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CFA Level I · CFA Level I Exam

Curve-Based and Empirical Fixed-Income Risk Measures

Curve-based and empirical risk measures estimate how a bond's price changes when yields move. Effective duration and convexity use a pricing model and a shifted benchmark curve. Key rate duration splits risk by maturity point. Empirical duration is estimated from observed data. You solve questions by picking the right measure, then applying the formula.

What this chapter covers

This chapter covers fixed-income risk measures that work when cash flows are not fixed. Simple Macaulay and modified duration assume a bond pays set cash flows. That fails for callable, putable and other embedded-option bonds. So you move to measures built from a pricing model: you shift the benchmark curve up and down, reprice the bond, and read the sensitivity from the price changes.

You start with the price-yield relationship, which is curved, not straight. Then you learn effective duration, which gives a first-order estimate of price change. Key rate duration shows exposure to a change at one maturity point, which matters when the curve twists instead of shifting in parallel. Effective convexity adds the second-order correction and shows how embedded options bend the price-yield curve. Empirical duration is the last step: it is estimated by regression from market data, and it can differ from model-based duration.

This chapter links to the rest of the Fixed Income topic, such as bond valuation, yield spreads, and bonds with embedded options. It also feeds Portfolio Construction and Derivatives and Risk Management, where you manage interest rate exposure. Fixed Income carries a weight of 11-14% in the 2027 curriculum, so these ideas repeat across many questions.

Fixed Income is one of the larger topics, and risk measures show up in both calculation and concept questions. Each question is a standalone three-option item with no penalty for wrong answers, so a clear grip on direction and size of effects lets you eliminate two options fast. Many questions ask which bond has more or less duration or convexity, and you can answer these without a calculator if you know the logic. The calculation items are short and use one formula, so they are quick marks once you practise.

Curve-Based and Empirical Fixed-Income Risk Measures: topics in the order to study them

  1. 1Interest Rate Risk and Bond Price-Yield RelationshipIt sets the base: price moves inversely to yield and the curve is convex, which every later measure builds on.
  2. 2Effective Duration and Curve Duration MeasuresIt is the main first-order measure and the formula you will use most, so learn it right after the price-yield idea.
  3. 3Key Rate Duration and Yield Curve ExposureIt extends effective duration to non-parallel curve shifts, so you need the duration formula first.
  4. 4Effective Convexity and Embedded Option BondsIt adds the second-order term to duration and explains negative convexity in callable bonds.
  5. 5Empirical Duration and Yield VolatilityIt comes last because it contrasts data-based estimates with the model-based measures you have just learned.

How to prepare Curve-Based and Empirical Fixed-Income Risk Measures

Aim to understand the logic first, then drill the formulas and direction questions. Short daily sessions suit a working schedule and a phone.

  1. Sketch the price-yield curve by hand. Mark the tangent line for duration and the gap above it for convexity.
  2. Learn the effective duration formula: (PV₋ − PV₊) ÷ (2 × ΔCurve × PV₀). Practise it with three or four numbers until it is automatic.
  3. Do the same for effective convexity: (PV₋ + PV₊ − 2 × PV₀) ÷ (ΔCurve² × PV₀). Check each sign and the squared term.
  4. Combine both in the price-change estimate: %ΔPrice ≈ (−EffDur × ΔCurve) + ½ × EffConvexity × (ΔCurve)². Use the TI BA II Plus or HP 12C only for the arithmetic, and keep ΔCurve in decimal form.
  5. Build a table for key rate durations: which maturity point drives the result under a steepening, flattening or parallel shift.
  6. Write one line each on callable, putable and straight bonds: what happens to duration and convexity when yields fall or rise, including when yields fall enough for the call option to be near or in the money.
  7. Finish with timed sets of standalone items at about 90 seconds each, and review every wrong answer for the rule you missed.

Common mistakes in Curve-Based and Empirical Fixed-Income Risk Measures

  • Using modified duration for a callable or putable bond.

    Fix: Check the stem for an embedded option. If there is one, use effective duration from a repricing model.

  • Entering ΔCurve as a whole number such as 25 instead of 0.0025.

    Fix: Convert to decimal first. 25 bps = 0.0025. Check that your answer is a sensible size before choosing.

  • Forgetting to square ΔCurve in the convexity term.

    Fix: Write the formula out first and mark the square. Because ΔCurve is squared, the convexity term always has the same sign as the convexity itself and is small relative to the duration term for small curve shifts.

  • Assuming all bonds gain price when yields fall, by the same pattern.

    Fix: For a callable bond, price gains are limited when yields fall, and effective duration shortens as the call becomes likely.

  • Treating key rate duration as only a number to sum.

    Fix: Read where the largest key rate duration sits. It tells you which part of the curve drives risk under a non-parallel shift.

  • Assuming empirical duration always equals analytical duration.

    Fix: Remember that empirical duration comes from regression on past data, so it depends on the sample, and it can differ when yield and spread changes are not perfectly linked.

Last-day revision: Curve-Based and Empirical Fixed-Income Risk Measures

  • Bond price and yield move in opposite directions, and the relationship is convex.
  • Effective duration = (PV₋ − PV₊) ÷ (2 × ΔCurve × PV₀), with ΔCurve as a decimal.
  • Effective duration works for bonds with embedded options; modified duration assumes fixed cash flows.
  • Price change ≈ −EffDur × ΔCurve, then add the convexity term for a better estimate.
  • Convexity adjustment = ½ × EffConvexity × (ΔCurve)².
  • Effective convexity = (PV₋ + PV₊ − 2 × PV₀) ÷ (ΔCurve² × PV₀).
  • A callable bond can show negative convexity when yields fall enough that the call option is near or in the money, so its price rise is capped.
  • A putable bond has positive convexity, and the put option puts a floor under the price when yields rise, so its price falls less than that of an otherwise identical straight bond.
  • Key rate duration measures price sensitivity to a change at one maturity point, with other points unchanged.
  • The sum of a bond's key rate durations approximately equals its effective duration (the parallel-shift sensitivity).
  • Empirical duration is estimated from observed price and yield data and can differ from model duration.
  • Higher convexity is a benefit to the holder, all else equal.

Curve-Based and Empirical Fixed-Income Risk Measures practice questions

Curve-Based and Empirical Fixed-Income Risk Measures in other exams

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

Curve-Based and Empirical Fixed-Income Risk Measures: frequently asked questions

What is the difference between effective duration and modified duration?

Modified duration assumes cash flows do not change when yields change. Effective duration reprices the bond under shifted benchmark curves, so it handles embedded options. Use effective duration whenever cash flows depend on the interest rate.

Why can a callable bond have negative convexity?

When yields fall enough that the call option is near or in the money, the issuer is more likely to call the bond, so its price rises less than a straight bond would. The price-yield curve bends the other way in that range. This is called negative convexity.

Do I need a calculator for these questions?

Only for the arithmetic. The TI BA II Plus or HP 12C is enough, since the formulas use the given prices and curve shift. Many questions on direction and comparison need no calculation at all.

What is key rate duration used for?

It shows how a bond's price responds to a change in the yield at one maturity point while other points stay fixed. It helps you assess exposure to twists and other non-parallel yield curve shifts.

How is empirical duration different from the other measures?

Empirical duration is estimated from observed market data, usually through regression of price changes on yield changes. The other measures come from a pricing model and assumed curve shifts. The two can give different answers.