Duration: Macaulay, Modified & DV01

You can price a bond exactly by discounting every cash flow — but if a trader asks “how much do I lose if yields jump 25 basis points?”, re-pricing the whole stream is slow and tells you nothing portable. Duration is the answer: a single number that captures a bond’s sensitivity to interest rates. It wears three closely related costumes — Macaulay duration (a time), modified duration (a percentage), and DV01 (a dollar amount) — and a desk fluently switches between all three. Master them and you can size, compare, and hedge rate risk in your head.

Before you read — take a guess

Which bond is MORE sensitive to a change in interest rates?

They are equally sensitive — maturity does not matterA 30-year zero-coupon bondThe one with the higher face valueA 2-year bond with a high coupon

Macaulay duration: the cash-flow balance point

Analogy. Lay a bond’s cash flows out along a time axis as little weights — each weight is the present value of that payment. Macaulay duration is the balance point of that seesaw: the single point in time where the cash-flow weights tip exactly even. Small early coupons sit near the fulcrum; the giant final coupon-plus-face sits way out at maturity and drags the balance point toward it.

The definition. Macaulay duration is the present-value-weighted average time to receive the bond’s cash flows:

D_{\text{Mac}} = \frac{\sum_{n=1}^{N} t_n \cdot \dfrac{CF_n}{(1+y)^{n}}}{\sum_{n=1}^{N} \dfrac{CF_n}{(1+y)^{n}}} = \frac{\sum_{n=1}^{N} t_n \cdot PV_n}{P},

where $t_n$ is the time (in years) of cash flow $n$ and $P$ is the bond’s price. The denominator is just the price; the numerator weights each cash flow’s time by its present value. It’s measured in years.

Macaulay duration: where the cash flows balanceMacaulay duration: 8.11 yr

Present value of each cash flowMacaulay duration

Coupon rate5.00%

Maturity (years)10

Market yield5.00%

Macaulay duration: 8.11 yr
Modified duration: 7.72 yr
Price change per +1% yield: -7.72%

Each bar is the present value of a cash flow; the triangle sits where they balance — that's Macaulay duration. Raise the coupon and weight shifts earlier, sliding the fulcrum left. A zero-coupon bond balances exactly at its maturity.

Worked example. A 3-year, 6% annual-coupon, $1,000 bond at a 6% yield (so it prices at par, $1,000). Cash flows are $60, $60, $1,060.

Year $t_n$	Cash flow	$PV_n = CF_n/1.06^n$	$t_n \cdot PV_n$
1	$60	$56.60	$56.60
2	$60	$53.40	$106.80
3	$1,060	$890.00	$2,670.00
		$1,000.00	$2,833.40

D_{\text{Mac}} = \frac{2{,}833.40}{1{,}000.00} \approx 2.83 \text{ years}.

The balance point is 2.83 years, not 3 — the early coupons pull it in from maturity. A zero-coupon bond, with no early coupons, would balance exactly at its maturity: its Macaulay duration equals its term.

Think first

A 10-year zero-coupon bond and a 10-year 8%-coupon bond — which has the longer Macaulay duration, and what is the zero's duration?

Hint: A zero has only one cash flow, at maturity. Coupons pull the balance point earlier.

Modified duration: the percentage price sensitivity

Analogy. Macaulay duration tells you where the weight sits in time; modified duration translates that into the answer to the trader’s real question — “what percent does my price move per 1% change in yield?” It’s the steering ratio: turn the yield wheel by a degree, and modified duration tells you how far the price swerves.

The definition. Modified duration adjusts Macaulay duration by one factor of the periodic yield:

D_{\text{mod}} = \frac{D_{\text{Mac}}}{1 + y/m},

and it gives the first-order percentage price change for a small yield move $\Delta y$ :

\frac{\Delta P}{P} \approx -D_{\text{mod}} \cdot \Delta y.

The minus sign encodes the seesaw: yields up, price down. Modified duration is a percentage per unit of yield — usually quoted as ”% per 1% (100 bp) move.”

Worked example. Our 3-year bond had $D_{\text{Mac}} = 2.83$ years and $y = 6\%$ annual ( $m = 1$ ):

D_{\text{mod}} = \frac{2.83}{1.06} \approx 2.67.

So if the yield rises by 50 bp ( $\Delta y = +0.005$ ):

\frac{\Delta P}{P} \approx -2.67 \times 0.005 = -0.01335 = -1.335\%.

On a $1,000 bond that’s a drop of about $13.35. The bond is now worth roughly $986.65 — a number you got without re-pricing a single cash flow. That’s the power of duration: instant, portable rate-risk.

Macaulay vs. modified — don't mix them up

Macaulay duration is a time (years); modified duration is a sensitivity (% per unit yield). They differ only by the $(1+y/m)$ factor, so for low yields they’re nearly equal — which is exactly why people sloppily say “duration” and let context sort it out. But the units are different. When someone says “this bond has a duration of 7,” they almost always mean modified duration: a 1% yield move shifts the price about 7%.

Fill in the duration relationships.

Pick the right option for each blank, then check.

Macaulay duration is measured in , while modified duration measures the price change per unit of yield. Modified duration equals Macaulay duration divided by , and the approximate percentage price change is .

DV01: the dollar value of one basis point

Analogy. Modified duration speaks in percentages, but a trading desk’s P&L is in dollars. DV01 — the dollar value of an 01 (one basis point) — converts duration into the only currency a P&L cares about: how many dollars you gain or lose if the yield moves a single basis point (0.01%). It’s the atomic unit of rate-risk on a desk.

The definition. DV01 (also called PV01 or the dollar duration of a basis point) is

\text{DV01} = D_{\text{mod}} \times P \times 0.0001,

i.e. the modified-duration percentage move applied to the dollar price, scaled to one basis point. It answers: “one basis point — how many dollars?”

Worked example. Our 3-year bond: $D_{\text{mod}} = 2.67$ and a price of $1,000 ( $P = 1000$ ).

\text{DV01} = 2.67 \times 1{,}000 \times 0.0001 = \$0.267 \text{ per basis point}.

Per $1,000 of face, you lose about 26.7 cents for every basis point yields rise. Now scale it: hold $10 million of this bond (10,000 of them) and your book’s DV01 is $10{,}000 \times 0.267 = 2670$ — about $2,670 per bp. A 25 bp sell-off costs you about $25 \times 2{,}670 = 66{,}750$ , i.e. $66,750. That’s the number a risk manager actually trades on.

A bond has a modified duration of 8 and a price of $1,200. Its DV01 (per bond) is closest to:

About $9.60About $96About $0.96About $0.10

What drives duration: maturity, coupon, and yield

Three levers move a bond’s duration, and a desk knows them in its sleep:

Maturity ↑ → duration ↑. Cash flows sit further out, so the balance point and the sensitivity both stretch. A 30-year bond is far more rate-sensitive than a 2-year.
Coupon ↑ → duration ↓. Bigger coupons front-load value, pulling the balance point earlier. A high-coupon bond gets more of its value back sooner, so it’s less sensitive. The zero-coupon bond is the extreme: no early coupons, longest duration for its maturity.
Yield ↑ → duration ↓. A higher discount rate shrinks the far cash flows relatively more, pulling weight toward the present. So as yields rise, duration falls modestly.

What moves the balance pointMacaulay duration: 8.11 yr

Present value of each cash flowMacaulay duration

Coupon rate5.00%

Maturity (years)10

Market yield5.00%

Macaulay duration: 8.11 yr
Modified duration: 7.72 yr
Price change per +1% yield: -7.72%

Crank the coupon and watch the fulcrum slide LEFT (shorter duration). Stretch maturity and it slides RIGHT (longer). Raise the yield and it creeps left as far cash flows shrink. Three levers, one balance point.

Match each duration measure to what it is and its units.

Pick a term, then click its definition.

Portfolio duration and the limits of the measure

Portfolio duration adds up — by market value. The modified duration of a portfolio is the market-value-weighted average of its bonds’ durations:

D_{\text{port}} = \sum_i w_i D_i, \qquad w_i = \frac{\text{market value of bond } i}{\text{total portfolio value}}.

Worked example. $6M in a bond with duration 3 and $4M in a bond with duration 9. Weights are 0.6 and 0.4:

D_{\text{port}} = 0.6 \times 3 + 0.4 \times 9 = 1.8 + 3.6 = 5.4.

The book behaves like a single bond of duration 5.4 — and its DV01 is just the sum of the two bonds’ DV01s. This additivity is what makes duration the lingua franca of rate-risk: you can roll a thousand positions into one number.

Duration is a straight-line approximation — it lies for big moves

Modified duration is the tangent line to the convex price–yield curve. For small yield moves it’s superb; for large ones it drifts off, because it ignores the curve’s bow. Specifically, for a big yield rise duration overstates the loss, and for a big yield fall it understates the gain — always erring against the holder’s actual (better) outcome. The fix is convexity, the second-order correction, which is the very next lesson. Until then, trust duration for small wiggles and distrust it for shocks.

Duration is the tangent; the real curve bows away from itPar

PricePar

Market yield5.0%

Price: 100.00%; $1,000.00
Market yield: 5.0%
Current yield: 5.00%

New bonds yield 5.0%, matching your 5% coupon — so your bond trades at par.

Duration is the straight slope of this curve at today's yield. It's a great local guide, but the price genuinely rides the curve — and the gap between the line and the curve is convexity, the subject of the next lesson.

A portfolio holds $8M of a duration-4 bond and $2M of a duration-14 bond. Its portfolio modified duration is:

4.56189

When to use which

Reach for Macaulay duration when you care about timing — matching a bond to a liability date, or building intuition about where a bond’s weight sits. Reach for modified duration for percentage sensitivity and for comparing rate risk across bonds of different prices. Reach for DV01 when you’re on a desk thinking in dollars — sizing a hedge, computing P&L for a rate move, or aggregating risk across a book. They’re three dialects of one language; fluency means switching without thinking.

Putting it together

Duration is the single most useful number in fixed income. Macaulay duration locates a bond’s cash-flow balance point in years; modified duration converts that to a percentage price move per unit of yield ( $\Delta P/P \approx -D_{\text{mod}}\Delta y$ ); and DV01 cashes it out to dollars per basis point ( $D_{\text{mod}} \cdot P \cdot 0.0001$ ). Longer maturity lengthens duration, bigger coupons and higher yields shorten it, and portfolio duration is just a market-value-weighted average. The one caveat — duration is a straight-line approximation that drifts for large moves — is exactly the gap convexity fills next.

Big picture

Duration — three costumes, one idea

Duration
- Macaulay
  - PV-weighted average time
  - Measured in years
  - Balance point of cash flows
  - Zero-coupon: = maturity
- Modified
  - D_mod = D_Mac / (1 + y/m)
  - ΔP/P ≈ −D_mod·Δy
  - % price move per unit yield
  - "Duration of 7" usually means this
- DV01
  - = D_mod · P · 0.0001
  - Dollars per basis point
  - The desk’s atomic risk unit
  - Scales linearly with position size
- What drives it
  - Maturity ↑ → duration ↑
  - Coupon ↑ → duration ↓
  - Yield ↑ → duration ↓
- Portfolio & limits
  - D_port = Σ wᵢDᵢ (value-weighted)
  - DV01s simply add up
  - Straight-line: lies for big moves
  - Convexity is the fix

Time (Macaulay), percentage (modified), and dollars (DV01) — all measuring how hard a bond's price reacts to a yield wiggle, with maturity, coupon and yield as the dials.

Recap: duration

Question 1 of 50 correct

Macaulay duration is best described as:

The number of coupons remainingThe present-value-weighted average time until the bond’s cash flows arrive, in yearsThe dollar loss per basis pointThe percentage price change for a 1% yield move

Check your answer to continue.

Next — convexity — we add the curve back in: the second-order term that corrects duration’s straight-line lie, why it always works in the holder’s favour, and how to price two bonds with identical duration but different convexity.