Why does duration alone get large rate moves wrong?

Every duration estimate you made in Unit 2 was slightly wrong, and always wrong in the same direction. That is not sloppiness — it is geometry, and it has a name.

The relationship is a curve, not a line

Plot a bond's price against its yield and you do not get a straight line. You get a curve that is convex — bowed toward the origin, falling steeply at low yields and flattening out at high ones.

Duration is the slope of that curve at one point: the tangent line drawn where the bond trades today, which is the straight overlay above. And a tangent to a convex curve always sits below the curve everywhere except at the point of contact.

That single geometric fact gives you the whole lesson:

• Rates fall → the true price rises faster than the tangent line → duration understates the gain.
• Rates rise → the true price falls more slowly than the tangent line → duration overstates the loss.

Both errors favour the bondholder. You make a little more than duration promised, and you lose a little less.

The evidence, on the house bond

Our 5-year, 5% par bond, modified duration 4.33:

• +100bp: duration says €956.71, truth is €957.88 — €1.17 better than predicted.
• −100bp: duration says €1,043.29, truth is €1,044.52 — €1.22 better than predicted.
• +300bp: duration says €870.12, truth is €880.22 — €10.10 better than predicted.

Look at the third line. Tripling the size of the move multiplied the error by roughly nine, not three. The error grows with the square of the yield change, which is why duration is nearly perfect for a 5bp wobble and materially wrong about a 300bp repricing — wrong in the bondholder's favour, as the three lines above show, but wrong by €10.10.

Why the curve bends

Duration is not a constant. As yields fall, the present values of distant cash flows swell relative to near ones, the balance point moves further out, and duration lengthens — so each additional fall in yield lifts the price by more than the last. As yields rise, duration shortens and each further rise costs less than the one before.

In other words: the bond automatically gets more sensitive when that helps you and less sensitive when that would hurt you. Convexity is the measure of how fast duration changes — and the next lesson turns it into a number you can add to your estimate.

A caution before you get comfortable

This pleasant asymmetry is a property of ordinary bonds with fixed cash flows. Attach an option to the bond — a call, a prepayment right on a mortgage pool — and the curve can bend the other way. The gift is not universal, and Unit 3's next lesson says exactly when it is withdrawn.

Try it now

1. Price the house bond at 4%, 5% and 6% (€1,044.52, €1,000.00, €957.88) and find those three points on the figure above. The bend is visible with three dots and a ruler, which is all the figure is.
2. Now read the vertical gap between the curve and the straight overlay at 4%, at 6% and at the right-hand edge. Three numbers, growing: that widening gap is convexity, and the next lesson gives it a coefficient.
3. Say the asymmetry in one sentence: "duration overstates my losses and understates my gains, and the bigger the move, the more it does so."