Two piles, one rate, different reactions
By now you know the move from the interest-theory rung cold: a bond's price is just the present value of its future cash flows, and **duration** measures how fast that value reacts when the rate used to discount it moves. The new idea on this rung is to point the *same lens* at the other side of the balance sheet. An insurer's promises — the death benefits, the annuity cheques, the pension payments — are also future cash flows. So they too can be discounted to a present value, and they too have a duration. Liabilities, it turns out, are bonds in disguise; just bonds the company has *sold* rather than bought.
Recall the working rule: the percentage change in a value is roughly minus its duration times the change in rate. A pile with duration 8 loses about 8% of its value when rates rise 1%, and gains about 8% when they fall 1%. That rule applies to *any* present value, so it applies to the asset pile and the liability pile alike. The crucial, unsettling fact is that the two piles rarely have the *same* duration — and that single mismatch is where this entire rung's drama lives.
The straight-line answer, and the bend it misses
Duration is a *first* approximation — it pretends the relationship between value and rate is a straight line, the tangent drawn at today's rate. For the small daily wiggles of the market that line is wonderfully accurate. But the true value-versus-rate relationship is gently bowed, curving away above the tangent line on both sides. **Convexity** is the name for that bend, and it is the *second* correction: a small extra push that duration alone leaves out.
The bend works in your favour when you *own* the cash flows. Because the curve bows upward, a large rate *fall* lifts your value by more than the straight line predicts, while an equal rate *rise* costs you less than the line predicts — heads you win a little extra, tails you lose a little less. High convexity is therefore a desirable property in an asset. But flip the logic for the side you have *sold*: a liability with high convexity bends *against* you, growing faster on a rate fall than its duration alone would warn. This asymmetry between the two sides is exactly why matching only the durations is not enough.
The duration gap: where surplus is born and lost
The number that captures the mismatch is the **duration gap**: the duration of the assets minus the duration of the liabilities (with a small adjustment for the fact that liabilities are usually a touch smaller than assets, since surplus is the sliver between them). When the gap is zero, the two piles move by the same percentage on a rate change and the surplus barely budges. When the gap is positive — assets longer than liabilities — the company is betting, perhaps unwittingly, that rates will *fall*. When it is negative, it is exposed to rates *rising*. The sign of the gap tells you which way the wind hurts.
Here is the part that surprises newcomers. Surplus — assets minus liabilities — is a *thin* layer sitting on top of two *thick* piles. A modest mismatch in how the thick piles react gets amplified ferociously when it lands on the thin layer. Picture a company with 100 of assets and 90 of liabilities, so surplus is 10. If assets have duration 5 and liabilities duration 9, a 1% rate fall lifts assets by about 5% (up 5, to 105) but lifts liabilities by about 9% (up 8.1, to 98.1). Surplus collapses from 10 to 6.9 — a 31% wound — from a rate move that touched neither pile by even a tenth.
rates -1% rates flat rates +1% Assets (D=5) 105.0 100.0 95.0 Liabs (D=9) 98.1 90.0 81.9 ----------------------------------------------------------- Surplus 6.9 10.0 13.1 change -31% -- +31%
Closing the gap — and why you never close it forever
The cure for a dangerous gap is to reshape the asset portfolio until its duration lines up with the liabilities'. If the promises are long-dated, lengthen the assets — buy longer bonds — so both sides rise and fall together. This deliberate alignment is the daily business of **asset-liability management**: not chasing the fattest yield, but choosing investments whose *interest-rate behaviour* mirrors the promises. The earlier guides on this rung showed why insurers invest at all and how to discount the liabilities; the duration gap is the dial that tells you whether the two sides are actually in step.
Matching durations alone is the blunt version. The sharper goal is **immunization** — the topic of the next guide — where you match the durations *and* arrange for the assets to carry at least as much convexity as the liabilities. Then for any small rate move, in either direction, the bend protects the surplus rather than draining it. It is the difference between standing the two piles back-to-back and gluing them together at the spine.
Honest limits — what duration quietly assumes
Duration's tidy single number rests on a heroic assumption: that the *whole yield curve shifts in parallel*, every maturity moving up or down by the same amount. Real curves twist and tilt — the short end can leap while the long end barely stirs. A portfolio that looks perfectly duration-matched against a single number can still spring a leak when the curve changes *shape* rather than just *level*. This is why sophisticated shops measure sensitivity to several separate curve movements, not just one.
Two more honest caveats. Duration and convexity describe only the *value* response to rates; they say nothing about whether the bonds will actually pay (credit risk) or whether you can sell them when you need cash. And even a well-matched book is exposed to **reinvestment risk** — when a falling rate lifts your liability values, the very same fall means the coupons you collect must be reinvested at a *lower* rate, a double squeeze that pure duration matching does not fully neutralise. The grown-up posture is this: duration and convexity are sharp, indispensable tools for the everyday move, but they are a model of reality, not reality itself. Pair them with scenario tests for the moves they were never built to see.