Bond duration and interest-rate risk

Price is not the whole story

Knowing what a bond is worth today is only half of a fixed-income decision. The other half is knowing how much that value moves when the market's required return changes. A treasury holding ৳40 million of a long-dated bond needs a number for that exposure, not a feeling about it.

A bond's price and its yield move in opposite directions, but the relationship is not a straight line. Duration measures the slope; convexity measures the curvature that the slope misses.

Duration: the weighted wait

Macaulay duration has a second reading that is easier to remember: it is the average time you wait to be paid. A five-year zero-coupon bond has a duration of exactly five years, because there is only one payment and it arrives at maturity. A coupon bond pays earlier as well, so its duration is shorter than its maturity.

Modified duration is the version you use. A modified duration of 7.36 years means a one percentage-point rise in yield lowers the price by roughly 7.36%.

Convexity: the correction

Duration assumes the price-yield relationship is a straight line. It is not: the curve bends upward, and that bend is convexity. Because of it, a rise in yield costs slightly less than duration predicts, and a fall in yield gains slightly more.

Convexity is a second-order effect, which is why it is invisible for a small yield move and material for a large one. Ignoring it is fine for a 10 basis-point move and misleading for a 200 basis-point move.

A worked example

Take a bond with a ৳1,000 face value, a 6% annual coupon, ten years to maturity, priced at a 6% yield. Because the coupon equals the yield, the price is exactly ৳1,000.

Macaulay duration is about 7.80 years, so modified duration is 7.80 ÷ 1.06 ≈ 7.36 years. Convexity is about 69.74. If the yield rises one percentage point to 7%, duration alone predicts a fall of about 7.36%, to ৳926. The convexity term adds back roughly ৳3.9 (0.5 × 69.74 × 0.01² × ৳1,000) and the estimate becomes about ৳929.89. The exact repriced value is ৳929.76, so the second-order estimate is accurate to about 13 paisa, while duration alone is off by about ৳3.9.

The same arithmetic runs in reverse for a fall in yield: the price gains a little more than duration predicts.

Using duration in practice

Duration turns a view about interest rates into an approximate profit and loss. A portfolio with a modified duration of 5 years and a market value of ৳10 million loses roughly ৳500,000 if yields rise one percentage point. That single multiplication is the core of interest-rate risk management.

It also lets a manager match assets to liabilities. If a pension must pay out in ten years, holding assets with a duration near ten years hedges the value of the funding against rate moves. This is immunisation, and duration is its unit of account.

The estimate has limits. Duration changes as time passes and as yields move, so the hedge must be rebalanced. Cash flows are not always certain: callable bonds, prepayments, and credit deterioration all break the assumption that the schedule is fixed.