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CFA Level II Exam · Measuring and Managing Market Risk

Sensitivity and Scenario Risk Measures for CFA Level II

Updated 7 October 2026 · Fact-checked

Sensitivity measures (duration, convexity, delta, gamma, vega, beta) show how a position's value changes for a small move in one risk factor. Scenario measures (historical, hypothetical, stress tests) revalue the portfolio under a set of large, joint moves. To solve questions, identify the factor, then apply the matching measure.

Understand Sensitivity and Scenario Risk Measures

A sensitivity measure answers one question: if one risk factor moves by a small amount and everything else stays fixed, how much does my position change in value? The risk factor can be an interest rate, a stock price, a volatility level or the market index. Each asset class has its own measure.

For bonds, duration gives the first-order price change for a yield change, and convexity adds the second-order correction. For options, delta is the change in option price per unit change in the underlying, gamma is the change in delta per unit change in the underlying, and vega is the change in option price per one percentage point change in volatility. For equities, beta is the sensitivity of a stock's return to the market's return.

Sensitivities are fast and easy to compute, but they have limits. They assume a small move in one factor. They ignore how factors move together. Delta alone is wrong for large moves, which is why gamma is added. They also say nothing about how likely the move is.

Scenario measures fix this by revaluing the whole portfolio under a defined set of moves in many factors at once. Historical scenarios replay a past event, such as the 2008 financial crisis or the 1997 Asian crisis. Hypothetical scenarios invent plausible future events, such as a sudden rate spike or a major currency collapse. They can capture events that never happened, but they depend on the judgment of the designer.

Stress testing is scenario analysis aimed at extreme but plausible events, often beyond what VaR captures. Reverse stress testing starts from an unacceptable loss and works backward to find which scenarios would cause it. Scenario results should supplement VaR, not replace it, because scenarios give no probability of occurrence.

Key formulas to remember

Price change using duration and convexity
%ΔP ≈ −ModDur × ΔY + ½ × Convexity × (ΔY)²
ΔY is in decimal form. Use effective duration and convexity for bonds with embedded options.
Option price change (delta-gamma)
ΔC ≈ Delta × ΔS + ½ × Gamma × (ΔS)²
Delta-only is the first-order estimate. Adding gamma improves it for larger moves.
Option price change from vega
ΔC ≈ Vega × Δσ
Vega is usually quoted per 1 percentage point change in volatility. Check the units in the vignette.
Beta
β = Cov(Ri, Rm) ÷ Var(Rm)
Expected stock move ≈ β × market move, ignoring the stock-specific part.
Delta of a position
Position delta = number of options × Delta × contract multiplier
Long calls and short puts have positive delta. Long puts and short calls have negative delta.

How to solve Sensitivity and Scenario Risk Measures questions

Use this method for any question on sensitivity or scenario measures in an item set.

  1. 1Read the question first, then find the risk factor in the vignette: yield, underlying price, volatility or market index.
  2. 2Pick the matching measure: duration/convexity for yields, delta/gamma for underlying price, vega for volatility, beta for market moves.
  3. 3Pull the exact inputs from the exhibit. Note the units, such as per 1 percentage point or per contract.
  4. 4Check the move size. If it is small, first-order is enough. If large, add the second-order term.
  5. 5Compute, keeping the sign. Note whether the position is long or short.
  6. 6For scenario questions, decide whether it is historical, hypothetical, stress or reverse stress, and note what each is designed to reveal.
  7. 7Check the answer: does the direction make sense, and is the unit right?

Quickest way: Direction first, then size

When to use it: When time is short and the options differ in sign or order of magnitude.

  1. Decide the direction of the value change from the sign of the sensitivity and the position (long or short).
  2. Convexity and gamma are positive for long option-free bonds and long options. For these holders the second-order term is positive, so they gain more or lose less than the first-order estimate suggests. Short option positions have negative gamma, so the second-order term works against the holder and adds to the loss. Bonds with embedded options, such as callable bonds, can have negative convexity.
  3. Eliminate options with the wrong sign, then compute only the first-order term.
  4. Add the second-order term only if two remaining options are close.
  5. For scenario questions, match the definition: past event is historical, invented event is hypothetical, start from a loss is reverse.

Common mistakes in Sensitivity and Scenario Risk Measures

  • Using ΔY in percent instead of decimal in the duration-convexity formula.

    Yields are quoted as percentages, so 1% is plugged in as 1.

    Fix: Convert to decimal first: 1% = 0.01. The convexity term uses (0.01)², which is 0.0001.

  • Forgetting the ½ in the convexity or gamma term.

    The formula is memorised from the Taylor series without the factor.

    Fix: Always write ½ × Convexity × (ΔY)² and ½ × Gamma × (ΔS)².

  • Confusing the sign of delta for short positions.

    Delta is quoted for one long option, and the position direction is missed.

    Fix: Multiply by the number of options, with a negative sign for short positions, before aggregating.

  • Treating stress testing and scenario analysis as the same thing.

    Both revalue a portfolio under big moves.

    Fix: Scenario analysis covers any defined set of moves, including mild ones. Stress testing focuses on extreme but plausible events. Reverse stress testing starts from a loss level.

  • Saying scenarios give probabilities of loss.

    VaR gives a probability, so people assume scenarios do too.

    Fix: Scenarios show the size of loss under a specified event. They do not assign a likelihood.

  • Applying duration alone to a bond with embedded options.

    Modified duration is used by habit.

    Fix: Use effective duration and effective convexity when cash flows change with yields.

Worked examples

Example 1

A bond portfolio has a value of ₹50,00,00,000, modified duration of 6.0 and convexity of 50. Yields are expected to rise by 1.00 percentage point. (1) Estimate the percentage price change using duration only. (2) Estimate it using duration and convexity. (3) Estimate the change in value using duration and convexity.

Show the solution
  1. ΔY = 0.01.
  2. Duration only: −6.0 × 0.01 = −0.06, or −6.00%.
  3. Convexity term: ½ × 50 × (0.01)² = 0.5 × 50 × 0.0001 = 0.0025, or +0.25%.
  4. Combined: −6.00% + 0.25% = −5.75%.
  5. Value change: −5.75% × ₹50,00,00,000 = −₹2,87,50,000.

Answer: (1) −6.00%. (2) −5.75%. (3) A fall of about ₹2,87,50,000. Convexity reduces the loss.

Example 2

A fund is short 200 call option contracts on a stock. Each contract covers 100 shares, so the position covers 20,000 shares. The call has delta 0.60, gamma 0.04 and vega 0.15 per 1 percentage point change in volatility. The stock is ₹1,000. (1) What is the position delta in shares? (2) Estimate the change in one call's price per share if the stock rises ₹5 using delta and gamma, and state the effect on the short position. (3) What is the effect on the call price and on the short position of a 2 percentage point rise in volatility?

Show the solution
  1. Position delta: −200 × 100 × 0.60 = −12,000 shares.
  2. One call: Delta term = 0.60 × 5 = 3.00.
  3. Gamma term = ½ × 0.04 × 5² = 0.5 × 0.04 × 25 = 0.50.
  4. Call price change ≈ 3.00 + 0.50 = +3.50 per share.
  5. The fund is short 20,000 shares' worth of calls, so the call's rise is a loss to the fund.
  6. Loss from delta: 12,000 × ₹5 = ₹60,000 (position delta of −12,000 shares times the ₹5 move). This equals 3.00 × 20,000.
  7. Loss from gamma: 0.50 × 20,000 = ₹10,000. The short position has negative gamma, so the second-order term adds to the loss.
  8. Total loss from the stock move: ₹60,000 + ₹10,000 = ₹70,000, which equals ₹3.50 × 20,000.
  9. Vega: 0.15 × 2 = +0.30 per share, so the call price rises by ₹0.30.
  10. Position effect of the volatility rise: the fund is short, so it loses about ₹0.30 × 20,000 = ₹6,000.

Answer: (1) −12,000 shares. (2) The call rises about ₹3.50 per share. The short position loses about ₹70,000: ₹60,000 from delta and ₹10,000 from negative gamma. (3) The call rises by ₹0.30 per share, so the short position loses about ₹6,000.

Exam tips

  • Read the vignette for units: vega per 1 percentage point and yield changes in basis points are common traps.
  • Questions often ask which measure is best for a situation. Sensitivity for small single-factor moves, scenarios for large multi-factor moves, reverse stress tests for finding vulnerabilities.
  • Know the strengths and limits of each scenario type: historical is realistic but backward-looking, hypothetical covers new risks but depends on judgment.
  • Always check long versus short before assigning the sign of delta, gamma or vega.

Sensitivity and Scenario 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.

Sensitivity and Scenario Risk Measures: frequently asked questions

What is the difference between sensitivity and scenario risk measures?

Sensitivity measures show the effect of a small change in one risk factor, such as duration or delta. Scenario measures revalue a portfolio under a set of large moves in several factors together. Sensitivities are quick, while scenarios capture combined and extreme events.

What is the difference between stress testing and scenario analysis?

Scenario analysis is the broad process of revaluing a portfolio under a defined set of factor moves. Stress testing is a form of it focused on extreme but plausible events, including rare ones that VaR may miss.

What is reverse stress testing?

You start with an outcome you cannot accept, such as a loss that breaks a capital limit. You then work backward to find the combinations of market moves that would cause it. It helps expose hidden vulnerabilities.

Why add gamma to delta?

Delta is only accurate for small price moves because the option price is curved, not straight. Gamma measures that curvature, so adding it gives a better estimate for larger moves.