FRM (Financial Risk Manager) Market Risk Measurement and Management (Part II) Syllabus

Market Risk Measurement and Management (Part II) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Market Risk Measurement and Management (Part II) in FRM (Financial Risk Manager), not a summary of it.

1. 1

   Advanced VaR and Risk Measurement

   4 topics

   • Non-Parametric and Parametric Estimation
     • Historical simulation with weighting schemes
     • Age-weighted and volatility-weighted approaches
   • Extreme Value Theory
     • Block maxima and peaks-over-threshold
     • Generalized Pareto distribution
   • Coherent Risk Measures and Spectral Measures
     • Expected shortfall properties
     • Risk measure backtesting
   • Mapping Positions to Risk Factors
     • Cash flow mapping for fixed income
     • Mapping linear and non-linear derivatives
2. 2

   Volatility and Correlation in Risk

   3 topics

   • Modeling Volatility Term Structures
     • Mean reversion in GARCH
     • Implied vs. realized volatility
   • Correlation Risk and Dependence
     • Correlation breakdown in stress periods
     • Empirical properties of correlation
   • Copulas in Risk Modeling
     • Tail dependence
     • Applications to portfolio risk
3. 3

   Fixed Income and Volatility Trading Risk

   3 topics

   • Term Structure and Rate Models
     • Vasicek and CIR models
     • Ho-Lee and Hull-White models
   • Volatility Smiles and Surfaces
     • Equity vs. currency smiles
     • Surface dynamics and skew
   • Hedging Non-Linear Risk
     • Dynamic delta hedging
     • Gamma and vega exposures
4. 4

   VaR Backtesting and Regulatory Capital

   3 topics

   • VaR Backtesting Methods
     • Unconditional and conditional coverage tests
     • Kupiec and Christoffersen tests
   • Fundamental Review of the Trading Book (FRTB)
     • Shift from VaR to Expected Shortfall
     • Standardized and internal models approach
   • Liquidity Horizons
     • Risk factor liquidity classification

Market Risk Measurement and Management (Part II) flashcards for FRM (Financial Risk Manager)

23 of 50 cards from the Market Risk Measurement and Management (Part II) deck — real questions with worked answers.

1. What is the key difference between non-parametric and parametric approaches to estimating VaR and Expected Shortfall?

   Non-parametric methods (e.g., historical simulation) estimate risk directly from the empirical distribution of past returns without assuming a specific distributional form. Parametric methods assume returns follow a specified distribution (e.g., normal or $t$) and estimate its parameters (mean, volatility) to compute risk analytically.

2. State the basic historical simulation (HS) procedure for estimating VaR at confidence level $c$.

   Collect $n$ historical returns, sort them from worst to best, and identify the loss at the $(1-c)$ quantile. The VaR is the loss not exceeded with probability $c$; e.g., for 95% VaR with 100 observations, it is roughly the 5th worst loss.

3. What are the main advantages of basic historical simulation?

   It is intuitive and simple, requires no distributional assumption, captures fat tails and skewness present in the data, can handle any instrument whose value can be revalued, and avoids estimating variance-covariance matrices.

4. What are the main disadvantages of basic historical simulation?

   It assumes the future resembles the past, weights all observations equally regardless of age, cannot extrapolate beyond the historical sample (ghost/plateau effects), is slow to reflect changing volatility, and the choice of window length involves a trade-off between data sufficiency and relevance.

5. Describe the 'ghost' or 'plateau' effect in historical simulation.

   A single extreme observation keeps VaR artificially elevated as long as it remains in the rolling window, then VaR drops abruptly when that observation drops out of the window. This produces step-like jumps unrelated to current market conditions.

6. In the age-weighted (BRW) approach to historical simulation, how are weights assigned to past observations?

   Recent observations receive higher weights that decay geometrically with age. The weight of an observation $i$ days old is $$w(i) = \frac{\lambda^{i-1}(1-\lambda)}{1-\lambda^{n}},$$ where $\lambda \in (0,1)$ is the decay factor; smaller $\lambda$ gives faster decay and more weight to recent data.

7. How does the age-weighted historical simulation approach improve on the basic method?

   By giving recent observations more weight, it makes VaR more responsive to current market conditions, reduces the ghost effect (an old extreme observation's influence decays smoothly), and lets older data fade gradually rather than dropping out abruptly.

8. Describe the volatility-weighted historical simulation approach (Hull-White).

   Each historical return is scaled by the ratio of current volatility to the volatility prevailing when that return occurred: $$r_{t}^{*} = r_{t}\cdot\frac{\sigma_{T}}{\sigma_{t}},$$ where $\sigma_{t}$ is the volatility forecast (e.g., EWMA/GARCH) at time $t$ and $\sigma_{T}$ is the current forecast. VaR is then computed from the adjusted returns.

9. What is a key benefit of volatility-weighted historical simulation over age-weighting?

   It directly incorporates current volatility, allowing the estimated VaR to exceed the maximum historical loss (extrapolation beyond the sample) when current volatility is higher than historical volatility, which age-weighting cannot do.

10. What is the idea behind correlation-weighted (and filtered) historical simulation?

    Correlation-weighted HS adjusts the historical returns matrix to reflect current correlations instead of historical ones, updating the variance-covariance structure. Filtered historical simulation (FHS) combines volatility-scaling (via GARCH) with bootstrapping of standardized residuals to generate forward-looking scenarios.

11. What does Extreme Value Theory (EVT) study, and why is it useful in risk management?

    EVT studies the asymptotic distribution of extreme (tail) values rather than the whole distribution. It is useful because it provides a theoretically grounded way to model rare, large losses in the tails where data is sparse, enabling estimation of VaR and ES at very high confidence levels.

12. What are the two principal approaches within Extreme Value Theory?

    (1) The Block Maxima approach, which models the maxima of data blocks using the Generalized Extreme Value (GEV) distribution; and (2) the Peaks-Over-Threshold (POT) approach, which models exceedances above a high threshold using the Generalized Pareto Distribution (GPD).

13. State the Generalized Extreme Value (GEV) distribution function used in the block maxima approach.

    $$H_{\xi}(x)=\exp\!\left[-\left(1+\xi\frac{x-\mu}{\sigma}\right)^{-1/\xi}\right],\quad \xi\neq 0,$$ where $\mu$ is location, $\sigma>0$ is scale, and $\xi$ is the tail (shape) index.

14. What do the three cases of the tail index $\xi$ in the GEV distribution correspond to?

    $\xi>0$: Fréchet (fat-tailed distributions such as Student-$t$, Pareto); $\xi=0$: Gumbel (thin-tailed distributions such as normal, lognormal); $\xi<0$: Weibull (bounded-tail distributions). Financial return tails typically exhibit $\xi>0$.

15. Describe the Peaks-Over-Threshold (POT) approach.

    Instead of taking block maxima, POT selects all observations exceeding a high threshold $u$ and models the distribution of the exceedances $(X-u)$. By the Gnedenko-Pickands-Balkema-de Haan theorem, these exceedances converge to a Generalized Pareto Distribution as $u$ becomes large.

16. State the cumulative distribution function of the Generalized Pareto Distribution (GPD).

    $$G_{\xi,\beta}(y)=1-\left(1+\frac{\xi y}{\beta}\right)^{-1/\xi},\quad \xi\neq 0,$$ for exceedance $y=x-u\geq 0$, where $\beta>0$ is the scale parameter and $\xi$ is the shape (tail) parameter.

17. Give the GPD-based formula for VaR at confidence level $c$ in the POT framework.

    $$\text{VaR}=u+\frac{\beta}{\xi}\left[\left(\frac{n}{N_{u}}(1-c)\right)^{-\xi}-1\right],$$ where $u$ is the threshold, $n$ is total observations, $N_{u}$ is the number of exceedances above $u$, and $\xi,\beta$ are GPD parameters.

18. Give the GPD-based formula for Expected Shortfall in the POT framework (for $\xi<1$).

    $$\text{ES}=\frac{\text{VaR}}{1-\xi}+\frac{\beta-\xi u}{1-\xi}.$$ Equivalently the ratio $\frac{\text{ES}}{\text{VaR}}\to\frac{1}{1-\xi}$ as the confidence level approaches 1.

19. What is the main trade-off in choosing the threshold $u$ in the POT approach?

    A higher threshold gives a better GPD approximation (less bias) but fewer exceedances, increasing parameter estimation variance. A lower threshold yields more data (lower variance) but worse approximation (more bias). Practitioners often use a mean-excess plot to select $u$.

20. What are the four properties that define a coherent risk measure?

    For a risk measure $\rho$: (1) Monotonicity: if $X_{1}\leq X_{2}$ then $\rho(X_{1})\geq\rho(X_{2})$; (2) Subadditivity: $\rho(X_{1}+X_{2})\leq\rho(X_{1})+\rho(X_{2})$; (3) Positive homogeneity: $\rho(\beta X)=\beta\rho(X)$ for $\beta\geq0$; (4) Translation invariance: $\rho(X+c)=\rho(X)-c$ for constant cash $c$.

21. Which coherence property does VaR generally violate, and why does it matter?

    VaR violates subadditivity: the VaR of a combined portfolio can exceed the sum of individual VaRs. This means VaR can fail to recognize diversification benefits and can discourage risk aggregation, which is a key motivation for using Expected Shortfall.

22. What is a spectral risk measure?

    A spectral risk measure is a weighted average of quantiles (VaR at all confidence levels), $$M_{\phi}=\int_{0}^{1}\phi(p)\,q_{p}\,dp,$$ where $\phi(p)$ is a weighting (risk-aversion) function. It is coherent if and only if $\phi(p)$ is non-negative, non-decreasing in $p$, and integrates to 1.

23. How is Expected Shortfall (ES) defined, and how does it relate to spectral measures?

    ES at confidence $c$ is the average of all losses in the tail beyond VaR: $$\text{ES}=\frac{1}{1-c}\int_{c}^{1}\text{VaR}_{u}\,du.$$ It is a special spectral measure assigning equal weight $\frac{1}{1-c}$ to all tail quantiles beyond $c$ and zero weight below.

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