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FRM Exam Part I · Applying Duration, Convexity, and DV01

Portfolio Duration and Hedging with DV01 for FRM Part I

Updated 11 October 2026 · Fact-checked

Portfolio duration is the market-value-weighted average of the durations of its holdings. DV01 is the dollar change in value for a one basis point yield move, and portfolio DV01 is the sum of position DV01s. To hedge, divide the portfolio DV01 by the hedge instrument's DV01 and take the opposite position.

Understand Portfolio Duration and Hedging with DV01

Duration tells you how much a bond's price changes for a small change in yield. Modified duration of 5 means the price falls about 5% if the yield rises by 1 percentage point. DV01 (dollar value of a basis point) turns that into money: the change in value for a 0.01% move in yield.

For a portfolio, you do not need to go bond by bond through cash flows. Portfolio duration is the average of the position durations, weighted by market value (not face value). Portfolio DV01 is simply the sum of the position DV01s. Both rely on one assumption: all yields move by the same amount, a parallel shift.

To hedge, you want a position whose value moves by the same amount as the portfolio, in the opposite direction. If a long bond portfolio has DV01 of $54,000 and one futures contract has DV01 of $85, you need about 635 contracts. You go short the futures. If yields rise, the portfolio loses and the short futures gain.

You can also hedge to a target duration rather than to zero. In that case you only offset the difference between current and target sensitivity. A short position lowers duration. A long position raises it.

Duration is a first-order measure. Convexity captures the curvature. A barbell (short and long bonds) and a bullet (bonds clustered around one maturity) can have the same duration. The barbell has higher convexity because its cash flows are more spread out. Higher convexity helps when yields move a lot in either direction. It usually costs something in yield.

Key formulas to remember

Portfolio duration
D_P = Σ (w_i × D_i), where w_i = MV_i ÷ total MV
Weights are market values (including accrued interest if given as dirty value). Use modified duration throughout, or effective duration for bonds with options.
DV01 of a position
DV01 ≈ D_mod × MV × 0.0001
Gives the dollar loss for a 1 bp rise in yield. Also equals the price change per 1 bp if you reprice the bond.
Portfolio DV01
DV01_P = Σ DV01_i
Valid for a parallel yield shift. Shorts carry negative DV01.
DV01 hedge ratio
N = DV01_P ÷ DV01_hedge per unit
Take the opposite position to the portfolio. Long portfolio means short hedge.
Duration-based futures hedge
N = (D_T − D_P) × P ÷ (D_F × F)
P is portfolio value, F is the futures contract price, D_T is the target duration. For a full hedge D_T = 0, so N = −D_P × P ÷ (D_F × F). A negative N means sell.
Yield beta adjustment
N_adjusted = N × β, where β = Δy_portfolio ÷ Δy_hedge
Use when the portfolio yield and the hedge yield do not move one-for-one.
Duration-convexity approximation
ΔP ÷ P ≈ −D_mod × Δy + ½ × C × (Δy)²
Portfolio convexity is also the market-value-weighted average of position convexities.

How to solve Portfolio Duration and Hedging with DV01 questions

Use this sequence for portfolio duration and DV01 hedge questions. Keep units consistent throughout.

  1. 1Write down each position's market value and duration (or DV01). Check whether the duration is modified, Macaulay or effective.
  2. 2If given Macaulay duration, convert: D_mod = D_Mac ÷ (1 + y ÷ m), where m is the compounding frequency.
  3. 3Compute portfolio duration as the market-value-weighted average, or compute each DV01 and add them.
  4. 4Convert to dollar risk: DV01_P = D_P × total MV × 0.0001.
  5. 5Find the DV01 of the hedge instrument per contract or per unit. For futures, use futures price × futures duration × 0.0001 if no DV01 is given.
  6. 6Compute N = DV01_P ÷ DV01_hedge. For a target duration, use the difference between target and current DV01 instead.
  7. 7Set the sign: short the hedge to cut a long portfolio's exposure, and round to a whole number of contracts.
  8. 8Sanity check: portfolio DV01 plus hedge DV01 (with sign) should be about zero, or equal to the target DV01.

Quickest way: DV01 matching in three lines

When to use it: Use when the question gives durations and values, and asks for contracts, hedge notional or the change in value for a small yield move.

  1. Compute DV01_P = D × MV × 0.0001 (add positions if there are several).
  2. Divide by the hedge DV01 per contract. Round.
  3. Check the sign: long portfolio means sell. For a 1 bp move, loss ≈ DV01_P. If the yield shift is larger, scale by the number of basis points and only consider convexity if it is given.

Common mistakes in Portfolio Duration and Hedging with DV01

  • Weighting durations by face value or by number of bonds

    Face value is easy to read from the question and looks like the size of the position.

    Fix: Always weight by market value. A bond trading at 90 has 90% of the weight its face value suggests.

  • Adding durations instead of averaging them, or averaging DV01s

    Mixing up the two additive quantities. DV01s add. Durations are weighted averages.

    Fix: Add DV01 (dollars). Average duration (years) with weights summing to 1.

  • Getting the hedge direction wrong

    Students compute the size correctly but forget that a long bond portfolio loses when yields rise.

    Fix: Long bonds, short futures. Check with a +1 bp move: the portfolio loses and the hedge must gain.

  • Using Macaulay duration in a DV01 formula

    Both are called duration and the question may give only one.

    Fix: Divide Macaulay by (1 + y ÷ m) before computing price sensitivity.

  • Ignoring the target duration when the question asks for a partial hedge

    Students default to a full hedge from habit.

    Fix: Use (D_T − D_P) in the numerator. The result is the change in duration you need, not the whole duration.

  • Assuming the hedge removes all interest rate risk

    DV01 matching feels complete.

    Fix: State the limits: it covers only parallel shifts, ignores convexity differences (for example barbell versus bullet) and basis between the portfolio and the hedge.

Worked examples

Example 1

A portfolio holds Bond A with market value $40 million and modified duration 3.0, and Bond B with market value $60 million and modified duration 7.0. A Treasury futures contract has DV01 of $85. How many contracts should you trade to hedge against a parallel yield shift?

Show the solution
  1. Total market value = $40m + $60m = $100m.
  2. Weights: w_A = 0.40, w_B = 0.60.
  3. Portfolio duration = 0.40 × 3.0 + 0.60 × 7.0 = 1.2 + 4.2 = 5.4.
  4. Portfolio DV01 = 5.4 × $100,000,000 × 0.0001 = $54,000.
  5. Check by position: A DV01 = 3.0 × 40,000,000 × 0.0001 = $12,000. B DV01 = 7.0 × 60,000,000 × 0.0001 = $42,000. Sum = $54,000.
  6. Number of contracts = 54,000 ÷ 85 = 635.29.
  7. The portfolio is long, so sell the futures. Round to whole contracts.

Answer: Sell about 635 futures contracts. Portfolio duration is 5.4 and portfolio DV01 is $54,000.

Example 2

A manager holds a $200 million bond portfolio with modified duration 6.5 and wants to cut the duration to 2.0 using Treasury futures. Each contract has a price of $120,000 and a duration of 8.0. How many contracts should be traded?

Show the solution
  1. Use N = (D_T − D_P) × P ÷ (D_F × F).
  2. D_T − D_P = 2.0 − 6.5 = −4.5.
  3. Numerator = −4.5 × 200,000,000 = −900,000,000.
  4. Denominator = 8.0 × 120,000 = 960,000.
  5. N = −900,000,000 ÷ 960,000 = −937.5.
  6. The negative sign means sell. Round to whole contracts.
  7. Check: portfolio DV01 before = 6.5 × 200,000,000 × 0.0001 = $130,000. Target DV01 = 2.0 × 200,000,000 × 0.0001 = $40,000. Reduction needed = $90,000. Futures DV01 per contract = 8.0 × 120,000 × 0.0001 = $96. 90,000 ÷ 96 = 937.5. This matches.

Answer: Sell about 938 contracts (937.5 before rounding). This lowers portfolio DV01 from $130,000 to about $40,000.

Exam tips

  • Read which duration is given. If it says Macaulay, convert before using it for price changes.
  • Check units: DV01 can be quoted per $100 face, per contract or for the whole position. Match the units before dividing.
  • Do the sign check with a +1 bp move. It catches the most common error in seconds.
  • When a question compares a barbell and a bullet with equal duration, answer with convexity: the barbell has higher convexity, so it gains more from large yield moves in either direction.
  • If a yield beta is given, multiply the hedge ratio by it. If it is not mentioned, assume a parallel shift.

Practice questions from Applying Duration, Convexity, and DV01

Portfolio Duration and Hedging with DV01 in other exams

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

Portfolio Duration and Hedging with DV01: frequently asked questions

How do you calculate the DV01 hedge ratio?

Divide the portfolio DV01 by the DV01 of one hedge contract. Take the opposite position to the portfolio. For example, $54,000 ÷ $85 per contract gives about 635 contracts to sell.

Is portfolio duration a simple average of bond durations?

No. It is a weighted average using market values as weights. A larger position has a bigger effect on the portfolio's sensitivity.

What is the difference between a barbell and a bullet portfolio?

A bullet concentrates holdings around one maturity. A barbell holds short and long maturities. With the same duration, the barbell has higher convexity because its cash flows are more dispersed.

Why can a duration hedge still fail?

It assumes a parallel yield shift and small moves. Non-parallel shifts, convexity differences and basis between the portfolio and the futures contract all leave residual risk.