Duration and Convexity: FRM Part 1 Notes with Worked Examples [2026]

Duration and convexity are the two numbers that tell you how a bond price reacts when yields move. Duration gives the first estimate: a straight-line approximation of the percentage price change for a small yield change. Convexity corrects that estimate for the curve in the price-yield relationship, and the correction grows quickly when the yield move is large. FRM Part 1 tests this topic from every angle: Macaulay duration from raw cash flows, modified duration and DV01, effective duration for bonds with embedded options, and the combined duration-convexity formula under time pressure. This note works through each calculation with full arithmetic, then closes with the exam traps and a one-page revision box.

Pressed for study time? Build a weekly plan with our free study timetable generator and slot these notes into your FRM schedule. This note is one of three new calculation-heavy FRM Part 1 notes, alongside option greeks explained and credit risk expected loss.

Table of Contents

• Duration and convexity: the definitions examiners expect
• Macaulay duration, calculated from raw cash flows
• Modified duration, dollar duration and DV01
• Effective duration and convexity for bonds with options
• Duration and convexity together: the price-change formula
• Portfolio duration in one line
• Duration and convexity: exam traps that cost marks
• One-page revision: duration and convexity
• Frequently asked questions
• Wrapping up

Duration and convexity: the definitions examiners expect

Duration measures the weighted average time to a bond cash flows and, in modified form, the percentage price change for a 1 percent yield move. Convexity measures how duration itself changes as yields move, correcting the straight-line duration estimate for the curved price-yield relationship between bond prices and yields.

Plot a bond price against its yield and you get a smooth, downward-sloping curve that bends toward the yield axis. Duration is the slope of that curve at the current yield. Convexity is the curvature. Because the curve bends, the straight-line duration estimate always misses after a large move: when yields fall, the actual price rises by more than duration predicts, and when yields rise, the actual price falls by less. That asymmetry is why investors like positive convexity: the holder gains more on a yield fall than the holder loses on an equal yield rise.

Three facts anchor the topic. Duration is quoted in years but used as a sensitivity. Convexity is a second-order effect, small for tiny moves and decisive for large ones. And both measures assume fixed cash flows, which is why bonds with embedded options need the effective versions below.

Macaulay duration, calculated from raw cash flows

Macaulay duration is the present-value-weighted average time at which the bond pays its cash flows:

Macaulay duration = sum of (t x w_t), where w_t = PV of the cash flow at time t, divided by the bond price.

Worked example. A 3-year bond pays a 6% annual coupon on a face value of USD 1,000, and the yield to maturity is 5%. The cash flows are USD 60, USD 60 and USD 1,060.

Year	Cash flow (USD)	PV at 5% (USD)	Weight	Year x weight
1	60	57.14	0.0556	0.0556
2	60	54.42	0.0530	0.1060
3	1,060	915.67	0.8914	2.6742
Total		1,027.23 (price)	1.0000	2.8358

Macaulay duration = 2.84 years. Sanity checks the examiners love: the answer sits below the 3-year maturity because coupons arrive earlier, and the year-3 weight dominates because the principal dwarfs the coupons. Two rules follow. A zero-coupon bond has Macaulay duration exactly equal to its maturity, since all the weight sits on one cash flow. And Macaulay duration rises with maturity, and falls with a higher coupon and a higher yield.

Modified duration, dollar duration and DV01

Modified duration converts the time measure into a price sensitivity by adjusting for the yield:

Modified duration = Macaulay duration / (1 + y)

For the bond above: 2.8358 / 1.05 = 2.70. Read it as a percentage: if the yield rises by 1%, the price falls by approximately 2.70%; if the yield falls by 1%, the price rises by approximately 2.70%, before the convexity correction. For semiannual bonds, run the same logic per period: divide the Macaulay duration in periods by (1 + y/2), and check whether the question wants the answer in periods or years.

Dollar duration scales the percentage into money:

Dollar duration = modified duration x price

Here: 2.7007 x 1,027.23 = USD 2,774. That is the approximate dollar price change for a full 1.00 (100 percentage point) yield move, which is an awkward unit, so practitioners divide it down to a single basis point:

DV01 = modified duration x price x 0.0001

DV01 for this bond = 2,774 x 0.0001 = USD 0.28 per basis point, per bond. Scale it up and DV01 becomes the risk number on every fixed income desk. A USD 10 million portfolio with modified duration 6.5 has DV01 = 10,000,000 x 6.5 x 0.0001 = USD 6,500 per basis point, so a 25 basis point rise in yields costs roughly 6,500 x 25 = USD 162,500. Examiners test exactly this chain: modified duration, then DV01, then a scenario loss.

Effective duration and convexity for bonds with options

Macaulay and modified duration assume the cash flows never change. For a callable bond, a putable bond or a mortgage-backed security, that assumption fails: when yields fall, the call moves into the money and the expected maturity shortens. The fix is to shock the yield up and down, reprice the bond each way with a proper model, and read the sensitivity off the three prices:

Effective duration = (P_– – P₊) / (2 x P₀ x delta y)

Effective convexity = (P₊ + P_– – 2 x P₀) / (P₀ x (delta y)²)

Worked example. A bond trades at 100. A 50 basis point fall in yield reprices it to 102.40, and a 50 basis point rise reprices it to 97.90, so delta y = 0.005.

Effective duration = (102.40 – 97.90) / (2 x 100 x 0.005) = 4.5 / 1.0 = 4.50.

Effective convexity = (97.90 + 102.40 – 200) / (100 x 0.005²) = 0.30 / 0.0025 = 120.

The convexity is positive but modest, typical near a call price. When a callable bond trades close to its call price, the price-yield curve flattens and can bend the other way: effective convexity turns negative, gains from falling yields get capped, and duration shortens as yields fall. Negative convexity in callable bonds and MBS is a recurring FRM exam theme.

Duration and convexity together: the price-change formula

The full second-order estimate combines both measures:

Percentage price change = -modified duration x delta y + 0.5 x convexity x (delta y)²

Worked example. A bond trades at USD 950 with modified duration 7.2 and convexity 68. Yields rise by 1.5%, so delta y = +0.015.

Duration alone: -7.2 x 0.015 = -0.108, a fall of 10.8%. Estimated price = 950 x 0.892 = USD 847.40.

Convexity adjustment: 0.5 x 68 x 0.015² = 0.5 x 68 x 0.000225 = +0.765%.

Combined: -10.8% + 0.765% = -10.035%. Estimated price = 950 x 0.89965 = USD 854.67.

Duration alone overstated the loss by about USD 7.27 per bond. Run the move the other way: if yields fall by 1.5%, duration alone predicts +10.8%, while the combined estimate gives +10.8% + 0.765% = +11.565%, a price of USD 1,059.87. The gain of 11.57% beats the loss of 10.04% for the same yield move. That gap is the convexity benefit, and it is why, of two bonds with equal duration, the one with higher convexity is worth more.

Portfolio duration in one line

Portfolio duration is the market-value-weighted average of the individual durations:

Portfolio duration = sum of (weight_i x duration_i), with weights based on market value, never face value.

Worked example. A portfolio holds USD 4 million (market value) of bonds with duration 5 and USD 6 million with duration 8. Portfolio duration = (4 x 5 + 6 x 8) / 10 = (20 + 48) / 10 = 6.8. One warning: a single portfolio duration only describes parallel shifts, where every yield moves by the same amount. For steepening or flattening curves, the exam expects key rate durations, which shock one maturity point at a time.

Duration and convexity: exam traps that cost marks

• Wrong duration in the formula. The price-change formula takes modified duration (or effective duration). Macaulay goes in only after dividing by (1 + y).
• DV01 scaling errors. One basis point is 0.0001. Forgetting the factor, or applying it twice, is the most common arithmetic slip here.
• Sign of the convexity term. For an option-free bond the convexity adjustment is always positive, whichever way yields move.
• Zero-coupon shortcuts. Macaulay duration equals maturity for a zero, but modified duration is still maturity divided by (1 + y), never maturity itself.
• Semiannual periods. With semiannual coupons, durations run per half-year period. Check which unit the answer choices use.
• Large moves, straight lines. Duration alone is a tangent line. For moves beyond roughly 100 basis points, an answer that skips convexity is usually the distractor.
• Fixed cash flow assumption. Callable bonds and MBS need effective duration. Plain modified duration on a callable bond near its call price overstates the upside.

Duration also feeds the interest-rate side of portfolio risk: a DV01 figure converts straight into a loss estimate under a yield shock, which is the same logic scaled up in our FRM value at risk cheat sheet.

One-page revision: duration and convexity

Measure	Formula	What it tells you
Macaulay duration	Sum of (t x wt), wt = PV(CFt) / price	Weighted average time to the cash flows, in years
Modified duration	Macaulay / (1 + y)	Approx. % price change for a 1% yield change
Dollar duration	Modified duration x price	Dollar change for a 1.00 yield change
DV01 (PVBP)	Modified duration x price x 0.0001	Dollar change for one basis point
Effective duration	(P– – P+) / (2 x P0 x delta y)	Sensitivity when cash flows change with yields
Effective convexity	(P+ + P– – 2P0) / (P0 x (delta y)2)	Curvature from shocked prices
Price change	-MD x delta y + 0.5 x C x (delta y)2	Second-order price estimate
Portfolio duration	Sum of (market-value weight x duration)	Whole-book sensitivity, parallel shifts only

• Zero-coupon bond: Macaulay duration = maturity.
• Duration rises with maturity; it falls with coupon and with yield.
• Positive convexity: gains from a yield fall exceed losses from an equal yield rise.
• Callable bonds and MBS can show negative convexity near the call price.
• DV01 of the example portfolio (USD 10m, MD 6.5) = USD 6,500 per basis point.

Frequently asked questions

What is the difference between Macaulay and modified duration?

Macaulay duration is a time measure: the present-value-weighted average years until the bond cash flows arrive. Modified duration is a sensitivity measure: Macaulay divided by (1 + y), read as the approximate percentage price change for a 1% yield change. The price-change formula always uses modified (or effective) duration.

Why does convexity benefit the bond holder?

Because the price-yield relationship is curved, not straight. With positive convexity, the price rises more when yields fall than it drops when yields rise by the same amount. In the worked example above, a 1.5% yield fall produced an 11.57% gain while a 1.5% rise produced only a 10.04% loss.

What is DV01 in plain words?

DV01 is the dollar value of one basis point: the approximate change in a bond or portfolio value when yields move by 0.01%. It converts an abstract duration number into money at risk, which is why trading desks quote DV01 rather than duration.

Does a higher coupon raise or lower duration?

It lowers duration. A higher coupon pulls more cash flow into the early years, so the weighted average time to the cash flows shortens. Higher yield lowers duration for the same reason: distant cash flows get discounted more heavily.

When is the duration-only estimate good enough?

For small yield moves, roughly up to 25 to 50 basis points on a typical bond, the convexity term is tiny and duration alone is close. For large moves or high-convexity bonds, the duration-only answer is systematically wrong in a predictable direction: it overstates losses and understates gains.

Duration and convexity: wrapping up

The reliable exam routine is: build the cash flow table, weight by present value for Macaulay duration, divide by (1 + y) for modified duration, scale by price and 0.0001 for DV01, and add the convexity term whenever the yield move is large. Keep going with the other notes in this series: option greeks explained covers delta hedging and gamma, and credit risk expected loss covers PD, LGD and EAD. The full set lives on the FRM notes hub.

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