FRM Exam Part I · Properties of Interest Rates

Duration and Convexity for FRM Part I

Updated 11 October 2026 · Fact-checked

Duration measures the sensitivity of a bond's price to yield changes; modified duration gives the approximate percentage price change per unit change in yield. Convexity corrects the error from curvature. Estimate the change with ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)². Hedge by matching dollar duration (DV01).

Understand Duration and Convexity

A bond's price falls when yields rise. Duration tells you how fast. It is a weighted average of the times of the bond's cash flows, where each weight is that cash flow's share of the bond's present value. This is Macaulay duration, measured in years. A zero-coupon bond has a Macaulay duration equal to its maturity. Coupons pull duration below maturity.

Modified duration turns Macaulay duration into a price sensitivity. It equals Macaulay duration divided by (1 + y/m), where y is the yield and m is the compounding frequency. A modified duration of 7 means the price falls about 7% for a 1% (100 bp) rise in yield. For continuous compounding, modified and Macaulay duration are the same.

Dollar duration (money duration) is modified duration times price. DV01 is the price change for a one basis point move, about dollar duration × 0.0001. Use DV01 when you compare or hedge positions of different sizes, because it is in currency units.

Duration is a straight-line approximation of a curved price-yield relationship. The curve bends upward for an option-free bond. Convexity measures that bend. Duration alone underestimates the price after a fall in yield and overestimates the loss after a rise. Adding the convexity term fixes most of this error, especially for large yield moves.

Duration assumes a small, parallel shift in yields. For portfolios, duration is the market-value-weighted average of bond durations. For hedging, you choose a position in a hedge instrument so that the total dollar duration (DV01) of the combined position is zero.

Key formulas to remember

Macaulay duration: D_Mac = Σ [t × PV(CF_t)] ÷ P; t in years; PV(CF_t) is the discounted cash flow at time t; P is the bond price.
Modified duration: D_mod = D_Mac ÷ (1 + y/m); y is the annual yield, m is the number of compounding periods per year. With continuous compounding, D_mod = D_Mac.
Price change from duration: ΔP/P ≈ −D_mod × Δy; Δy in decimals: 25 bp = 0.0025.
Convexity: C = (1/P) × d²P/dy² = Σ [t(t + 1/m) × PV(CF_t)] ÷ [P × (1 + y/m)²]; Positive for option-free bonds. Units are years squared.
Duration plus convexity approximation: ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)²; The convexity term is always positive when C is positive.
Dollar duration and DV01: Dollar duration = D_mod × P; DV01 ≈ D_mod × P × 0.0001; DV01 is the price change for a 1 bp rise in yield, quoted as a positive number.
Portfolio duration: D_port = Σ w_i × D_i; w_i is the market value weight. Dollar durations and DV01s add directly.
Zero-coupon bond duration: D_Mac = T; Modified duration is T ÷ (1 + y/m).
Hedge with DV01: N = −DV01_position ÷ DV01_hedge; Choose N units of the hedge instrument so net DV01 is zero. A long bond is hedged by a short position.

How to solve Duration and Convexity questions

Follow this order for most duration and convexity questions. It keeps units and signs straight.

1Identify what is given: Macaulay or modified duration, price, yield, compounding frequency, convexity, and the size of the yield change.
2Convert to modified duration if needed: D_mod = D_Mac ÷ (1 + y/m). Use the correct m.
3Convert the yield change to decimals. 50 bp is 0.005.
4Compute the duration effect: −D_mod × Δy. Keep the sign: a yield rise gives a price fall.
5If convexity is given or the move is large, add ½ × C × (Δy)². Check that it adds to the price.
6Multiply the percentage change by the price to get the currency change. Add it to the original price for the new price estimate.
7For hedging, compute the DV01 or dollar duration of the position and of the hedge instrument, then solve N = −DV01_position ÷ DV01_hedge.
8Sanity check: sign, size and direction. Convexity should help the holder of a positive convexity bond.

Quickest way: Percent change in four lines

When to use it: Use for any multiple-choice question asking for an approximate price change or new price after a yield move.

Write Δy in decimals, for example +0.01.
Duration part: −D_mod × Δy gives the percent change.
Convexity part: 0.5 × C × Δy². For a 1% move this is 0.5 × C × 0.0001, so C = 80 gives +0.40%.
Add the two parts, multiply by the price, and eliminate options with the wrong sign.

Common mistakes in Duration and Convexity

Using Macaulay duration directly in the price change formula.
The two durations look alike and the question may give only one.
Fix: Divide Macaulay duration by (1 + y/m) first. Use that modified duration in ΔP/P ≈ −D_mod × Δy.
Forgetting the minus sign or mixing up the direction.
Students focus on the size of duration and not on its inverse link to price.
Fix: Write the sign before computing. Yield up means price down for an option-free bond.
Entering basis points as whole numbers, such as Δy = 25.
Rushing under time pressure.
Fix: Convert immediately: 25 bp = 0.0025. Convexity needs (Δy)², so an error becomes huge.
Forgetting the ½ in the convexity term.
The formula is memorised as C × (Δy)².
Fix: Remember it comes from the second-order Taylor expansion: ½ × C × (Δy)².
Subtracting the convexity term for a yield rise.
Students think convexity follows the sign of the yield change.
Fix: Convexity adjustment is always positive when C is positive, for both rises and falls in yield.
Hedging with the wrong ratio, for example the ratio of durations instead of DV01s.
Ignoring that the position and the hedge have different prices.
Fix: Use dollar duration or DV01 for both. N = −DV01_position ÷ DV01_hedge.

Worked examples

Example 1

A bond priced at $98.00 per $100 face has a Macaulay duration of 6.30 years and a convexity of 55. It pays semiannual coupons and its yield is 5% with semiannual compounding. Yields rise by 40 bp. Estimate the new price using duration and convexity.

Show the solution

Modified duration = 6.30 ÷ (1 + 0.05/2) = 6.30 ÷ 1.025 = 6.1463.
Δy = 0.0040.
Duration effect = −6.1463 × 0.0040 = −0.024585, or −2.4585%.
Convexity effect = 0.5 × 55 × (0.0040)² = 0.5 × 55 × 0.000016 = 0.00044, or +0.044%.
Total change = −2.4585% + 0.044% = −2.4145%.
Price change = 98.00 × (−0.024145) = −2.366.
New price ≈ 98.00 − 2.366 = 95.634.

Answer: The estimated new price is about $95.63 per $100 face.

Example 2

A portfolio manager holds a bond position worth $50 million with a modified duration of 8.0. She hedges with a futures contract whose DV01 is $850 per contract. Assume parallel yield shifts. How many futures contracts should she short, and what is the position's DV01?

Show the solution

Position DV01 = D_mod × P × 0.0001 = 8.0 × 50,000,000 × 0.0001 = $40,000.
Hedge ratio N = −DV01_position ÷ DV01_hedge = −40,000 ÷ 850 = −47.06.
A negative sign means a short position in the futures.
Rounded to whole contracts, short about 47 contracts.

Answer: The position DV01 is $40,000. Short about 47 futures contracts.

Exam tips

Check whether the question gives Macaulay or modified duration. The exam often gives the wrong one on purpose.
Look at the compounding frequency before dividing by (1 + y/m).
For hedging questions, compute DV01 for both instruments in the same currency units before dividing.
If an option gives a convexity-corrected answer, expect it to be slightly better than the duration-only answer for the holder of a positive-convexity bond. Use this to remove wrong options.
Remember duration is only a good guide for small parallel shifts. Questions on non-parallel moves point to key rate durations.

Practice questions from Properties of Interest Rates

Duration and Convexity in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Duration and Convexity: frequently asked questions

What is the difference between Macaulay duration and modified duration?

Macaulay duration is the weighted average time to receive cash flows, in years. Modified duration is Macaulay duration divided by (1 + y/m) and measures the percentage price sensitivity to a yield change. With continuous compounding they are equal.

How do I calculate the price change using convexity?

Use ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)². Enter Δy in decimals. Multiply the result by the bond price to get the change in currency.

What is DV01 and why does it matter for hedging?

DV01 is the price change for a one basis point change in yield. It is stated in currency, so you can compare a bond with a futures contract and size a hedge so the total DV01 is zero.

Is convexity always positive?

For option-free bonds, yes. Bonds with embedded options, such as callable bonds or mortgage-backed securities, can have negative convexity in some yield ranges.