Actuarial Mathematics for Modelling · Term structure of interest rates

Duration, Convexity and Redington Immunisation Explained

Updated 11 October 2026 · Fact-checked

Duration measures how sensitive a cash flow's present value is to interest rates. Macaulay duration is the present-value-weighted mean term, Σ t·v^t·c_t ÷ Σ v^t·c_t. Convexity measures the curvature of that relationship. Redington immunisation matches present values and durations, and needs assets with higher convexity, so surplus stays non-negative after small yield changes.

Understand Duration, Convexity and Immunisation

When interest rates change, the present value of a set of cash flows changes. Cash flows far in the future lose or gain more value than near ones. You need a single number that says how exposed a set of cash flows is. That number is duration.

Macaulay duration (also called the discounted mean term) is the average time of the payments, where each time t is weighted by the present value of the payment made then. A zero-coupon bond has duration equal to its term. A coupon bond has a duration shorter than its term, because the coupons come earlier.

Volatility (modified duration) turns this into a price sensitivity. It tells you the proportional fall in value for a small rise in the interest rate. Duration is a first-order measure, so it is only accurate for small changes. The value-versus-interest-rate curve is bent, not straight. Convexity measures that bend. It improves the estimate for larger changes.

Immunisation uses these ideas to protect an insurer or pension fund. Suppose assets and liabilities are both sensitive to interest rates. You choose assets so that a small change in interest rates cannot reduce the surplus. Redington's conditions say how: equal present values, equal durations, and asset convexity greater than liability convexity. The first two make the surplus flat at the current rate. The third makes the surplus curve bend upwards, so any small move raises it.

Redington's theory assumes the same small change in the interest rate applies to all terms (a flat yield curve and a parallel shift), that it happens immediately, and that the cash flows are fixed. You must be able to state these assumptions and the limits of the method.

Key rules to remember

Present value at effective annual rate i: V(i) = Σ c_t · v^t, where v = 1 ÷ (1 + i); c_t is the cash flow at time t. This is the value you differentiate.
Macaulay duration (discounted mean term): D = Σ t · c_t · v^t ÷ Σ c_t · v^t; Measured in years. For a zero-coupon bond, D equals its term.
Volatility (modified duration): ν = −V′(i) ÷ V(i) = D ÷ (1 + i); Use (1 + i) for an effective annual rate. If the rate is a force of interest δ, then −V′(δ) ÷ V(δ) = D exactly.
Convexity: C = V″(i) ÷ V(i) = Σ t(t + 1) · c_t · v^(t+2) ÷ Σ c_t · v^t; With respect to i. With respect to δ, it is Σ t² · c_t · v^t ÷ V. State which version you use.
Change in value for a small change Δi: ΔV ÷ V ≈ −ν · Δi + ½ · C · Δi²; Using duration alone gives the first term only. Convexity corrects the estimate.
Redington condition (i): V_A(i₀) = V_L(i₀); Present value of assets equals present value of liabilities at the current rate i₀.
Redington condition (ii): V_A′(i₀) = V_L′(i₀); Given (i), this means the discounted mean terms of assets and liabilities are equal.
Redington condition (iii): V_A″(i₀) > V_L″(i₀); Asset convexity exceeds liability convexity. This is a strict inequality and guarantees surplus does not fall for small changes in i.

How to solve Duration, Convexity and Immunisation questions

Use this method for any question on duration, convexity or immunisation. Set out a table so that every figure can be checked.

1Write down the cash flows and their times. State the interest rate and whether it is effective annual or a force of interest.
2Calculate v^t for each time and the present value of each cash flow. Sum these to get V.
3Add a column for t × PV and sum it. Divide by V to get the Macaulay duration D.
4Divide D by (1 + i) to get volatility if asked. Use it to estimate the percentage change in value for a small Δi, with the sign negative for a rise in i.
5For convexity, add a column for t(t + 1) × PV (or t² × PV for the δ version). Sum and divide by V, then add the ½ · C · Δi² term if the question needs more accuracy.
6For an immunisation question, test or impose the three Redington conditions in order: equal present values, equal durations, then asset convexity greater than liability convexity.
7If you are designing the assets, set unknown amounts using the first two conditions, then check the third using convexity figures.
8State the assumptions and conclusion in words: immunised against small, immediate, parallel changes in the rate.

Quickest way: Table method with mean-term shortcut

When to use it: Use when cash flows are few (up to about four or five) and the question asks for D, volatility or a Redington check.

Make one table with columns t, cash flow, PV, t × PV, t² × PV.
Fill PV first, then multiply by t and t² in the next columns. This avoids repeated discounting.
Compute D = Σ(t × PV) ÷ ΣPV and convexity (δ basis) = Σ(t² × PV) ÷ ΣPV.
For Redington with two asset payments around a single liability, put proportions w and (1 − w) of the PV at the two times and solve w·t₁ + (1 − w)·t₂ = liability mean term.
Check convexity: spread-out asset payments around the liability date always give larger Σ t² × PV than the liability, so condition (iii) holds. Show the numbers anyway for marks.

Common mistakes in Duration, Convexity and Immunisation

Forgetting to divide by (1 + i) when asked for volatility or modified duration.
Students stop at Macaulay duration because it is the number they just computed.
Fix: Read the question for the word volatility or modified. Then compute ν = D ÷ (1 + i) and state it separately.
Weighting by cash flow instead of present value when computing duration.
It feels natural to use the payments as weights because they are the numbers given.
Fix: Always discount first. Duration is Σ t × PV ÷ Σ PV. The weights must sum to 1 after dividing by V.
Equating durations of assets and liabilities without first checking that present values are equal.
Students remember the duration condition and treat it as the whole of Redington.
Fix: Check condition (i) first. Equal D is the same as V_A′ = V_L′ only when V_A = V_L.
Writing condition (iii) as equality or reversing it, so that liability convexity exceeds asset convexity.
Equality in (i) and (ii) leads students to expect equality in (iii).
Fix: Remember the surplus must curve upwards. Assets need the larger convexity, strictly. Check the direction against a spread-out asset portfolio and a single liability.
Claiming immunisation works for any change in interest rates, or that it is permanent.
The word immunised sounds like complete protection.
Fix: State that it holds for small immediate changes, in a flat yield curve with a parallel shift. As time passes the durations drift, so you must rebalance.
Mixing convexity formulas, using t(t + 1) with a force-of-interest answer or t² with an i-based approximation.
Both versions appear in notes and look similar.
Fix: Write down whether you differentiate with respect to i or δ, then use the matching formula consistently.

Worked examples

Example 1

A 3-year bond pays an annual coupon of 5 per 100 nominal, and is redeemed at par at the end of year 3. The yield is 6% per annum effective. Calculate (a) the price, (b) the Macaulay duration, (c) the volatility, and (d) the approximate percentage change in price if the yield rises to 6.5%.

Show the solution

Cash flows are 5 at t = 1, 5 at t = 2, and 105 at t = 3. Here v = 1 ÷ 1.06 = 0.943396, v² = 0.889996 and v³ = 0.839619.
Present values: 5 × 0.943396 = 4.7170; 5 × 0.889996 = 4.4500; 105 × 0.839619 = 88.1600.
Price V = 4.7170 + 4.4500 + 88.1600 = 97.327.
t × PV: 1 × 4.7170 = 4.7170; 2 × 4.4500 = 8.9000; 3 × 88.1600 = 264.4800. Sum = 278.097.
Macaulay duration D = 278.097 ÷ 97.327 = 2.857 years.
Volatility ν = 2.857 ÷ 1.06 = 2.696.
A rise of Δi = 0.005 gives ΔV ÷ V ≈ −2.696 × 0.005 = −0.01348, which is a fall of about 1.35%.

Answer: (a) Price ≈ 97.33. (b) Macaulay duration ≈ 2.857 years. (c) Volatility ≈ 2.696. (d) Price falls by approximately 1.35% (to about 96.0).

Example 2

An insurer must pay ₹2,00,000 in 10 years. The interest rate is 5% per annum effective. It will hold two zero-coupon bonds, one maturing at time 5 and one at time 15. Find the maturity amounts that satisfy Redington's first two conditions and show that the third condition holds.

Show the solution

Liability present value: 2,00,000 × 1.05⁻¹⁰ = 2,00,000 × 0.613913 = ₹1,22,782.65.
Condition (i): total asset PV must equal ₹1,22,782.65. Let a fraction w of the PV be in the time-5 bond and (1 − w) in the time-15 bond.
Condition (ii): the asset mean term must equal the liability mean term, which is 10 (a single payment at time 10). So 5w + 15(1 − w) = 10, which gives 15 − 10w = 10 and w = 0.5.
Each bond therefore has present value ₹61,391.33.
Maturity amounts: time-5 bond = 61,391.33 × 1.05⁵ = 1,00,000 × 1.05⁻⁵ = ₹78,352.6. Time-15 bond = 61,391.33 × 1.05¹⁵ = 1,00,000 × 1.05⁵ = ₹1,27,628.2.
Condition (iii), using the δ basis: let P = 1,22,782.65. Asset Σ t² × PV ÷ P = 0.5 × 25 + 0.5 × 225 = 125. Liability = 10² = 100.
Since 125 > 100, asset convexity exceeds liability convexity.

Answer: Buy about ₹78,352.6 maturing at time 5 and about ₹1,27,628.2 maturing at time 15. Present values and mean terms match, and asset convexity (125) exceeds liability convexity (100), so all three Redington conditions hold. The fund is immunised against small, immediate, parallel changes in the interest rate.

Exam tips

Show the table. Even if you slip on arithmetic, the method marks are available only when your working is visible.
Always state the three Redington conditions in words and symbols, then check each one explicitly. Examiners award a mark for each condition.
In immunisation design questions, solve the PV equation and the mean-term equation as two simultaneous equations, then check convexity at the end.
Add a line on assumptions and limits: small changes, parallel shift, immediate change, rebalancing needed as time passes. These are frequent short-answer marks.
In Paper B (computer-based), build the t, PV, t × PV and t² × PV columns in Excel or R with a clear formula so you can change the interest rate and re-run quickly.

Practice questions from Term structure of interest rates

Duration, Convexity and Immunisation in other exams

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

FRM Part IDuration and Convexity

Duration, Convexity and Immunisation: frequently asked questions

What is the difference between duration and convexity?

Duration is the first-order sensitivity of present value to interest rates, a measure of the slope. Convexity is the second-order measure, the curvature of the value curve. Duration gives a straight-line estimate of the change in value and convexity corrects it for larger changes.

How do I calculate Macaulay duration?

Discount each cash flow, multiply each present value by its time t, add these up, and divide by the total present value. This is the discounted mean term. For a zero-coupon bond the answer is just the term.

What are Redington's conditions for immunisation?

The present value of assets equals that of liabilities at the current rate. The first derivatives with respect to interest are equal, which means equal discounted mean terms. The second derivative of assets exceeds that of liabilities. Together they ensure surplus does not fall for a small immediate change in the rate.

Why must asset convexity be greater than liability convexity?

With conditions (i) and (ii) the surplus has zero value and zero slope at the current rate. Its shape is then decided by the second derivative. If assets are more convex, the surplus curve bends upward, so any small rate change leaves surplus positive.

Is volatility the same as modified duration?

In the IAI treatment, yes. Volatility is −V′(i) ÷ V(i), which equals Macaulay duration divided by (1 + i) for an effective annual rate. Read the question to confirm which interest rate basis is used.