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FRM Part I · FRM Exam Part I

Applying Duration, Convexity, and DV01 for FRM Part I

Duration, convexity and DV01 measure how a bond's price changes when yields move. Duration gives the first-order percentage change, DV01 gives the dollar change per basis point, and convexity corrects the estimate for large moves. To solve questions, pick the right measure, apply the formula, and check the sign and units.

What this chapter covers

This chapter is about interest rate sensitivity. You start with the price-yield relationship, then learn to measure sensitivity three ways: DV01 (dollar change for a 1 basis point move), duration (percentage change per unit change in yield) and convexity (the curvature that duration misses).

The core formulas are short. Modified duration = Macaulay duration ÷ (1 + y/k), where y is the yield and k is the compounding frequency. The first-order estimate is ΔP ≈ −D_mod × P × Δy. Adding convexity gives ΔP ≈ −D_mod × P × Δy + ½ × C × P × (Δy)². DV01 ≈ D_mod × P × 0.0001. Effective duration is computed from prices at shifted yields, so it also works for bonds with embedded options.

The chapter connects to the rest of the paper in several ways. It builds on bond pricing and the time value of money from Financial Markets and Products. It feeds into hedging with futures and swaps, into fixed income risk in Valuation and Risk Models, and into key rate and curve risk. It also supports Foundations of Risk Management, where interest rate risk is a core market risk. Expect numerical questions that need a calculator and a clear head.

This chapter is worth the effort because the questions are quantitative, formula-based and very learnable. With 100 equally weighted questions in 4 hours, calculation items you can finish in two or three minutes are the best use of your time. The same ideas also appear in hedging, portfolio risk and VaR questions, so one solid understanding pays off in several places. Mastering the unit logic (percent versus dollar, per 1% versus per 1 bp) also protects you from the traps the exam sets.

Applying Duration, Convexity, and DV01: topics in the order to study them

  1. 1Price-Yield Relationship and Bond Pricing BasicsEverything else describes the slope and curve of this relationship, so you need to price a bond and see its convex shape first.
  2. 2DV01 and Basis Point ValueDV01 is the simplest sensitivity measure, a dollar change per basis point, and it makes the idea of slope concrete before percentages appear.
  3. 3Macaulay, Modified and Effective DurationDuration turns the slope into a percentage measure, and you need the links between Macaulay, modified and effective versions before you use them.
  4. 4Convexity and Second-Order Price ApproximationConvexity refines the duration estimate, so it only makes sense after you are comfortable with first-order changes.
  5. 5Portfolio Duration and Hedging with DV01Here you apply the measures: weight durations across a portfolio and size hedges by matching DV01.
  6. 6Limitations of Duration and Key Rate MeasuresLast, you learn where the single-number approach fails, such as non-parallel shifts and embedded options, and how key rate measures help.

How to prepare Applying Duration, Convexity, and DV01

Treat this chapter as a set of formulas you can apply fast and a set of ideas you can explain. Build in that order, and practise with a financial calculator.

  1. Price a few bonds by hand and with your calculator (N, I/Y, PMT, FV, CPT PV). Note how price falls as yield rises, and that the curve is convex.
  2. Write the formulas on one page: modified duration, DV01, the first-order estimate and the convexity term. Say aloud what each symbol means and its units.
  3. Compute DV01 two ways: from modified duration (D_mod × P × 0.0001) and from repricing the bond at yield ± 1 bp. Check that they agree closely.
  4. Practise estimating price changes for both a small and a large yield move. Compare the duration-only answer with the duration plus convexity answer and see where the gap grows.
  5. Do hedging problems: find each position's DV01, then choose the hedge size so that total DV01 is zero. Check the direction (long or short) of the hedge.
  6. Finish with concept questions on limitations: parallel shift assumptions, embedded options, negative convexity, and key rate durations. Then mix timed questions across all six topics.

Common mistakes in Applying Duration, Convexity, and DV01

  • Using Macaulay duration where modified duration is needed in the price change formula.

    Fix: Convert first: D_mod = D_Mac ÷ (1 + y/k). Use D_mod in ΔP ≈ −D_mod × P × Δy.

  • Mixing up units for yield changes, such as entering 50 bp as 0.5 instead of 0.005.

    Fix: Write Δy as a decimal every time: 1 bp = 0.0001, 50 bp = 0.0050. Check that your answer is plausible in size.

  • Forgetting the sign, or adding the convexity term with the wrong sign.

    Fix: Remember that the convexity adjustment is always positive for a positive-convexity bond, so it raises the estimated price in either direction.

  • Hedging with the wrong ratio, such as matching notional amounts instead of DV01.

    Fix: Hedge ratio = DV01 of the position ÷ DV01 of the hedge instrument. Then decide whether to go long or short.

  • Treating duration as an exact measure for large moves or non-parallel shifts.

    Fix: Remember that duration is a first-order, parallel-shift approximation. Add convexity for large moves, and use key rate measures for curve changes.

  • Applying modified duration to bonds with embedded options.

    Fix: Use effective duration, which reprices the bond under up and down yield shifts and so captures changing cash flows.

Last-day revision: Applying Duration, Convexity, and DV01

  • Bond price and yield move in opposite directions, and the price-yield curve is convex for plain bonds.
  • Modified duration = Macaulay duration ÷ (1 + y/k), where k is the number of compounding periods per year.
  • First-order estimate: ΔP/P ≈ −D_mod × Δy.
  • DV01 ≈ D_mod × P × 0.0001, the dollar price change for a 1 bp yield move.
  • With convexity: ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)².
  • Convexity is positive for plain bonds, so duration alone underestimates the price after a yield change in either direction.
  • Effective duration = (P₋ − P₊) ÷ (2 × P₀ × Δy), using prices from yield shifts up and down.
  • Use effective duration for bonds with embedded options, since cash flows change with yield.
  • Portfolio duration is the market-value-weighted average of the durations of its holdings (for a parallel yield shift).
  • A DV01 hedge sets the position's DV01 plus the hedge's DV01 equal to zero, which protects only against small, parallel shifts.
  • Callable bonds can show negative convexity when yields fall.
  • Key rate durations measure sensitivity to shifts at specific maturities and capture non-parallel curve moves.

Applying Duration, Convexity, and DV01 practice questions

Applying Duration, Convexity, and DV01 in other exams

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

Applying Duration, Convexity, and DV01: frequently asked questions

What is the difference between duration and DV01?

Duration is a percentage sensitivity, such as the percent price change per 1% yield change. DV01 is a dollar sensitivity, the price change in currency for a 1 basis point move. They are linked by DV01 ≈ D_mod × P × 0.0001.

When do I need convexity in an FRM question?

Use it when the yield move is large or when the question asks for the most accurate estimate. For small moves, duration alone is usually close enough. Convexity matters more as the move grows because the price-yield curve bends away from the duration line.

Do I need a financial calculator for this chapter?

It helps a lot for pricing bonds and repricing at shifted yields to find effective duration and convexity. Learn the time value keys and practise until the steps are automatic. Many questions can also be done with formulas and a basic calculation.

How is a DV01 hedge set up?

Compute the DV01 of the position and the DV01 of the hedge instrument. Divide one by the other to get the hedge amount, and take the opposite side so the DV01s cancel. This protects only against small parallel shifts.

Why does the order of study matter here?

Each topic builds on the last: pricing leads to DV01, then duration, then convexity, then portfolio use and finally limitations. Following that order means each new formula has a clear meaning when you meet it.