Duration

The Fed hints at a rate move this afternoon, and the head of the bond desk wants an answer in the next thirty seconds: "if yields jump 25 basis points, how much do we lose?" There is no time to reprice every bond in the book from scratch by summing discounted cash flows off the zero curve again. What the desk needs is a single number per bond that answers exactly this question, to a good first approximation — that number is duration.

Macaulay duration: a weighted-average maturity

Every cash flow a bond pays arrives at its own time t_i, and contributes its own present value PV_i = C_i e^{-y t_i} to the bond's price B (here y is the bond's yield — we treat it as one flat rate for this lesson, for a bond priced off a fairly flat stretch of curve). Macaulay duration is the present-value-weighted average of those payment times — literally, "on average, how long do you wait for your money back?"

Macaulay duration: \displaystyle D = \sum_{i=1}^n t_i\,\frac{PV_i}{B}, \qquad PV_i = C_i\,e^{-y t_i} — a weighted average of payment times, in years, with weights summing to 1.
Under continuous compounding, D is exactly the (negative, proportional) sensitivity of price to yield: \displaystyle \frac{dB}{dy} = -\sum_i t_i C_i e^{-y t_i} = -D\,B.
So the first-order approximation to a price change from a small yield move \Delta y is \displaystyle \frac{\Delta B}{B} \approx -D\,\Delta y.
Under the market's usual m-times-a-year compounding convention, the same sensitivity needs a small correction, called modified duration: \displaystyle D^{*} = \frac{D}{1 + y/m}, with \Delta B/B \approx -D^{*}\Delta y. Continuous compounding is exactly the m \to \infty limit, where D^{*} = D — one more reason this course works in continuous compounding throughout.

Two useful sanity checks before the worked example: a zero-coupon bond has only one cash flow, so its Macaulay duration is trivially its own maturity, D = T. A coupon-paying bond's duration is always shorter than its maturity, because some cash (the coupons) arrives earlier and pulls the weighted average in. Higher coupons, or higher yields, both shorten duration for the same reason — more weight moves to the earlier, closer-in cash flows.

Worked example: duration of a 2-year bond, and a price-move estimate

Take a 2-year bond, face $100, a 6% coupon paid semiannually ($3 every six months), yielding y = 6.5\% (continuously compounded, flat across its short life). Build the duration table cash flow by cash flow:

t_i	C_i	PV_i = C_i e^{-y t_i}	t_i \times PV_i
0.5	3	2.904	1.452
1.0	3	2.811	2.811
1.5	3	2.721	4.082
2.0	103	90.444	180.889
Sum		98.881	189.234

The price is B = 98.881 (the sum of the PV_i column), and the Macaulay duration is the ratio of the two sums:

D = \frac{189.234}{98.881} \approx 1.914 \text{ years}.

Sensible: shorter than the 2-year maturity, because roughly $9 of the $118 total cash paid arrives before the end. Since we are already in continuous compounding, modified duration equals this same number, D^* = 1.914. Now use it: estimate the price change if the yield jumps by \Delta y = +1\% (100 basis points):

\frac{\Delta B}{B} \approx -D\,\Delta y = -1.914 \times 0.01 = -1.914\%,

\Delta B \approx -0.01914 \times 98.881 \approx -\$1.89, \qquad B_{\text{new}} \approx \$96.99.

For a move this small, that estimate is very close to the true reprice — the desk's thirty-second answer is essentially the right one. Whether it stays that close for a much larger move is the subject of the next lesson.

Duration was introduced in 1938 by the economist Frederick Macaulay, in a study of long-run interest-rate movements in the United States — decades before it became the risk-management tool a trading floor lives by every day. Macaulay's original motivation was almost philosophical: a bond's quoted maturity is a poor summary of "how long" your money is really tied up, since a low-coupon, long-dated bond returns almost nothing until the very end, while a high-coupon bond of the same maturity returns a good chunk of its value much sooner. His weighted-average-time measure sat as a theoretical curiosity for decades. It took until the 1970s, once interest-rate volatility turned bond-price risk into something desks actively had to manage, for someone to notice Macaulay's number was, algebraically, exactly the price-sensitivity a hedger needed — and duration became indispensable practically overnight.

Two classic slips:

Macaulay vs. modified. Macaulay duration is measured in years — a weighted average time. Modified duration is a sensitivity — percent price change per unit yield change — and is what actually belongs in the hedging formula \Delta B/B \approx -D^*\Delta y. They are numerically close (and under continuous compounding, identical), but conceptually distinct, and using the wrong one in the wrong place is a common, silent error.
Trusting duration for a large move. Duration is a tangent line — a first-order (linear) approximation to a curve that is not actually a straight line. For a small rate move it is excellent; for a large one it visibly understates how much a bond price actually rises when yields fall, and overstates how much it falls when yields rise. That gap is not a flaw to be ignored — it is exactly what the next lesson measures and corrects for.