How do you build a bond portfolio that pays a bill in eight years?

This is the problem duration was invented to solve. An insurer owes €1,000,000 in eight years. It has money today. What does it buy?

The two risks that fight each other

You met them separately; now watch them collide.

• Price risk — if rates rise, the bonds are worth less when the bill comes due.
• Reinvestment risk — if rates fall, the coupons compound at less than assumed and the pot falls short.

Rates can't do both. Whichever way they go, one risk hurts and the other helps.

Redington's insight

In 1952 the actuary Frank Redington noticed that if you set the duration of the assets equal to the duration of the liability, the two effects approximately cancel. Rates rise: you lose on price, you gain on reinvestment. Rates fall: the reverse. The value of the portfolio at the horizon is roughly unchanged either way. He called it immunisation.

Three conditions have to hold together:

1. Present value of assets = present value of the liability, discounted at the same rate.
2. Duration of assets = duration of the liability. For a single payment in eight years, the liability's duration is exactly 8.
3. The assets' cash flows are more dispersed than the liability's — equivalently, asset convexity ≥ liability convexity. This is where convexity finally earns its keep: given equal duration and present value, the more convex portfolio does at least as well under a small parallel shift in either direction.

The honest baseline first

The cleanest answer is not clever at all: buy an eight-year zero-coupon bond with €1,000,000 face. Duration 8 by construction, no coupons to reinvest, no rebalancing, no model. This is cash-flow matching, and it is the benchmark every sophisticated alternative should be measured against. If a strategy can't beat the strip after costs, it is complexity for its own sake.

When you must use coupon bonds

Suitable zeros are not always available, cheap, or permitted. Then you duration-match with coupon bonds — and the first thing to get right is that the weighting equation takes durations, not maturities. A coupon bond's duration is well short of its term. At a 5% yield, 5% par bonds run like this:

• 2-year → duration 1.95
• 20-year → duration 13.09

With the real numbers in hand you have a choice:

• Bullet — a single bond with duration 8. That is roughly a 10-year 5% par bond, whose duration is 8.11 — not an 8-year one, which only reaches 6.79.
• Barbell — the 2-year and the 20-year, weighted so their durations average to 8: solve w × 1.95 + (1 − w) × 13.09 = 8, giving 45.7% in the 2-year and 54.3% in the 20-year. Same duration, far more dispersed cash flows, therefore far more convexity — so for a small parallel shift it does better in either direction. That is a second-order, local result: matching duration and comparing convexity says what happens near today's yield, not what happens for an arbitrarily large move, and it says nothing at all about a shift that is not parallel.

Feed the maturities 2 and 20 into that same equation and you get two-thirds/one-third — a portfolio whose real duration is 5.66, short of the liability by more than two years. The immunisation would be broken before it started, and this is exactly why the last three lessons kept insisting that maturity and duration are not interchangeable.

And here is the professional point, the one that stops this from being a free lunch: the barbell's advantage assumes the curve moves in parallel. Twist the curve instead and the ranking can reverse — a steepening, where long yields rise relative to short, hurts the barbell's big long leg far more than it hurts a bullet sitting in the belly. So does a shock confined to the short end while the belly stays pinned: the barbell takes the hit, the bullet does not. Convexity is not given away; the market prices it, and you pay for it in curve risk. Unit 4 gives you the vocabulary for exactly that exposure.

It is not set-and-forget

Immunisation drifts, for two reasons:

• Time passes unevenly. After one year the liability has seven years to run, but a coupon portfolio's duration falls by less than one year. The match slips. (A zero's duration falls exactly one year per year — one more reason it is the clean instrument.)
• Yields move. Duration itself changes when the yield changes, and the assets' duration and the liability's duration do not change by the same amount.

So the portfolio must be rebalanced periodically, and the transaction costs of rebalancing are a real, recurring cost of the strategy.

What immunisation does and does not do

It converts interest-rate risk into a smaller set of other risks: model risk (the parallel-shift assumption), credit risk (a defaulted bond pays no liability), rebalancing cost, and liquidity. That is a real improvement for a known future obligation — and it is not the elimination of risk. This is a description of a technique used by insurers and pension funds; it is not a recommendation, and nothing here forecasts where rates will go.

Try it now

1. Work out the barbell weights for a 6-year liability using 1-year and 15-year 5% par bonds — using their durations, which are 1.00 (one payment left, so duration equals the remaining term) and 10.90. Solve w × 1.00 + (1 − w) × 10.90 = 6. (You should get about 49.5% short, 50.5% long. Put the maturities 1 and 15 in instead and you get 64.3% — a wrong answer that looks exactly as tidy as the right one.)
2. Price the exact cash-flow match instead: the high-quality corporate spot curve is a zero-coupon curve, and discounting the liability at its own date's spot rate is the honest cost of certainty. The newest month is below. For this lesson's €1,000,000 due in eight years, there is no 8-year row: interpolate a third of the way from the 7-year rate to the 10-year rate, then compute 1,000,000 ÷ (1 + s)⁸. That is what the zero-coupon match would cost today at high-quality corporate rates.

3. Name the two risks that cancel under immunisation, the assumption that makes them cancel, and the one reason the portfolio still needs looking after.