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FRM Exam Part II · Portfolio Risk: Analytical Methods

Limitations of Analytical VaR and Stress Testing

Updated 11 October 2026 · Fact-checked

Analytical (parametric) VaR assumes normal returns and linear exposures. Real returns have fat tails and options are nonlinear, so it can understate tail losses. You fix this with delta-gamma adjustments, stress tests that show losses beyond the VaR cutoff, and expected shortfall, which averages losses in the tail.

Understand Stress Testing and Limitations of Analytical VaR

Analytical VaR turns a portfolio into a formula. You take the portfolio's standard deviation, assume returns are normal, and read off a quantile. For a 99% one-day VaR, you multiply the standard deviation by 2.33. It is fast and needs few inputs. That is its appeal.

The cost is two big assumptions. The first is normality. Real returns have fat tails (excess kurtosis) and are often skewed. Extreme days happen far more often than a normal curve says. A normal-based VaR at 99% can be too low, and the error grows as you move further into the tail.

The second is linearity. The delta-normal method approximates every position by its delta, so portfolio P&L is a linear function of risk factors. That works for stocks and forwards. It fails for options, where P&L curves with the underlying. A long option position has positive gamma, so delta-normal overstates its loss on a big move. A short option position has negative gamma, so delta-normal understates the loss. The delta-gamma method adds a second-order term to capture curvature. It improves accuracy for moderate moves but is still an approximation. It also breaks the normality of P&L, so the quantile is no longer a simple multiple of the standard deviation.

VaR has a further limit: it only gives a threshold. It says nothing about how bad losses are beyond it. Expected shortfall (ES) is the average loss given that the loss exceeds VaR. It sees the shape of the tail and is subadditive, so it behaves as a coherent measure. VaR is not subadditive in general.

Stress tests complement both. They apply specific shocks or scenarios, historical or hypothetical, without relying on a distribution. They show what a severe but plausible event does to the portfolio. Their weakness is subjectivity: scenarios have no probability attached and depend on the designer's imagination. So you use VaR, ES and stress tests together.

Key formulas to remember

Delta-normal parametric VaR (normal, zero mean)
VaR = z(α) × σ(P) × V
z = 1.645 at 95% and 2.326 at 99% (one-tailed). σ(P) is portfolio return volatility, V is portfolio value.
Delta approximation for an option
ΔP ≈ δ × ΔS
Linear in the underlying move. Ignores curvature, so it errs for large moves.
Delta-gamma approximation
ΔP ≈ δ × ΔS + ½ × Γ × (ΔS)²
Positive gamma adds to P&L in both directions. Negative gamma subtracts from it.
Expected shortfall under normality
ES = σ × φ(z) ÷ (1 − α) (zero mean)
φ is the standard normal density. At 99%, ES ≈ 2.665σ versus VaR ≈ 2.326σ.
Expected shortfall definition
ES(α) = E[Loss | Loss ≥ VaR(α)]
Always at least as large as VaR at the same confidence level.

How to solve Stress Testing and Limitations of Analytical VaR questions

Use this sequence for any question on limitations of analytical VaR, approximations or stress testing.

  1. 1Identify the portfolio: linear (equities, forwards, bonds for small moves) or nonlinear (options, convertibles, mortgage products).
  2. 2Identify the distribution assumption: normal, or fat-tailed and skewed. Check any kurtosis or skewness given.
  3. 3Decide what breaks. Nonlinear positions break delta-normal. Fat tails break the normal quantile, especially at 99% and beyond.
  4. 4If a calculation is needed, compute the delta or delta-gamma P&L first, then the VaR or ES with the right z-value.
  5. 5Compare delta-normal with delta-gamma. Positive gamma means delta-normal overstates the loss. Negative gamma means it understates it.
  6. 6State what the measure misses. VaR ignores losses beyond the cutoff, so name ES or stress tests as the complement.
  7. 7Match the remedy to the weakness: curvature needs delta-gamma or full revaluation, fat tails need ES, EVT or historical data, unmodelled events need stress tests.

Quickest way: Weakness-to-fix matching

When to use it: For conceptual MCQs where you must pick the correct statement fast.

  1. Ask: is the problem the shape of the distribution, the shape of the payoff, or the tail beyond the cutoff?
  2. Distribution shape (fat tails, skew): normal VaR too low far in the tail. Think ES, EVT, historical simulation.
  3. Payoff shape (options): linear delta-normal is wrong. Think gamma, full revaluation.
  4. Beyond the cutoff or no historical precedent: think ES and stress tests.
  5. Eliminate options that say stress tests assign probabilities or that ES is smaller than VaR.

Common mistakes in Stress Testing and Limitations of Analytical VaR

  • Saying delta-normal always understates option risk.

    Students remember short-option blowups and generalise.

    Fix: Check the sign of gamma. Long gamma means delta-normal overstates the loss. Short gamma means it understates.

  • Treating delta-gamma VaR as exact.

    Adding a second-order term feels like a complete fix.

    Fix: It is still a Taylor approximation. It degrades for very large moves, and P&L is no longer normal. Full revaluation is the accurate alternative.

  • Claiming ES is just a bigger confidence level of VaR.

    Both are tail measures, so they look similar.

    Fix: ES is the average of losses beyond VaR. It depends on the whole tail shape, not a single quantile.

  • Assuming normal VaR is wrong at every confidence level.

    Students overstate the fat-tail rule.

    Fix: Fat tails mean more mass in the tails and more near the centre, with less in the shoulders. Normal VaR is typically too low far into the tail, but the effect varies by level and data.

  • Giving stress tests a probability.

    Students confuse them with VaR scenarios.

    Fix: Stress tests are not tied to a probability. That is both their strength, since no distribution is needed, and their weakness, since results are subjective.

  • Using the wrong z-value for ES under normality.

    Students plug the VaR z into the ES step.

    Fix: ES = σ × φ(z) ÷ (1 − α). At 99%, φ(2.326) ≈ 0.0267, so ES ≈ 2.67σ.

Worked examples

Example 1

A long call position has delta 0.60 and gamma 0.04 per USD² on a stock at USD 100. Over the horizon the stock falls USD 5. Compare the delta-only and delta-gamma estimates of the P&L change, and say which is more accurate directionally.

Show the solution
  1. Delta-only: ΔP ≈ 0.60 × (−5) = −USD 3.00.
  2. Gamma term: ½ × 0.04 × (−5)² = 0.02 × 25 = +USD 0.50.
  3. Delta-gamma: ΔP ≈ −3.00 + 0.50 = −USD 2.50.
  4. Gamma is positive (long option), so the curvature cushions the loss.
  5. Delta-only overstates the loss by USD 0.50.

Answer: Delta-only gives −USD 3.00 and delta-gamma gives −USD 2.50. With positive gamma, delta-normal overstates the loss.

Example 2

A portfolio worth USD 200 million has daily return volatility of 1%. Assuming normal returns with zero mean, compute the 1-day 99% VaR and the 99% expected shortfall. Use z = 2.326 and φ(2.326) = 0.0267.

Show the solution
  1. Daily σ in USD: 0.01 × 200 million = USD 2 million.
  2. VaR = 2.326 × 2 million = USD 4.652 million.
  3. ES multiplier = 0.0267 ÷ 0.01 = 2.67.
  4. ES = 2.67 × 2 million = USD 5.34 million.
  5. ES exceeds VaR by about USD 0.69 million, the average extra loss in the worst 1% of days under normality.

Answer: VaR ≈ USD 4.65 million and ES ≈ USD 5.34 million. With fat tails the true ES would likely be larger than this normal estimate.

Exam tips

  • Read the sign of gamma before choosing which method over- or understates risk.
  • Expect conceptual questions that ask what stress tests and ES add that VaR lacks. The core answer is tail information and no distribution assumption.
  • Remember ES ≥ VaR at the same confidence level, and ES is subadditive while VaR is not in general.
  • For normal ES, you will usually be given φ(z). Do not rely on remembering it.
  • Watch for options saying a method is exact. Every analytical approximation has a stated limit.

Practice questions from Portfolio Risk: Analytical Methods

Stress Testing and Limitations of Analytical VaR: frequently asked questions

What are the main limitations of parametric VaR?

It assumes normal returns and linear exposures. Fat tails and skew make normal VaR too low far into the tail, and options break the linear approximation. It also says nothing about losses beyond the VaR cutoff.

What is the difference between delta-normal and delta-gamma?

Delta-normal uses only the first-order sensitivity, so P&L is linear in the risk factor. Delta-gamma adds a second-order term, ½ × Γ × (ΔS)², to capture curvature. It is more accurate for options but still approximate.

How does expected shortfall differ from VaR?

VaR is the loss threshold at a confidence level. ES is the average loss given that the loss exceeds that threshold. ES captures tail severity and is subadditive, so it is a coherent measure.

Why do we still need stress tests if we have VaR and ES?

Stress tests do not depend on a fitted distribution or past data. They examine specific severe scenarios, including ones with no historical precedent. They have no probability attached, so they complement rather than replace VaR and ES.