Market Risk Measurement & Management

FRM® Part II, published weight 20%. VaR and ES estimation, backtesting, EVT, GARCH, copulas, FRTB.

What the outline covers

VaR estimation approaches and their limitations; backtesting VaR
Expected Shortfall and other coherent measures; parametric and non-parametric estimation
Extreme value theory and tail risk
Volatility modelling: GARCH, EWMA, implied volatility surfaces
Correlation and copulas
Fixed-income and term-structure models; interest rate risk hedging
Volatility smiles; exotic and mortgage-backed products
Fundamental Review of the Trading Book (FRTB)

Sample questions

These 5 questions are free in the app too. Try each one before opening the answer.

Question 1

A portfolio has a 1-day 99% VaR of $2.5m. Assuming i.i.d. returns, the 10-day 99% VaR is closest to:

VaR(1) = $2.5m · T = 10

A. $7.9m
B. $25.0m
C. $2.5m
D. $5.0m

Show the answer

A. √10 ≈ 3.162, so 2.5 × 3.162 ≈ $7.9m. Square-root-of-time scaling requires independent, identically distributed returns.

Reading: Market Risk — VaR Measurement

Question 2

A bank’s 99% 1-day VaR model produced 9 exceptions in 250 trading days. This most likely indicates:

Expected exceptions = 0.01 × 250 = 2.5

A. The model overstates risk
B. The model is well calibrated
C. The model understates risk
D. Nothing without expected shortfall data

Show the answer

C. Nine breaches against 2.5 expected is far more than chance would suggest — the model understates risk and falls into the Basel red zone.

Reading: Market Risk — Backtesting VaR

Question 3

Breaches of a VaR model cluster in a single stressed week rather than occurring evenly. The most appropriate test is:

A. Kupiec’s unconditional coverage test
B. Christoffersen’s conditional coverage test
C. A Jarque–Bera normality test
D. A Dickey–Fuller unit root test

Show the answer

B. Kupiec tests only the number of exceptions. Christoffersen adds an independence test, which is what detects clustering.

Reading: Market Risk — Backtesting VaR

Question 4

In extreme value theory using the peaks-over-threshold approach, a positive shape parameter ξ indicates:

A. Tails thinner than the normal distribution
B. Tails fatter than the normal distribution
C. A bounded loss distribution
D. That the threshold was set too low

Show the answer

B. A positive ξ corresponds to the Fréchet domain — heavy tails, decaying more slowly than the normal distribution.

Reading: Market Risk — Extreme Value Theory

Question 5

The Gaussian copula was widely criticised after 2008 primarily because it:

A. Cannot join different marginal distributions
B. Exhibits zero tail dependence
C. Requires equal marginals for all variables
D. Cannot be calibrated to market data

Show the answer

B. The Gaussian copula implies assets become independent in the extreme tail, exactly the opposite of observed crisis behaviour. Student t copulas admit tail dependence.

Reading: Market Risk — Correlation and Copulas

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