Duration is the bond number that answers what if rates move

Macaulay, modified, and DV01

Macaulay duration is the weighted average time until you receive the bond’s cash flows, measured in years, with each payment weighted by its present value. It is a description of the bond, and on its own it does not predict anything. Modified duration divides it by one plus the yield per period, and that is the number that estimates a price change: multiply modified duration by the yield move, flip the sign, and you have the approximate percentage price change.

DV01 does the same thing in currency for one basis point, and it is the version to use when you are comparing positions of different sizes rather than bonds of different structures. Percentage sensitivity tells you about the instrument. Dollar sensitivity tells you about your exposure, and only one of those can be added up across a portfolio.

The coupon matters more than the maturity

A zero-coupon bond has a duration equal to its maturity, because there is exactly one cash flow and it is at the end. Add coupons and duration falls below maturity, because money is arriving sooner and the weighted average time shortens. Two bonds maturing on the same day with different coupons are therefore different instruments in the only sense that matters here.

Push that to the end of the curve and you get the most rate-sensitive instrument in the market: a long-dated bond with a low coupon. This is how a portfolio of long government bonds, an asset class sold on its safety, can deliver a decline of a size people associate with equities. Nothing defaulted, no credit event occurred, and the loss was entirely the duration doing exactly what it says on the label.

Convexity, and when the estimate stops working

The full approximation is that the percentage price change equals minus modified duration times the yield change, plus half the convexity times the yield change squared. Because the correction term is squared, it is positive whichever way yields move: a normal bond gains slightly more than duration predicts when yields fall, and loses slightly less when they rise. Positive convexity is a small gift, and it is priced.

For a 25 basis point move you can ignore the second term. For 300 basis points you cannot, and using duration alone will materially overstate the loss. Some instruments have the gift in reverse: callable bonds and mortgage-backed securities shorten when yields fall, because the borrower refinances, and extend when yields rise, because they do not. Negative convexity means the position gets shorter exactly when length would have paid, which is why yield to worst exists as a separate number.

• Use modified duration for moves inside about 50 basis points.
• Add the convexity term for anything larger.
• Use DV01 when combining positions of different sizes.
• For a callable bond, price to the worst call date, not to maturity.

Use it as a hedge ratio, not as a fact about the bond

Duration exists so that positions can be compared and offset. Matching the DV01 of a hedge against the DV01 of an exposure neutralises the first-order rate risk between them, and the residual is convexity and whatever the curve does that a single yield number cannot capture. A parallel shift is an assumption, and the curve rarely obliges.

Our position: for anyone holding bonds rather than trading them, the entire subject collapses to one sentence. Duration times the yield move is your loss, and the yield you were quoted is the compensation for accepting exactly that. If you do not want the number, buy a shorter one and accept less yield, because those two facts are the same fact.