Duration: How Much Bonds Move

You already know bonds move opposite to yields: when market interest rates rise, the price of a bond you’re holding falls. But by how much? Two bonds can both drop when rates climb — one barely flinches, the other craters. The single number that tells you which is which is called duration, and despite being quoted in years it’s really a measure of price sensitivity. It’s the bond market’s answer to “how hard will this thing whip around when rates move?” — and once you can read it, you can size up a bond’s risk at a glance. Let’s find the balance point.

Duration as the balance point of your cash flows

Picture a bond’s cash flows laid out on a see-saw. The horizontal plank is time, running from today out to maturity. At each date the bond pays you something — a small coupon (the periodic interest payment) here, another coupon there, and at the very end the big lump of your face value (the principal repaid at maturity) plus the final coupon. Stack a weight at each payment date sized by how valuable that payment is today. Where does the plank balance? That balance point is the Macaulay duration.

Formally, Macaulay duration is the present-value-weighted average time, in years, until you receive the bond’s cash flows. Each future payment is discounted back to its present value (what it’s worth today), those present values become the weights, and you take the weighted-average of the payment times:

$D_{\text{Mac}} = \sum_{t} t \cdot w_t, \qquad w_t = \frac{\text{PV of cash flow at time } t}{\text{total price}}$

The weights $w_t$ add up to 1 (they’re shares of the bond’s total value), so the result lands somewhere between today and maturity — exactly where the cash-flow see-saw balances.

The big idea hiding in that pretest: a bond that pays you sooner has a shorter duration, because early cash flows tug the balance point left. Maturity is just the latest possible date; duration is the average date your money actually shows up, weighted by value.

When it matters

Whenever you need to compare the risk of two bonds with different coupons or maturities, duration — not maturity — is the honest yardstick. A 30-year bond with huge coupons can have a shorter duration (and so less rate risk) than a 15-year zero-coupon bond. If you ranked them by maturity you’d get the risk ordering backwards.

Modified duration: turning years into a percent

Macaulay duration is a lovely “average time,” but you can’t use a number in years to predict a price move directly. You need to convert it into a sensitivity. The analogy: Macaulay duration is the length of the wrench; modified duration is how much torque one turn of the rate dial actually applies.

Modified duration rescales Macaulay duration by the per-period yield, and it’s the number that actually predicts price moves:

$D_{\text{mod}} = \frac{D_{\text{Mac}}}{1 + y/n}$

where $y$ is the annual yield (the bond’s market return) and $n$ is the number of coupon payments per year. The division shrinks it slightly below the Macaulay figure. Then comes the rule that makes the whole lesson worth it:

$\%\Delta\text{Price} \approx -D_{\text{mod}} \times \Delta y$

Read it out loud: the percentage change in price is roughly minus the modified duration times the change in yield. The minus sign is the price-yield seesaw (rates up, price down). The size of the move is the modified duration.

Worked example — modified duration 7

Suppose a bond has a modified duration of 7. Yields jump by 1 percentage point, so $\Delta y = 0.01$:

$\%\Delta\text{Price} \approx -7 \times 0.01 = -0.07 = -7\%$

The price falls about 7%. On a $1,000 bond that’s a drop of roughly $70 (that’s $0.07 \times 1000$), leaving you near $930. Flip the move — yields fall 1% — and the price rises about 7%, to roughly $1,070. A duration of 7 means “expect about 7% of price movement for every 1% the yield moves.” That’s the entire skill: read the duration, multiply by the rate move, get the price hit.

When it matters

Modified duration is what every bond desk actually quotes, because it answers the only question a holder cares about: if rates move, how much do I make or lose? Use it for small, realistic rate moves (a quarter-point, a half-point, a point). For large shocks, treat it as a first guess that overstates the loss and understates the gain — and reach for convexity.

Working it out: a 3-year, 5% bond, step by step

Time to actually grind the see-saw arithmetic. Take a tiny bond so every number is visible: face value $100, a 5% annual coupon (so $5 each year), 3 years to maturity, and a market yield of 5%. Because the coupon rate equals the yield, this bond is priced exactly at par — $100 — which keeps the bookkeeping clean.

Each year’s cash flow gets discounted by $(1.05)^t$ to its present value, each present value becomes a weight $w_t = \text{PV} / 100$, and we sum $t \cdot w_t$:

The present values total $100 (the price — good, the math is consistent), the weights total 1, and the weighted-time column sums to about 2.86 years. So:

$D_{\text{Mac}} \approx 2.86 \text{ years}$

Notice it’s less than the 3-year maturity — the two early $5 coupons drag the balance point left of the final payment, but only a little, because that year-3 cash flow (coupon + face) carries over 90% of the weight. Now convert to modified duration with $y = 0.05$ and $n = 1$:

$D_{\text{mod}} = \frac{2.86}{1 + 0.05/1} = \frac{2.86}{1.05} \approx 2.72$

And put it to work. If yields rise from 5% to 6% ($\Delta y = 0.01$):

$\%\Delta\text{Price} \approx -2.72 \times 0.01 = -2.72\%$

The $100 bond should fall to about $97.28. (Discount the real cash flows at 6% and you’d get $97.33 — the straight-line estimate is off by a nickel, exactly the small-move accuracy we promised.)

The four drivers — what lengthens and shortens duration

Duration isn’t a mystery dial; four levers move it in predictable directions. Here’s the whole cause-and-effect map in one table, then we’ll feel each one on the see-saw.

The headline cases:

• A zero-coupon bond has duration equal to its maturity. One cash flow, at the end — the see-saw has a single weight, so it balances right under that weight. A 10-year zero has a Macaulay duration of exactly 10 years. It’s the most rate-sensitive bond of its maturity, because nothing arrives early to soften the blow.
• A higher coupon shortens duration. Fat coupons deliver more cash up front, pulling the balance point toward today. Same maturity, more coupon, shorter duration, less price sensitivity.
• A longer maturity lengthens duration — the giant final face-value payment sits further out, pulling the balance point right.
• A lower yield lengthens duration. Lower discounting means far-off cash flows lose less value, so they carry more weight, tugging the balance point outward.

Drag the sliders below. Crank the coupon up and watch the triangle (the balance point) slide left as duration shortens; drop the coupon toward zero and watch it slide all the way out to maturity. Push maturity out or pull the yield down and the balance point drifts right:

When it matters

These drivers are how you engineer rate risk. Scared of rising rates? Shorten duration — hold shorter maturities or higher coupons. Betting rates will fall and want to maximize the price pop? Lengthen duration — long maturities, low or zero coupons. Pension funds and insurers go further and match the duration of their bonds to the duration of their future payouts, so a rate move hits both sides equally and cancels out.

Putting it together

One number — duration — collapses a bond’s coupons, maturity and yield into a single “how much will this move?” Chunk the whole idea:

A mixed recap pulling from everything above:

Key Takeaways