Bond Duration

Bond duration is a measure of a bond's sensitivity to changes in interest rates. The Municipal Securities Rulemaking Board defines it as "a measure of the timing of cash flows (i.e., the interest payments and the principal repayment) to be received from a given fixed income security," used "to assess price volatility for given changes in interest rates."

That definition points at the naming problem, which is worth clearing up before anything else. Macaulay duration is the timing measure itself: the weighted average number of years until the bond's payments arrive, with each payment weighted by its present value. Modified duration is derived from it and answers a different question: approximately what percentage of its price the bond gains or loses for each percentage point change in yield. Consumer writing uses the bare word for both, and a bond fund reporting "a duration of 6" is nearly always reporting the second. The two are close in size, which is precisely why the confusion survives, and they are not the same quantity: one is measured in years and the other in percent per percentage point.

Why maturity is not enough, which is the reason the measure exists at all. A maturity date says when the last payment arrives. It says nothing about how much of the bond's value arrives before then. Two bonds maturing in ten years, one paying a high coupon and one paying almost nothing, are very different investments: the high-coupon bond returns a large share of its value along the way, while the low-coupon bond concentrates almost everything in a single payment ten years out. Money further in the future is more sensitive to the rate used to value it, so the low-coupon bond moves more when rates move. Duration captures that, and a maturity date on its own cannot.

What pushes duration up, and why each one does. A longer maturity extends the whole payment schedule further out. A lower coupon shifts the weight of the schedule toward the final repayment, because less is being paid along the way. A lower yield raises duration too, because when the discount rate is low, distant payments retain more of their value and therefore carry more weight in the average. All three work through the same channel, which is how much of the bond's value sits far in the future.

The one exact case. For a zero-coupon bond, Macaulay duration equals the maturity, because there is only one payment and its weighted average timing can only be the date it arrives. No coupon-paying bond shares that property: every interest payment pulls the average forward, so a coupon bond's duration is always shorter than its maturity. That single fact is the cleanest way to see what duration is measuring.

The estimate is a straight line, and the line is drawn against a curve. Multiplying modified duration by a rate change gives an approximation, and it is a good one for small moves and progressively worse for large ones. The direction of the error is consistent and favors the bondholder. As the MSRB puts it in describing convexity, "prices rise at increasing rates as yields fall and prices decline at decreasing rates as yields rise." So for a large move, duration overstates the loss when rates rise and understates the gain when they fall. Convexity is the name for the second-order term that corrects it, and it is a refinement rather than a different idea.

What duration does not measure. It is a rate-sensitivity number and nothing else. It says nothing about whether the issuer will pay, so a long-duration Treasury and a long-duration bond from a struggling company are identical on this measure and not remotely comparable as investments. It also becomes unreliable on a bond the issuer can repay early, because a call option truncates the payment schedule at a date the issuer chooses. Funds and dealers report an effective duration for those, which models the call rather than ignoring it.

Modified duration is used as a multiplier. Take the number, multiply it by the change in yield in percentage points, and the result is the approximate percentage change in price, in the opposite direction to the rate move.

A hypothetical example. Amara holds $10,000 of a bond with a modified duration of 7.

Yields rise by 1 percentage point. The estimated price change is 7 × 1 = 7 percent, so the value falls by roughly $700, to about $9,300.

Yields fall by 1 percentage point instead. The same arithmetic gives a gain of roughly $700, to about $10,700.

Yields rise by a quarter of a point. The estimate is 7 × 0.25 = 1.75 percent, or about $175.

Now compare a second holding. Bilal has the same $10,000 in a short bond with a modified duration of 2. The same one point rise costs him 2 × 1 = 2 percent, about $200, against Amara's $700. Neither of them chose a different market. They chose a different position on the same one, and duration is the number that says by how much.

Two limits on the arithmetic. It is an approximation that degrades as the rate move gets larger, and it describes the price only. Amara still receives every interest payment she was promised, and still receives face value at maturity if the issuer pays, so the $700 is a real loss on a sale and a change in comparison rather than a loss for a holder who waits.

Pros (of using duration)

• It converts a vague worry about rising rates into a number, and one that can be compared directly across bonds and funds.
• It is additive across a portfolio in the usual case, so a mix of holdings has a duration and it can be aimed at a horizon.
• It exposes the difference between two bonds that look identical on a maturity date, which is the comparison a maturity date hides.
• Funds publish it, so the interest rate risk inside a fund is visible without inspecting what the fund holds.

Cons (and limits of the measure)

• It is an approximation, and the error grows with the size of the rate move.
• It says nothing at all about credit. Two bonds with the same duration can have completely different chances of being paid.
• It assumes yields move by the same amount at every maturity, which real rate moves rarely do.
• It is unreliable on callable bonds unless an effective duration that models the call is used instead.
• The two measures share one word, so a number quoted without saying which one it is invites a misreading, in years or in percent.