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

Market Risk Measurement and Management 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 in Financial Risk Manager (FRM), not a summary of it.

1. 1

   Advanced VaR and Risk Measures

   4 topics

   • Estimating volatility and correlation for VaR
     • EWMA, GARCH, and the constant-correlation model
     • Correlation breakdown in stressed markets
   • Non-parametric and semi-parametric approaches
     • Bootstrap historical simulation
     • Filtered historical simulation
   • Extreme Value Theory
     • Block maxima and peaks-over-threshold methods
     • Generalized Pareto distribution and tail estimation
   • Quantifying volatility in VaR models
2. 2

   Fixed Income and Term Structure Risk

   4 topics

   • Modeling non-parallel curve shifts with key rates
   • Single-factor short-rate models
     • Vasicek and Cox-Ingersoll-Ross models
     • Mean reversion and term structure dynamics
   • Volatility, drift, and arbitrage-free term structure models
   • The art of term structure modeling: time-dependent volatility
3. 3

   Mortgage and Securitized Products Risk

   4 topics

   • Mortgage pools and pass-through securities
   • Prepayment modeling and option-adjusted spread
   • Collateralized mortgage obligations and tranching
   • Measuring interest rate risk of MBS
4. 4

   Volatility, Correlation, and Trading Risk

   4 topics

   • Implied volatility term structure and skew
   • Correlation trading and basket products
   • Risk of volatility and variance swaps
   • Hedging higher-order risks in option books
5. 5

   Regulatory Capital for Market Risk

   4 topics

   • The Basel market risk framework evolution
   • Fundamental Review of the Trading Book (FRTB)
     • Expected Shortfall replacing VaR
     • Standardized versus internal models approach
     • Boundary between trading and banking book
   • Liquidity horizons and non-modellable risk factors
   • Backtesting and P&L attribution under FRTB

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

20 of 60 cards from the Market Risk Measurement and Management deck — real questions with worked answers.

1. In the EWMA model for estimating variance, what is the recursive update formula for $\sigma_n^{2}$?

   $$\sigma_n^{2} = \lambda \sigma_{n-1}^{2} + (1-\lambda)u_{n-1}^{2}$$ where $\lambda$ is the decay (persistence) factor and $u_{n-1}$ is the most recent return. RiskMetrics uses $\lambda = 0.94$ for daily data.

2. Write the GARCH(1,1) variance equation and state its mean-reversion/stationarity condition.

   $$\sigma_n^{2} = \omega + \alpha u_{n-1}^{2} + \beta \sigma_{n-1}^{2}$$ The long-run variance is $V_L = \frac{\omega}{1-\alpha-\beta}$. Stationarity (mean reversion) requires $\alpha + \beta < 1$; if $\alpha+\beta \geq 1$ the model is non-stationary (IGARCH when $=1$).

3. How does EWMA relate to GARCH(1,1) as a special case?

   EWMA is GARCH(1,1) with $\omega = 0$, $\alpha = 1-\lambda$, and $\beta = \lambda$. Since $\alpha+\beta = 1$, EWMA has no mean reversion (it is the IGARCH case).

4. In GARCH(1,1), what is the expected forecast of variance $t$ days ahead, $E[\sigma_{n+t}^{2}]$?

   $$E[\sigma_{n+t}^{2}] = V_L + (\alpha+\beta)^{t}\,(\sigma_n^{2} - V_L)$$ where $V_L = \frac{\omega}{1-\alpha-\beta}$. Forecasts revert toward $V_L$ at rate $(\alpha+\beta)$.

5. How is the EWMA recursion used to update a covariance estimate between assets $X$ and $Y$?

   $$\mathrm{cov}_n = \lambda\,\mathrm{cov}_{n-1} + (1-\lambda)\,x_{n-1}y_{n-1}$$ The correlation estimate is then $\rho_n = \frac{\mathrm{cov}_n}{\sigma_{x,n}\sigma_{y,n}}$.

6. What condition must a variance-covariance matrix satisfy to be internally consistent, and how can EWMA/GARCH violate it?

   It must be positive-semidefinite, i.e. $\vec{w}^{T}\Sigma\vec{w}\geq 0$ for all weight vectors $\vec{w}$. Estimating each variance/covariance separately (rather than jointly) can produce a non-positive-semidefinite matrix, implying impossible negative portfolio variances.

7. Describe the basic historical simulation approach to computing VaR.

   Collect $n$ historical daily returns/changes, apply each to current positions to build $n$ hypothetical P&L scenarios, sort them, and read VaR off the empirical distribution's tail quantile. It is non-parametric: no distributional assumption, and it captures actual fat tails and correlations. Each historical day is given equal weight.

8. What is the age-weighted (BRW) historical simulation approach and its weight formula?

   Boudoukh-Richardson-Whitelaw assign exponentially declining weights to older observations. The weight of a return $i$ days old is $$w_i = \frac{\lambda^{i-1}(1-\lambda)}{1-\lambda^{n}}$$ Recent observations matter more, so VaR reacts faster to volatility changes than equal-weighted HS.

9. What is volatility-weighted (Hull-White) historical simulation?

   Each past return is rescaled by the ratio of current to historical volatility: $$u_t^{*} = u_t \cdot \frac{\sigma_{\text{today}}}{\sigma_t}$$ This updates old returns to current volatility conditions, letting VaR exceed the historical range and incorporating GARCH/EWMA vol forecasts.

10. Name the two semi-parametric VaR refinements besides age- and volatility-weighting, and one weakness of basic historical simulation.

    Correlation-weighted HS (adjusts past returns for current correlations using updated covariance matrices) and filtered historical simulation (FHS, combines GARCH bootstrap with HS). Weakness of basic HS: it cannot produce losses larger than the worst observed historical loss and reacts slowly to volatility regime changes (ghosting/plateau effects).

11. State the two main theorems/distributions underlying Extreme Value Theory.

    (1) Block Maxima / Fisher-Tippett-Gnedenko: normalized block maxima converge to the Generalized Extreme Value (GEV) distribution. (2) Peaks-Over-Threshold (POT): exceedances over a high threshold $u$ converge to the Generalized Pareto Distribution (GPD).

12. Write the Generalized Pareto Distribution (GPD) CDF used in the POT approach to EVT.

    $$F(x) = 1 - \left(1 + \frac{\xi x}{\beta}\right)^{-1/\xi}$$ for $\xi \neq 0$, where $\xi$ is the tail (shape) index and $\beta>0$ is the scale parameter. Larger $\xi$ means heavier tails; financial returns typically have $\xi > 0$ (Frechet-type).

13. In EVT, what does the tail index $\xi$ (shape parameter) signify across the three GEV cases?

    $\xi > 0$: Frechet — heavy/fat tails (typical for financial returns, power-law decay). $\xi = 0$: Gumbel — thin tails (normal, lognormal). $\xi < 0$: Weibull — finite-tail / bounded distributions. Higher $\xi$ = fatter tail.

14. Give the POT (GPD) closed-form expressions for VaR and Expected Shortfall.

    $$\mathrm{VaR} = u + \frac{\beta}{\xi}\left[\left(\frac{n}{N_u}(1-\text{cl})\right)^{-\xi} - 1\right]$$ $$\mathrm{ES} = \frac{\mathrm{VaR}}{1-\xi} + \frac{\beta - \xi u}{1-\xi}$$ where $u$ = threshold, $N_u$ = exceedances, $n$ = total observations, $\text{cl}$ = confidence level.

15. Why is EVT preferred over the normal distribution for estimating extreme (e.g. 99.9%) VaR?

    The normal distribution underestimates tail risk because real returns are leptokurtic (fat-tailed). EVT models only the tail directly, providing more accurate estimates of very high-confidence VaR and Expected Shortfall and allowing extrapolation beyond the largest observed loss.

16. What is the square-root-of-time rule for scaling VaR, and its key assumption/limitation?

    $$\mathrm{VaR}(T\text{ days}) = \mathrm{VaR}(1\text{ day}) \times \sqrt{T}$$ It assumes i.i.d. returns with zero autocorrelation and constant volatility. It fails (overstates or understates) when returns are autocorrelated or volatility mean-reverts; positive autocorrelation makes true multi-day VaR larger than the rule implies.

17. Define a power law tail and its parameter, as used to model fat tails in VaR.

    A power law states the probability of exceeding a large value $x$ behaves as $$P(X > x) \approx K x^{-\alpha}$$ where $\alpha$ is the tail index. Smaller $\alpha$ means fatter tails. This contrasts with exponential decay of the normal distribution and underlies EVT's Frechet case.

18. Define key rate ('01) duration and how key rates model non-parallel yield curve shifts.

    A key rate duration (KR01/DV01) measures price sensitivity to a 1bp shift in one specific benchmark rate (e.g. 2y, 5y, 10y, 30y) while holding all other key rates fixed. Summed key rate shifts approximate any non-parallel curve move; the sum of all key rate durations equals the total effective duration.

19. How do key rate exposures relate to the total DV01 of a bond, and what is a key shortcoming of key rate analysis?

    $$\text{DV01}_{\text{total}} = \sum_i \text{KR01}_i$$ Shortcoming: results depend on the (somewhat arbitrary) choice of key rates and the linear interpolation between them; adjacent key rate shifts overlap, and key rates do not correspond to independent statistical factors (unlike PCA-based forward-bucket or principal component '01s).

20. Contrast key rate shifts with forward-bucket '01 and principal component analysis for curve risk.

    Key rate '01: shift one par rate, interpolate to neighbors. Forward-bucket '01: shift the forward rate over a maturity bucket, giving localized, non-overlapping exposures. PCA: decomposes historical curve moves into orthogonal factors — level (~80-90% of variance), slope, and curvature — providing statistically independent risk factors.

See more Market Risk Measurement and Management flashcards →

Planning Market Risk Measurement and Management for Financial Risk Manager (FRM)

Market Risk Measurement and Management (Financial Risk Manager (FRM)) FAQ