Actuarial Mathematics for Modelling · Duration, convexity and immunisation

Full Immunisation and Its Practical Limitations

Updated 11 October 2026 · Fact-checked

Full immunisation sets up assets so that a small change in the interest rate cannot cause a loss against a liability. It needs three Redington conditions: equal present values, equal discounted mean terms, and assets with greater convexity. In practice it needs rebalancing and fails for large or non-parallel yield changes.

Understand Full Immunisation and Practical Limitations

A company owes a fixed payment at a future date. It holds assets to meet it. The risk is that interest rates move, so the asset value and the liability value change by different amounts. Immunisation chooses assets so that the surplus (asset value minus liability value) cannot fall after a small rate change.

Think of the surplus as a function of the force of interest δ. Call it S(δ) = V_A(δ) − V_L(δ). You want S to have a local minimum at the current rate, and that minimum must be zero. Zero surplus at the current rate gives condition (i). A zero first derivative gives condition (ii). A positive second derivative gives condition (iii). Then any small change in rate, up or down, leaves the surplus positive.

For a single liability due at time t, condition (ii) says the discounted mean term of the assets must equal t. Condition (iii) says the asset cashflows must be more spread out in time than the liability. In practice you hold assets that pay both before and after t, such as two zero-coupon bonds. A single zero-coupon bond maturing exactly at t is not a valid immunising portfolio on its own, because its convexity equals that of the liability and it satisfies conditions (i) and (ii) only. It is a matching portfolio instead.

Matching means asset cashflows equal the liability cashflows in amount and timing, so interest rate risk is removed whatever happens to rates. Immunisation only protects against small changes in the interest rate and needs the portfolio to be reviewed over time. Immunisation is easier to build, because assets need not pay at exactly the liability dates.

The limits matter as much as the method.

The result holds for small changes in a single yield. Large changes can break it.
It assumes a flat yield curve with parallel shifts. Real curves twist and change shape.
Conditions drift as time passes and as rates change, so the portfolio needs rebalancing, which costs money through transaction costs.
Suitable assets may not exist, and default risk and uncertain liability amounts or timing are ignored.

Key rules to remember

Surplus function: S(δ) = V_A(δ) − V_L(δ); V_A and V_L are the present values of asset and liability cashflows at force of interest δ. Immunisation concerns the behaviour of S near the current δ.
Condition (i): equal present values: V_A(δ₀) = V_L(δ₀); Assets must be worth the same as liabilities at the current rate.
Condition (ii): equal discounted mean terms: V_A'(δ₀) = V_L'(δ₀), i.e. DMT_A = DMT_L; Valid because condition (i) holds, so equal first derivatives mean equal DMTs. DMT = Σ t·PV(t) ÷ Σ PV(t).
Condition (iii): convexity: V_A''(δ₀) > V_L''(δ₀); With equal present values this means Σ t²·PV_A(t) ÷ V_A > Σ t²·PV_L(t) ÷ V_L. Asset cashflows are more spread out than liability cashflows.
Single liability test: DMT_A = t and Σ t_k²·PV_k ÷ V_A > t²; Here t is the liability date and PV_k are present values of asset payments at times t_k.
Two zero-coupon assets: w·t₁ + (1 − w)·t₂ = t, with t₁ < t < t₂; w is the proportion of the present value in the earlier bond. Then the nominal amount at time t_k is PV_k × (1 + i)^(t_k).

How to solve Full Immunisation and Practical Limitations questions

Use this order for any question on full immunisation of a single liability, or on discussing its limits.

1Write down the liability amount, its date t and the interest rate. Find its present value V_L.
2Set the total present value of the assets equal to V_L (condition i).
3Choose asset times t₁ < t < t₂ if the question gives them. Solve w·t₁ + (1 − w)·t₂ = t for the proportion w (condition ii).
4Split V_L into PV_1 = w·V_L and PV_2 = (1 − w)·V_L. Convert to nominal amounts by accumulating each at the interest rate to its payment date.
5Test condition (iii): compute Σ PV_k·t_k² ÷ V_L and compare it with t². The assets must give the larger value.
6State the conclusion: any small change in interest rate leaves a positive surplus. Quote the assumptions.
7For discussion parts, link each limit to a failed assumption: non-parallel shift, large change, passage of time, transaction costs, asset availability, default, uncertain liability.

Quickest way: Two zero-coupon bonds around the liability date

When to use it: Use when a question asks you to build an immunised portfolio for one liability using two assets that pay on either side of the liability date.

Compute V_L = L × v^t.
Solve for w from the mean term equation. If the two times are equally spaced about t, w = 0.5.
PV in each asset = w·V_L and (1 − w)·V_L.
Nominal amount = PV × (1 + i)^(time of that payment).
Check convexity using Σ PV·t² against V_L·t². Write one line stating asset > liability.

Common mistakes in Full Immunisation and Practical Limitations

Stopping after matching present value and mean term and not checking convexity.
The first two conditions give the equations to solve, so the third feels like an afterthought.
Fix: Always compute Σ PV·t² for assets and compare with the liability. Without it, the surplus could have a maximum instead of a minimum.
Choosing an asset that matures exactly at the liability date and calling it immunised.
The mean term matches, so it looks right.
Fix: That is exact matching, not immunisation. For immunisation the asset cashflows must be spread both before and after the liability date so that convexity is larger.
Using nominal amounts, not present values, as the weights in the mean term equation.
The nominal payments are what the question finally asks for, so they are used too early.
Fix: The weights in the mean term equation are present values. Find PVs first, then convert to nominal amounts.
Saying immunisation protects against any change in interest rates.
Students remember that the surplus is positive and drop the word small.
Fix: State that the result holds for small changes in the interest rate, with the same rate applied to all terms (a parallel shift).
Saying immunisation never needs rebalancing once set up.
The conditions are checked once, so they seem permanent.
Fix: Explain that as time passes and rates change, the present value weights and the mean terms change. Asset and liability durations move differently, so the conditions must be restored by rebalancing, which brings transaction costs.
Treating immunisation and matching as the same thing.
Both aim to reduce interest rate risk.
Fix: Matching copies the liability cashflows in amount and timing and removes the risk. Immunisation only guards against small, parallel yield shifts and needs monitoring.

Worked examples

Example 1

A fund must pay ₹10,00,000 in 10 years. The effective annual interest rate is 5%. It will hold two zero-coupon bonds, maturing in 5 years and 15 years, so that the liability is immunised against small changes in the interest rate. Find the nominal amounts of the two bonds and check that the convexity condition holds.

Show the solution

V_L = 10,00,000 × 1.05^(−10) = 10,00,000 × 0.613913 = ₹6,13,913.
Let w be the proportion of present value in the 5-year bond. Mean term: 5w + 15(1 − w) = 10, so 15 − 10w = 10 and w = 0.5.
PV in each bond = 0.5 × 6,13,913 = ₹3,06,957 (rounded). Total equals V_L, so condition (i) holds.
5-year bond nominal amount = 3,06,957 × 1.05^5 = 3,06,957 × 1.276282 ≈ ₹3,91,763.
15-year bond nominal amount = 3,06,957 × 1.05^15 = 3,06,957 × 2.078928 ≈ ₹6,38,141.
Convexity check, dividing by V_L: assets give 0.5 × 5² + 0.5 × 15² = 12.5 + 112.5 = 125. Liability gives 10² = 100.
125 > 100, so condition (iii) holds. Check at 6%: asset value ≈ 2,92,748 + 2,66,278 = ₹5,59,026, liability value = 10,00,000 × 1.06^(−10) ≈ ₹5,58,395. The surplus is about ₹630, positive, as expected.

Answer: Buy about ₹3,91,763 nominal of the 5-year bond and about ₹6,38,141 nominal of the 15-year bond. All three Redington conditions hold, so small changes in the rate leave a positive surplus.

Example 2

A liability of ₹2,00,000 is due in 6 years. The effective annual interest rate is 4%. It is immunised using zero-coupon bonds maturing in 3 and 9 years, with ₹88,900 nominal in the 3-year bond and ₹1,12,486 nominal in the 9-year bond. One year later the interest rate suddenly falls to 3% effective, with no other change. Show that the portfolio now needs rebalancing.

Show the solution

After one year the bonds have 2 and 8 years left. The liability has 5 years left.
At 3%, PV of the first bond = 88,900 ÷ 1.03² = 88,900 ÷ 1.0609 ≈ ₹83,797.
PV of the second bond = 1,12,486 ÷ 1.03^8 = 1,12,486 ÷ 1.266770 ≈ ₹88,797.
Total asset value ≈ ₹1,72,594.
Discounted mean term of assets = (2 × 83,797 + 8 × 88,797) ÷ 1,72,594 = (1,67,594 + 7,10,376) ÷ 1,72,594 = 8,77,970 ÷ 1,72,594 ≈ 5.09 years.
The liability has a single payment in 5 years, so its mean term is exactly 5 years.
5.09 ≠ 5, so condition (ii) no longer holds. The weights moved towards the longer bond because it gained more in value when rates fell.

Answer: The asset mean term is about 5.09 years against 5 years for the liability, so the portfolio must be rebalanced. Rebalancing involves buying and selling assets and so incurs transaction costs.

Exam tips

In written questions, write all three Redington conditions by name and show a check of each. Marks are usually split across them.
Show the algebra for the mean term equation and the convexity comparison. A correct final amount with no working loses marks.
For discussion questions, give at least four distinct limitations and tie each to a reason: non-parallel shifts, large changes, rebalancing and transaction costs, asset availability, default and liability uncertainty.
For multiple-choice questions, check whether the question asks about immunisation or matching. A portfolio whose cashflows copy the liability is matched, and it has no interest rate risk.
In computer-based papers, build the asset and liability present values as functions of the rate and compute the surplus at a few nearby rates to demonstrate the local minimum.

Practice questions from Duration, convexity and immunisation

Full Immunisation and Practical Limitations: frequently asked questions

What is full immunisation for a single liability?

It is a portfolio of assets whose present value equals the liability, whose discounted mean term equals the liability date, and whose convexity is larger. With these three conditions, a small change in the interest rate leaves the surplus positive.

Why does immunisation require rebalancing?

As time passes, the discounted mean terms of assets and liabilities shrink at different rates. Changes in interest rates also change the present value weights of the asset cashflows. The conditions then stop holding, so assets must be bought or sold to restore them.

What is the difference between immunisation and matching assets and liabilities?

Matching sets asset cashflows equal to liability cashflows in amount and timing, which removes interest rate risk. Immunisation only protects against small changes in the interest rate, and uses assets that need not pay on the liability dates. It needs monitoring and rebalancing.

What are the main limitations of Redington immunisation?

It covers only small changes in interest rates and assumes parallel shifts of a flat yield curve. It needs regular rebalancing with transaction costs. Suitable assets may not be available, and it ignores default risk and uncertainty in the liability.