Actuarial Mathematics for Modelling · Allowing for inflation

Index-Linked Cash Flows and Annuities Explained

Updated 11 October 2026 · Fact-checked

An index-linked cash flow is a payment whose amount rises with a price index. To value it, project each payment using the index, then discount at the money rate. If inflation is constant at e and the money rate is i, you can instead discount the base payments at the real rate j, where 1 + j = (1 + i) ÷ (1 + e).

Understand Index-Linked Cash Flows and Annuities

A normal fixed payment loses buying power when prices rise. An index-linked payment fixes this. Its amount moves with an index such as a consumer price index, so its real value stays roughly constant.

The payment at time t is the base payment times an index ratio: payment = C × Q(t) ÷ Q(0), where Q is the index. Money cash flows are then discounted at the money rate i, the rate the market pays on ordinary fixed-interest money.

If inflation runs at a constant rate e per year, the index ratio is (1 + e)^t. Each payment is then C(1 + e)^t, discounted by (1 + i)^t. The two growth factors combine into one: (1 + e) ÷ (1 + i) = 1 ÷ (1 + j). That is why you can treat the cash flows as level and discount at the real rate j. A constant-real-value annuity is exactly this case.

Real indexed payments are rarely linked to the index on the payment date. Most are linked to the index a few months earlier. This is the indexation lag, and it exists because inflation figures are published late. With a lag L, the payment at time t depends on the index at time t − L. The lag changes which index values you use. It does not change the discounting, which still runs from the actual payment date.

Watch the base date too. If the base index is also taken L earlier than issue, the lag cancels under constant inflation. If the base is the index at issue, it does not cancel. Read the question to see which applies.

Key rules to remember

Index-linked payment: Payment at time t = C × Q(t − L) ÷ Q(base date); L is the indexation lag, which is 0 if there is no lag. Q is the price index. C is the payment in base-date money.
Real rate of interest: 1 + j = (1 + i) ÷ (1 + e), so j = (i − e) ÷ (1 + e); Valid for constant annual inflation e and money rate i. j ≈ i − e is only an approximation, so do not use it for exact answers.
Annuity-immediate rising with inflation, first payment 1 at time 1 unindexed: PV = (1 ÷ (1 + e)) × a_n at rate j, where payment at time t is (1 + e)^(t − 1); Use when the first payment is 1 and later payments rise by e a year.
Annuity where payment at time t is (1 + e)^t: PV = a_n at rate j; Use when the first payment already includes one year of indexation.
Lag with constant inflation, base at issue: PV = C × (1 + e)^(−L) × Σ [(1 + e) ÷ (1 + i)]^t; Payment at time t is C(1 + e)^(t − L), with L in years. The sum is a_n at rate j.
Annuity-immediate at rate j: a_n = (1 − (1 + j)^(−n)) ÷ j; Use the real rate j in place of i in the usual annuity formulas.

How to solve Index-Linked Cash Flows and Annuities questions

Use this method for any question on index-linked payments, annuities or bonds.

1Write down the payment dates, the base payment and the index used. Note whether payments are annual, monthly or continuous.
2Find the base date of the index and the indexation lag. Decide which index value drives each payment.
3Write the payment at time t as C × (index ratio). If inflation is constant, write it as C(1 + e)^(time gap).
4Check what the question gives: money rate i, real rate j, or an implied inflation rate. Convert using 1 + j = (1 + i) ÷ (1 + e).
5If payments grow at e from the first payment, factor out the constants and reduce the sum to an annuity at rate j. Adjust for any unindexed first payment or lag.
6If inflation is not constant, project each payment from the given index values and discount each at the money rate i.
7Evaluate with the annuity formula and keep at least five decimals in the factors.
8State the answer in rupees and check that it is sensible. The real-rate value should exceed the value of the same level payments discounted at i.

Quickest way: Real-rate shortcut

When to use it: Use it when inflation is constant, payments are indexed at every payment date, and the question gives or lets you find i and e.

Compute j = (1 + i) ÷ (1 + e) − 1 once.
Decide the first payment in index terms: is it 1 at time 1, or (1 + e) at time 1?
If it is (1 + e) at time 1, the PV is C × a_n at j.
If it is 1 at time 1, the PV is C × a_n at j ÷ (1 + e).
For a lag of L years with base at issue, multiply by (1 + e)^(−L).
Check with the first term only: C × (1 + e)^(t − 1) ÷ (1 + i)^t for t = 1.

Common mistakes in Index-Linked Cash Flows and Annuities

Using j = i − e as an exact real rate
The approximation is common in textbooks and is quick to type.
Fix: Use j = (1 + i) ÷ (1 + e) − 1 unless the question tells you to approximate.
Treating the first payment as indexed when it is not, or the reverse
Students memorise 'PV = a_n at j' without checking how the first payment is defined.
Fix: Write the payment at time 1 explicitly. If it is 1, divide a_n at j by (1 + e). If it is (1 + e), use a_n at j directly.
Discounting the payment at the index date instead of the payment date
With a lag, the index date and the payment date differ and get mixed up.
Fix: Use the index date only to find the amount. Always discount from the actual payment date.
Assuming the lag always cancels
Students remember that a lagged base index gives no lag effect.
Fix: Check the base date. If the base index is dated at issue and payments use index at t − L, include (1 + e)^(−L).
Mixing money and real cash flows in one calculation
Students project in real terms but discount at the money rate i, or the reverse.
Fix: Either use money cash flows with i, or real cash flows with j. Never mix them.
Using the wrong time units for the lag
A lag of 8 months is entered as 8 in the power.
Fix: Convert to years first: 8 months is 2/3 of a year.

Worked examples

Example 1

An annuity pays ₹10,000 at the end of the first year, and each later annual payment is 4% higher than the one before. There are 5 payments. Find the present value at a money rate of 7.12% a year.

Show the solution

Payment at time t is 10,000 × 1.04^(t − 1).
Real rate: 1 + j = 1.0712 ÷ 1.04 = 1.03, so j = 3%.
PV = 10,000 × Σ 1.04^(t − 1) ÷ 1.0712^t = 10,000 × (1 ÷ 1.04) × Σ (1.04 ÷ 1.0712)^t.
This is 10,000 × (1 ÷ 1.04) × a_5 at 3%.
1.03^5 = 1.159274, so v^5 = 0.862609.
a_5 = (1 − 0.862609) ÷ 0.03 = 4.57971.
Divide by 1.04: 4.57971 ÷ 1.04 = 4.40357.
PV = 10,000 × 4.40357 = 44,036 (to the nearest rupee).

Answer: ₹44,036 (approximately)

Example 2

A 3-year contract pays ₹1,000 at the end of each of years 1, 2 and 3, multiplied by the price index 8 months before the payment date divided by the index at time 0. Assume prices grow at 3% a year and the money rate is 8% a year. Find the present value at time 0.

Show the solution

The lag is 8 months = 2/3 of a year. The payment at time t is 1,000 × 1.03^(t − 2/3).
PV = 1,000 × 1.03^(−2/3) × Σ (1.03 ÷ 1.08)^t for t = 1, 2, 3.
1.03^(−2/3) = exp(−(2/3) × ln 1.03) = exp(−0.0197059) = 0.980487.
Let r = 1.03 ÷ 1.08 = 0.953704. Then r² = 0.909551 and r³ = 0.867442.
Σ r^t = 0.953704 + 0.909551 + 0.867442 = 2.730697. This equals a_3 at the real rate j = 1.08 ÷ 1.03 − 1 = 4.8544%.
PV = 1,000 × 0.980487 × 2.730697 = 2,677.41.

Answer: ₹2,677 (approximately)

Exam tips

Write the payment at time t in full on the first line. Marks are given for a correct cash flow even if the arithmetic slips later.
Check the first payment and the base date before choosing a formula. Examiners often vary these slightly from standard cases.
Keep at least five decimal places in discount factors. Small rounding errors grow when you multiply by 1,000 or more.
In the computer-based paper, build the money cash flows in a column, discount each at i, and compare with the real-rate shortcut as a check.
For written answers, state your assumptions: constant inflation, the index used, and the lag in years.

Practice questions from Allowing for inflation

Index-Linked Cash Flows and Annuities: frequently asked questions

How do I value an index-linked annuity using the real rate?

Find j from 1 + j = (1 + i) ÷ (1 + e). Then value the annuity as a level annuity at rate j, adjusting for whether the first payment is indexed. If the first payment is 1 at time 1 and later payments rise by e, divide a_n at j by (1 + e).

What is an indexation lag?

It is the gap between the date of the index value used and the date the payment is made. Index figures are published late, so a payment at time t often uses the index at t − L. The lag changes the amount of each payment, not the discounting date.

Does the indexation lag change the present value?

It can. If the base index is dated at issue and payments use the index at t − L, each payment is smaller by (1 + e)^(−L) under constant inflation. If the base index is lagged by the same amount, the effect cancels.

Is the real rate the same as money rate minus inflation?

Only approximately. The exact relationship is 1 + j = (1 + i) ÷ (1 + e). The approximation j ≈ i − e is close when both rates are small, but exam answers should use the exact formula.