Actuarial Mathematics for Modelling · Duration, convexity and immunisation

Redington's Immunisation Conditions: The Three Rules Explained

Updated 11 October 2026 · Fact-checked

Redington's immunisation protects a portfolio against a small, immediate change in the interest rate. You need three conditions at the starting rate: the present value of assets equals that of liabilities, their discounted mean terms are equal, and the assets are more spread out in time, giving higher convexity. Solve the first two for amounts, then check the third.

Understand Redington's Immunisation Conditions

A life office or pension fund owes money in the future. It holds assets to pay those liabilities. If interest rates move, the present value of both assets and liabilities changes. The risk is that the liabilities rise in value more than the assets. Immunisation is a way of choosing assets so that this does not happen for small rate changes.

Write V_A(i) for the present value of assets and V_L(i) for the present value of liabilities, both at interest rate i. The surplus is V_A(i) − V_L(i). Redington wants this surplus to be zero at today's rate i₀, and to be at a local minimum there. If it has a local minimum of zero, any small move in the rate in either direction leaves a surplus of zero or more.

A minimum of a function needs two things: the first derivative is zero, and the second derivative is positive. That gives the three conditions. First, V_A(i₀) = V_L(i₀), so the surplus starts at zero. Second, V_A′(i₀) = V_L′(i₀), so the surplus has zero slope. Given the first condition, this is the same as saying the discounted mean terms (DMT) of assets and liabilities are equal. Third, V_A″(i₀) > V_L″(i₀), so the surplus curves upward. Given the first condition, this means asset convexity exceeds liability convexity.

In plain terms, the assets must be more spread out around the DMT than the liabilities. A common set-up is to match a single liability with two zero-coupon bonds, one maturing before the liability and one after. The spread of the two bonds gives the extra convexity.

Know the assumptions. The theory covers a small change in rate, happening once and straight away, and applying equally to all terms (a flat yield curve that shifts in parallel). Changes in the cash flows themselves, or a changing yield curve shape, are not covered. The portfolio also needs rebalancing as time passes, because the DMTs drift apart.

Key rules to remember

Condition 1: equal present values: V_A(i₀) = V_L(i₀); Assets and liabilities have the same present value at the current rate i₀.
Condition 2: equal slopes: V_A′(i₀) = V_L′(i₀); Derivatives taken with respect to the interest rate (or force of interest). With condition 1, this is the same as equal DMT.
Condition 3: convexity: V_A″(i₀) > V_L″(i₀); Strict inequality. With condition 1, this means asset convexity is greater than liability convexity.
Discounted mean term: DMT = Σ t · CF_t · vᵗ ÷ Σ CF_t · vᵗ; The present-value-weighted average time of the cash flows. Use v = 1 ÷ (1 + i₀).
Convexity: c = Σ t² · CF_t · vᵗ ÷ Σ CF_t · vᵗ; Equals V″(δ) ÷ V(δ) when differentiating with respect to the force of interest δ. The common divisor V means the comparison is valid once present values are equal.
Volatility (modified duration): ν = −V′(i) ÷ V(i) = DMT ÷ (1 + i); Equal DMT with equal present values gives equal volatility.
Two-bond matching of one liability: x + y = V_L and t₁x + t₂y = t_L(x + y); x and y are the present values spent on bonds maturing at t₁ and t₂. Then check that t₁ < t_L < t₂.

How to solve Redington's Immunisation Conditions questions

Use this method for any question that asks you to immunise, or to check whether a portfolio is immunised.

1Write down the interest rate i₀, v = 1 ÷ (1 + i₀), and list all asset and liability cash flows with their times.
2Calculate the present value of the liabilities. This is the target present value for the assets.
3Calculate the DMT of the liabilities, Σ t · CF · vᵗ ÷ Σ CF · vᵗ.
4If assets are unknown, let their present values be x and y (or similar). Set total present value equal to V_L and set the present-value-weighted mean term equal to the liability DMT. Solve the two equations.
5Convert present values back into the nominal amounts or units of bond by multiplying by (1 + i₀) raised to the bond's term. For coupon bonds, divide by the present value per unit.
6Check condition 3. Calculate Σ t² · PV for assets and for liabilities, divide each by its total present value, and compare. Assets must be strictly larger.
7State the conclusion in words: conditions 1, 2 and 3 hold, so the portfolio is immunised against small changes in the interest rate. Mention the assumptions if the question asks for limitations.

Quickest way: Two-bond matching for a single liability

When to use it: Use this when one liability payment at time t_L is matched with two zero-coupon bonds at times t₁ < t_L < t₂.

Find V_L = L × v^t_L.
The weights on the bonds follow from the mean-term equation. The proportion of present value in the later bond is (t_L − t₁) ÷ (t₂ − t₁), and in the earlier bond (t₂ − t_L) ÷ (t₂ − t₁).
Multiply each proportion by V_L to get x and y.
Convert to the nominal payment: x × (1 + i₀)^t₁ and y × (1 + i₀)^t₂.
Convexity check is automatic when t₁ < t_L < t₂ because the assets are spread either side of the DMT. Just say so, and confirm with t² numbers if marks are given for it.

Common mistakes in Redington's Immunisation Conditions

Writing the second condition as equal durations of the cash flows without equal present values.
Students remember 'match the term' and forget it is a present-value-weighted term, or that condition 1 must hold too.
Fix: Always state condition 1 first. DMT equality and equal slopes are equivalent only when present values are equal.
Putting the nominal amounts, not the present values, into the mean-term equation.
The unknowns look like the bond payments, so they go straight into the equation.
Fix: Let x and y be present values, solve, then convert back to payments by accumulating at i₀ for the bond's term.
Reversing the convexity inequality, or using a non-strict one.
Students think 'less risk means lower convexity'.
Fix: The surplus must curve upward, so V_A″ > V_L″ strictly. Assets need the larger convexity.
Using only bonds on one side of the liability date.
The two simple conditions can be solved with bonds both shorter than the liability, but the solution has a negative holding or fails condition 3.
Fix: Put one bond before and one after the liability term. Check that both holdings are positive.
Treating the portfolio as immunised for ever.
The conditions are verified once and then forgotten.
Fix: Say that DMTs change as time passes and rates change, so the portfolio needs regular rebalancing. The result holds only for small, immediate, parallel changes in yield.
Mixing effective rate and force of interest derivatives.
Both versions appear in notes, and the convexity formulas differ.
Fix: Pick one basis and stay with it. For a comparison with equal present values, using t² · PV sums relative to total PV (force of interest basis) is simplest.

Worked examples

Example 1

A fund must pay ₹5,00,000 in 10 years. The interest rate is 5% per year effective. It will buy two zero-coupon bonds, one maturing in 5 years and one in 15 years, to satisfy Redington's first two conditions. Find the nominal amount of each bond, and show that the third condition holds. (Use v⁵ = 0.783526 and 1.05⁵ = 1.276282.)

Show the solution

The liability present value is V_L = 5,00,000 × v¹⁰ = 5,00,000 × 0.783526² = 5,00,000 × 0.613913 = ₹3,06,956.
Let x and y be the present values of the 5-year and 15-year bonds. Condition 1: x + y = 3,06,956.
Condition 2: the asset DMT equals the liability DMT of 10. So (5x + 15y) ÷ (x + y) = 10, which gives 5x + 15y = 10x + 10y, so y = x.
Hence x = y = ₹1,53,478.
Nominal payment of the 5-year bond: 1,53,478 × 1.05⁵ = 1,53,478 × 1.276282 = ₹1,95,882 (approximately). Equivalent check: 2,50,000 × v⁵ = 2,50,000 × 0.783526 = ₹1,95,882.
Nominal payment of the 15-year bond: 1,53,478 × 1.05¹⁵. This equals 2,50,000 × 1.05⁵ = 2,50,000 × 1.276282 = ₹3,19,070.
Condition 3: asset Σ t² · PV ÷ total PV = (25 × 0.5) + (225 × 0.5) = 125. Liability value = 10² = 100.
Since 125 > 100, asset convexity exceeds liability convexity, so condition 3 holds.

Answer: Buy about ₹1,95,882 nominal of the 5-year bond and ₹3,19,070 nominal of the 15-year bond. Each has present value ₹1,53,478. The convexity measure is 125 for assets against 100 for liabilities, so all three conditions hold.

Example 2

A company has a liability of ₹1,21,000 payable in 2 years. It holds zero-coupon bonds paying ₹55,000 in 1 year and ₹66,550 in 3 years. The interest rate is 10% per year effective. Show whether the portfolio satisfies Redington's conditions for immunisation.

Show the solution

Liability present value: 1,21,000 × v² = 1,21,000 ÷ 1.21 = ₹1,00,000.
Asset present values: 55,000 ÷ 1.1 = ₹50,000, and 66,550 ÷ 1.331 = ₹50,000. Total ₹1,00,000.
Condition 1 holds: 1,00,000 = 1,00,000.
Asset DMT = (1 × 50,000 + 3 × 50,000) ÷ 1,00,000 = 2 years. Liability DMT = 2 years.
Condition 2 holds, since the DMTs are equal and the present values are equal.
Asset convexity measure = (1² × 50,000 + 3² × 50,000) ÷ 1,00,000 = (50,000 + 4,50,000) ÷ 1,00,000 = 5.
Liability convexity measure = 2² = 4.
Since 5 > 4, condition 3 holds.

Answer: All three conditions hold at 10%, so the portfolio is immunised against a small, immediate change in the interest rate. It would need rebalancing as time passes.

Exam tips

Write the three conditions in full at the start of any written answer. Marks are usually given for naming each one, even before the calculation.
Define your unknowns as present values, solve the two linear equations, then convert back. This keeps the algebra short and avoids errors.
Always finish with the convexity check and a one-line conclusion. Many students lose the final mark by stopping at the DMT equation.
Be ready to state the limitations: small changes only, parallel shift, rebalancing needed, and no allowance for changes in the cash flows themselves.
In Paper B (computer-based), set up the cash flows in a sheet, use the same present value and t² · PV formulas, and show the check cells for all three conditions.

Practice questions from Duration, convexity and immunisation

Redington's Immunisation Conditions: frequently asked questions

What are Redington's three conditions for immunisation?

First, the present value of assets equals the present value of liabilities at the current rate. Second, the discounted mean terms are equal, which is the same as equal first derivatives. Third, the convexity of assets is greater than the convexity of liabilities, so the surplus has a local minimum.

Why must asset convexity be greater, not just equal?

With equal present values and equal slopes, the surplus is flat at the current rate. Equal convexity would leave its behaviour undetermined. Greater asset convexity makes the surplus curve upward, so any small change in rate gives a gain.

How do I immunise a liability using bonds?

Work out the liability present value and DMT. Choose assets, often two zero-coupon bonds on either side of the liability date, with the same total present value and the same DMT. Then check that asset convexity exceeds liability convexity.

Does Redington immunisation protect against any change in interest rates?

No. It protects against a small, immediate, one-off change that applies equally across all terms. Large moves, changes in the shape of the yield curve and changes over time can break it, so the portfolio needs regular rebalancing.