IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Duration, Convexity and Immunisation Explained for IAI Actuarial

Duration measures how sensitive the present value of cash flows is to interest rate changes. Convexity measures the curvature of that relationship. Immunisation uses both: you match present values, match durations, and make asset convexity exceed liability convexity, so small interest rate shifts do not hurt the surplus.

What this chapter covers

This chapter asks one question: how does the value of a set of cash flows change when interest rates change? You answer it with three tools. Discounted mean term (Macaulay duration) is the present-value-weighted average time of the cash flows. Modified duration (volatility) gives the proportional price change for a small change in the rate. Convexity corrects for the curvature that duration ignores.

The second half applies these tools to Redington's theory of immunisation. An institution holds assets to meet liabilities. If interest rates move, both change in value. Redington's three conditions ensure that, for a small change in the interest rate, the asset value stays at or above the liability value. You then see why this protection is limited in practice: it holds only for small, parallel shifts, and it needs rebalancing over time.

The chapter builds directly on the theory of interest rates and the equation of value, which together carry a large share of the CM1 syllabus. It also links to pricing and reserving, where assets back liabilities, and to CM2, where you study measures of investment risk and asset-liability matching further. If your present value and force of interest skills are solid, this chapter is mostly careful algebra and calculus.

Immunisation questions are a favourite for written papers because one question can test present values, derivatives, interpretation and judgement together. You may be asked to compute duration or convexity, check Redington's conditions for a given portfolio, or comment on practical limitations. The calculations are short once you know the method, so this chapter rewards practice. Interpretation parts also give marks to students who can explain results in plain words. It also feeds into Paper B work, where you may need to set up cash flows and compute these measures efficiently.

Duration, convexity and immunisation: topics in the order to study them

1Discounted Mean Term and Macaulay DurationStart here because it defines duration as a weighted average time, which every later topic builds on.
2Volatility and Modified DurationIt links duration to the derivative of present value with respect to the interest rate, giving you the first-order price change.
3ConvexityIt adds the second derivative, so you need the first-order idea from modified duration in place first.
4Redington's Immunisation ConditionsThe three conditions use present value, duration and convexity, so you can only apply them once those are clear.
5Full Immunisation and Practical LimitationsStudy this last because it extends Redington's idea and asks you to judge when the theory fails in practice.

How to prepare Duration, convexity and immunisation

Treat this chapter as a derivation-plus-calculation topic. Understand where each formula comes from, then drill the numerical method until it is quick.

Revise present value and the force of interest first. Duration and convexity are derivatives of present value, so weak basics will slow you down everywhere.
Write the definitions in your own words: discounted mean term as a weighted average time, volatility as a proportional rate of change of value, convexity as a scaled second derivative.
Derive the key relationships yourself at least once, such as the link between volatility and discounted mean term. Then you can rebuild them in the exam instead of relying on memory.
Practise computing duration and convexity for small sets of cash flows by hand, laying out the working in a table of time, cash flow, discount factor and present value.
Work through Redington questions in a fixed order: check present values equal, check durations equal, then compare convexities. State each condition clearly before you test it.
Write short answers on practical limitations and on what happens when interest rates change by a large amount or non-parallel shifts occur. Keep each point to one or two sentences.
Redo a few calculations in R or Excel for Paper B practice, and check your hand results against the spreadsheet.

Common mistakes in Duration, convexity and immunisation

Mixing up Macaulay duration and modified duration, or dividing by the wrong factor.
Fix: Write which measure you are computing at the top of your working. Remember that volatility comes from differentiating present value, and convert only once.
Using the wrong interest rate or mixing rate types in the present value table.
Fix: Convert to the effective annual rate or the force of interest first. Say which one you use and keep it the same through the whole question.
Checking only two of Redington's three conditions.
Fix: Use a fixed checklist: present values, durations, convexities. Write a one-line conclusion that the asset convexity is greater or not.
Comparing convexity of assets and liabilities using different bases.
Fix: Because present values are equal under condition 1, the comparison is consistent. Check that you used the same definition for both sides, and say so.
Claiming immunisation protects against any change in interest rates.
Fix: State that the conditions give protection against small changes, usually parallel shifts, and that portfolios need rebalancing as time passes and rates move.
Losing marks on interpretation parts by giving only a number.
Fix: Add one sentence on what the result means, for example that a higher duration means greater sensitivity of value to rate changes.

Last-day revision: Duration, convexity and immunisation

Discounted mean term = Σ t·v^t·cash flow at t ÷ Σ v^t·cash flow at t, the present-value-weighted average time.
Macaulay duration is the discounted mean term; state the interest rate you use.
Volatility = −(1 ÷ V) × dV/di, where V is the present value at interest rate i.
Modified duration equals Macaulay duration ÷ (1 + i) when interest is effective annual and cash flows are discrete.
Convexity = (1 ÷ V) × d²V/di², using the same V and i as in duration.
A first-order estimate of the change in V is −V × volatility × Δi; the second-order term adds ½ × V × convexity × (Δi)².
Redington condition 1: present value of assets equals present value of liabilities at the current rate.
Redington condition 2: the duration of assets equals the duration of liabilities, so the first derivative of the surplus is zero.
Redington condition 3: convexity of assets exceeds convexity of liabilities, so the surplus has a local minimum at the current rate.
The three conditions protect against small changes in the interest rate only.
Full immunisation requires the asset cash flows to surround the liability cash flows, at least in the classic one-liability case.
Practical limits: non-parallel yield curve shifts, rebalancing needs, transaction costs and uncertain liability cash flows.

Duration, convexity and immunisation practice questions

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.

Duration, convexity and immunisation: frequently asked questions

What is the difference between Macaulay duration and modified duration?

Macaulay duration, or discounted mean term, is the present-value-weighted average time of the cash flows, measured in years. Modified duration, or volatility, measures the proportional fall in value for a rise in the interest rate. For effective annual rates with discrete cash flows, modified duration equals Macaulay duration divided by (1 + i).

What are Redington's three conditions for immunisation?

First, the present value of assets equals the present value of liabilities at the current interest rate. Second, the duration of assets equals the duration of liabilities. Third, the convexity of assets is greater than the convexity of liabilities. Together they make the surplus have a local minimum at the current rate.

Why is convexity needed if I already have duration?

Duration gives only a first-order, straight-line estimate of how value changes. Real value changes follow a curve, so the estimate drifts for larger rate moves. Convexity captures that curvature and, in Redington's theory, ensures the surplus rises whichever way rates move.

Do I need to know the limitations of immunisation for the exam?

Yes. Written questions often ask you to comment on practical problems. Know that the theory covers small changes, assumes the same change applies to all maturities, and needs regular rebalancing, with costs and uncertainty in liability timing also reducing its effectiveness.