FRM Exam Part I · Modeling Non-Parallel Term Structure Shifts and Hedging

Duration, DV01 and Convexity Review for FRM Part I

Updated 11 October 2026 · Fact-checked

Duration, DV01 and convexity measure how a bond's price reacts to a parallel yield shift. Modified duration gives the percentage price change per unit yield change. DV01 gives the money change per 1 basis point. Convexity corrects the second-order error. All three assume every yield moves by the same amount.

Understand Duration, DV01 and Convexity Review

A bond's price falls when yields rise. The price-yield curve is not a straight line. It bends. Duration, DV01 and convexity describe that curve around the current yield.

Modified duration is the first-order measure. It tells you the approximate percentage price change for a 1.00 (100%) change in yield, with the sign reversed. A modified duration of 7 means a 1% (100 bp) rise in yield cuts the price by about 7%.

DV01 (dollar value of a basis point) turns that into money. It is the price change for a 1 bp yield change. It equals modified duration × price × 0.0001. It is the number traders use to size hedges.

Convexity is the second-order measure. The price-yield curve is convex, so duration overstates losses when yields rise and understates gains when yields fall. Adding the convexity term improves the estimate, especially for large yield moves.

All three measures share one limit. They assume a parallel shift: every point on the yield curve moves by the same amount. Real curves steepen, flatten and twist. Two portfolios with the same DV01 can behave very differently when the curve changes shape. That is why this chapter moves on to key rate durations and multi-factor hedging.

Key formulas to remember

Modified duration from Macaulay duration: D_mod = D_Mac ÷ (1 + y/m); y is the annual yield, m is compounding periods per year. With continuous compounding, modified duration equals Macaulay duration.
First-order price change: ΔP ÷ P ≈ −D_mod × Δy; Δy in decimals. A 25 bp move is 0.0025.
DV01: DV01 = D_mod × P × 0.0001; Quoted as a positive number per 1 bp. The price falls by this amount when yield rises 1 bp. Here P is the full (dirty) price of the position.
Duration and convexity approximation: ΔP ÷ P ≈ −D_mod × Δy + ½ × C × (Δy)²; C is convexity. Convexity is positive for plain bonds, so the second term adds to price.
Portfolio duration: D_p = Σ (w_i × D_i); w_i are market value weights. Portfolio DV01 is simply the sum of position DV01s.
DV01 hedge ratio: N = −DV01_target ÷ DV01_hedge; N is the number of hedge instrument units. The negative sign means take the opposite position.

How to solve Duration, DV01 and Convexity Review questions

Use this order for any question on price sensitivity to a parallel yield shift.

1Identify what is asked: percentage change, money change, DV01, or a hedge size.
2Write down the price P, the duration type given (Macaulay or modified) and the yield change Δy.
3Convert to modified duration if you were given Macaulay duration, using the compounding frequency.
4Convert Δy to decimals: 1 bp = 0.0001, 1% = 0.01.
5Apply the first-order formula. Add the convexity term if the yield move is large or convexity is given.
6For DV01, multiply modified duration by price (or position value) by 0.0001.
7For hedging, divide the target DV01 by the hedge DV01 and take the opposite position.
8Check the sign: rising yields lower prices. Then note any assumption that the shift is parallel.

Quickest way: DV01 shortcut for sizing and checking

When to use it: Use when the question asks for a money change for a small yield move or a hedge size.

Compute DV01 once: D_mod × value × 0.0001.
Multiply DV01 by the number of basis points moved.
Skip convexity for moves under about 25 bp unless the option gives a convexity figure and asks for it.
For hedges, divide DV01s. Do not recompute prices.
Eliminate options with the wrong sign before calculating fully.

Common mistakes in Duration, DV01 and Convexity Review

Using Macaulay duration directly in the price change formula.
The two durations sound alike and both are in years.
Fix: Divide Macaulay duration by (1 + y/m) first. Only modified duration gives the price change per unit yield.
Entering basis points as whole numbers, for example Δy = 50 instead of 0.005.
Rushing under time pressure.
Fix: Convert immediately: bp ÷ 10,000. Write the decimal on your paper.
Forgetting the ½ in the convexity term or squaring the wrong quantity.
The formula is memorised loosely.
Fix: Write ½ × C × (Δy)² every time. Square the decimal yield change, not the basis point count.
Treating DV01 as a percentage.
DV01 comes from duration, which is a percentage measure.
Fix: DV01 is a money amount per 1 bp. Multiply by price or position size.
Assuming equal DV01 means a position is fully hedged.
DV01 matching is the standard textbook hedge.
Fix: It protects only against parallel shifts. Non-parallel moves leave residual risk, so key rate exposures are needed.
Applying convexity with the wrong sign when yields fall.
Students think convexity only matters for yield rises.
Fix: The term is positive in both directions because (Δy)² is positive. It adds to price either way.

Worked examples

Example 1

A bond has a price of $98.50 per $100 face value, a modified duration of 6.2 and a convexity of 55. Estimate the percentage price change if yields rise by 50 bp, using duration and convexity.

Show the solution

Δy = 50 bp = 0.005.
Duration term: −6.2 × 0.005 = −0.031 = −3.10%.
Convexity term: ½ × 55 × (0.005)² = 0.5 × 55 × 0.000025 = 0.0006875 = +0.06875%.
Total: −3.10% + 0.06875% = −3.03125%.

Answer: The price falls by about 3.03%. Duration alone gives −3.10%.

Example 2

A bank holds $20 million market value of bonds with a modified duration of 5.0. It wants to hedge using futures with a DV01 of $85 per contract. How many contracts, and in which direction, neutralise a parallel shift?

Show the solution

Portfolio DV01 = 5.0 × 20,000,000 × 0.0001 = $10,000.
Hedge ratio N = −10,000 ÷ 85 = −117.65.
The portfolio loses value when yields rise, so the hedge must gain when yields rise: sell futures.
Round to the nearest whole contract: 118.

Answer: Sell about 118 futures contracts. This hedges only parallel yield shifts.

Exam tips

Check whether the question gives Macaulay or modified duration before doing anything else.
Expect questions that ask why a DV01 hedge fails under a curve twist. The answer is the parallel shift assumption.
Use DV01 to compare positions. Portfolio DV01 is additive, so you can add positions directly.
If an option sign looks wrong (a price rise when yields rise for a plain bond), discard it.
Convexity corrections are small for small yield moves. Do not spend time on them unless asked.

Practice questions from Modeling Non-Parallel Term Structure Shifts and Hedging

Duration, DV01 and Convexity Review in other exams

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

Duration, DV01 and Convexity Review: frequently asked questions

What is the difference between modified duration and DV01?

Modified duration is a percentage sensitivity per unit yield change. DV01 is the money change for a 1 bp move. DV01 = modified duration × price × 0.0001, so DV01 depends on position size and duration measures do not.

How do you calculate the DV01 of a bond?

Multiply modified duration by the bond's full price (or position value) and by 0.0001. You can also reprice the bond at a yield 1 bp lower and higher and take half the difference. Both methods give nearly the same answer.

Why does duration hedging fail for non-parallel shifts?

Duration and DV01 assume every maturity's yield moves by the same amount. If the curve steepens, flattens or twists, different parts of the portfolio and the hedge move by different amounts. The hedge then leaves residual risk.

Does convexity always help the bondholder?

For plain bonds with positive convexity, yes. The price rises more than duration predicts when yields fall and falls less when yields rise. Bonds with embedded options such as callable bonds can have negative convexity.