part is dropped.
Fix: Remember (Ia)(n) = (ä(n) − n vⁿ) ÷ i. Test with n = 1: (1 − v) ÷ i = v. Correct.
Calculating loan interest on the original loan every period.
Flat-rate thinking is carried over.
Fix: Interest is always i times the balance outstanding at the start of the period. Capital repaid is the instalment minus that interest.
Rounding v or a(n) early.
Intermediate results are rounded to save time.
Fix: Keep full calculator precision and round only the final answer, as the question requires.

Worked examples

Example 1

A loan of ₹5,00,000 is repaid by 5 level annual instalments, the first one year after the loan is made. The effective annual rate is 8%. Find the instalment and the capital repaid in the second instalment.

Show the solution

v = 1 ÷ 1.08 = 0.925926. v⁵ = 0.680583.
a(5) = (1 − 0.680583) ÷ 0.08 = 0.319417 ÷ 0.08 = 3.992710.
Instalment X = 5,00,000 ÷ 3.992710 = ₹1,25,228 (to the nearest rupee).
Interest in the first year = 0.08 × 5,00,000 = ₹40,000. Capital repaid = 1,25,228 − 40,000 = ₹85,228. Balance B1 = 5,00,000 − 85,228 = ₹4,14,772.
Interest in the second year = 0.08 × 4,14,772 = ₹33,182. Capital in the second instalment = 1,25,228 − 33,182 = ₹92,046.

Answer: Instalment ₹1,25,228; capital repaid in the second instalment ₹92,046.

Example 2

An annuity pays ₹10,000 at the end of each year for 10 years, with the first payment at the end of year 4. Find its present value at 6% effective per year.

Show the solution

Payments run from time 4 to time 13, so there are 10 payments and the annuity is deferred 3 years.
Value at time 3 (one period before the first payment) = 10,000 × a(10) at 6%.
v¹⁰ = 1.06⁻¹⁰ = 0.558395. a(10) = (1 − 0.558395) ÷ 0.06 = 7.360087.
Value at time 3 = ₹73,600.87.
Discount back 3 years: v³ = 1.06⁻³ = 0.839619. PV = 73,600.87 × 0.839619 = ₹61,797 (to the nearest rupee).

Answer: Present value ≈ ₹61,797.

Exam tips

Draw the timeline in every written question. Examiners give marks for correct timing even when the arithmetic slips.
In MCQs, test an answer with n = 1 or a simple limiting case. This quickly exposes wrong versions of the increasing annuity formula.
For loan schedules, state the instalment, then the interest and capital for the rows asked. Show the balance as a check. A loan balance can be found directly as X × a(n−t).
In computer-based Paper B, build the annuity as a column of cashflows and discount it. Compare with the closed-form formula to verify your work.
State your assumptions, such as payments in arrears and the effective rate per period.

Practice questions from Equation of value

Annuities and Level Payment Cashflows: frequently asked questions

What is the difference between an annuity-immediate and an annuity-due?

An annuity-immediate pays at the end of each period. An annuity-due pays at the start. The due value is (1 + i) times the immediate value, so ä(n) = (1 + i) a(n).

How do I value an increasing annuity?

For payments 1, 2, ..., n at the end of each year, use (Ia)(n) = (ä(n) − n vⁿ) ÷ i. For other patterns, split the payments into level pieces and add the values, or use the first-principles sum.

How do I find the outstanding balance on a loan?

Use the prospective method. The balance after t instalments is the instalment multiplied by the annuity factor for the remaining payments, at the loan rate. For level year-end instalments, Bₜ = X × a(n−t).

Does a perpetuity have a finite value?

Yes, when i > 0. A perpetuity-immediate of 1 a year is worth 1 ÷ i, since vⁿ tends to zero as n grows. A perpetuity-due is worth 1 ÷ d.