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

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.

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.

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.

List the key properties of Expected Shortfall as a risk measure.

ES is a coherent risk measure (satisfies all four coherence axioms including subadditivity), accounts for the magnitude of tail losses beyond VaR, is more conservative than VaR, and gives a smoother, more informative picture of tail risk. Its main drawbacks are harder backtesting and greater estimation noise in the tail.

Why is Expected Shortfall always greater than or equal to VaR at the same confidence level?

Because ES is the average of all losses that are at least as large as the VaR threshold. Since every loss in the tail is $\geq$ VaR, their average must satisfy $\text{ES}\geq\text{VaR}$, with strict inequality whenever tail losses vary.

What is the goal of backtesting a risk measure, and what is an 'exception'?

Backtesting compares predicted risk estimates (e.g., VaR) against actual realized losses to validate model accuracy. An exception (or exceedance) occurs when the actual loss exceeds the predicted VaR. The expected exception rate equals $(1-c)$ for a VaR at confidence $c$.

Describe Kupiec's unconditional coverage test for VaR backtesting.

It tests whether the observed proportion of exceptions equals the expected proportion $p=1-c$ using the likelihood ratio statistic $$LR_{uc}=-2\ln\!\left[\frac{(1-p)^{T-N}p^{N}}{(1-\hat{p})^{T-N}\hat{p}^{N}}\right],$$ where $N$ exceptions occur in $T$ observations and $\hat{p}=N/T$. It is $\chi^{2}(1)$ distributed.

What does Christoffersen's conditional coverage test add to Kupiec's test?

It adds an independence test checking that exceptions are not clustered in time (no serial dependence). The combined statistic $LR_{cc}=LR_{uc}+LR_{ind}$ jointly tests correct unconditional coverage and independence of exceptions, and is $\chi^{2}(2)$ distributed.

Why is Expected Shortfall harder to backtest than VaR?

ES is not 'elicitable' in the same simple way as VaR (which only counts exceedance hits). Backtesting ES requires estimating the magnitude of tail losses, not just their frequency, making tests more sensitive to tail estimation error and requiring more data and more complex procedures.

What does it mean to 'map' positions to risk factors?

Mapping replaces actual portfolio positions with positions in a smaller set of standardized, primitive risk factors (e.g., benchmark interest rates, exchange rates, equity indices). This simplifies VaR computation by reducing dimensionality and enabling use of a common variance-covariance matrix.

What are the three main types of mapping for fixed-income portfolios?

(1) Principal mapping: maps the bond to a single risk factor based on the timing of the principal (average maturity); (2) Duration mapping: maps to a single zero with maturity equal to the bond's duration; (3) Cash-flow mapping: decomposes each cash flow and maps it to adjacent standardized vertices.

Describe cash-flow mapping for a fixed-income instrument.

Each cash flow is assigned (present-valued) to the standardized maturity vertices (RiskMetrics buckets) nearest to its actual timing. A cash flow between two vertices is split so as to preserve its present value and match the interpolated volatility (variance) of the original cash flow, maintaining risk equivalence.

In cash-flow mapping, how is a cash flow falling between two vertices allocated to preserve risk?

The allocation weight $\alpha$ to the nearer vertex is chosen so the mapped position's variance matches the original. Using $\sigma$ interpolated between the two vertices, $\alpha$ solves $$\alpha^{2}\sigma_{1}^{2}+(1-\alpha)^{2}\sigma_{2}^{2}+2\alpha(1-\alpha)\rho_{12}\sigma_{1}\sigma_{2}=\sigma^{2},$$ preserving both present value and volatility.

Compare principal mapping, duration mapping, and cash-flow mapping in terms of accuracy.

Cash-flow mapping is the most accurate because it captures the full term structure of each payment; duration mapping is intermediate, capturing interest-rate sensitivity via a single duration vertex; principal mapping is the simplest and least accurate, ignoring intermediate coupons.

How is a forward foreign exchange contract mapped to risk factors?

A forward FX contract is decomposed into three positions: a long/short position in the foreign currency zero-coupon bond, a position in the domestic currency zero-coupon bond, and a spot FX exposure. Each component is mapped to its respective interest-rate and FX risk factors.

How are linear derivatives (e.g., forwards, futures) mapped to risk factors compared with non-linear derivatives (options)?

Linear derivatives are mapped by decomposing them into equivalent positions in underlying risk factors with constant (delta = 1-like) sensitivities. Non-linear derivatives (options) are mapped using delta (and sometimes gamma) to approximate exposure: the delta-normal method treats the option as a linear position of size $\Delta\times$ underlying, which is inaccurate for large moves.

What is the limitation of delta-normal mapping for options, and how is it addressed?

Delta-normal mapping is a first-order (linear) approximation that ignores the convexity (gamma) of option payoffs, understating risk for large price moves. It is improved by delta-gamma approximations that add a second-order term, or by full revaluation (Monte Carlo / historical simulation).

What does it mean for volatility to exhibit mean reversion, and how does GARCH capture it?

Mean reversion means volatility tends to revert toward a long-run average level over time. In GARCH(1,1), $$\sigma_{t}^{2}=\omega+\alpha u_{t-1}^{2}+\beta\sigma_{t-1}^{2},$$ the long-run variance is $V_{L}=\frac{\omega}{1-\alpha-\beta}$, and forecasts pull toward $V_{L}$ at a rate governed by $(\alpha+\beta)$.

State the GARCH(1,1) variance forecast for $k$ periods ahead and its mean-reversion interpretation.

$$E[\sigma_{t+k}^{2}]=V_{L}+(\alpha+\beta)^{k}\left(\sigma_{t}^{2}-V_{L}\right).$$ Since $\alpha+\beta<1$, the term $(\alpha+\beta)^{k}\to0$ as $k\to\infty$, so the forecast reverts to the long-run variance $V_{L}$. The persistence $(\alpha+\beta)$ controls the speed of reversion.

In GARCH(1,1), what condition ensures stationarity and mean reversion, and what does persistence near 1 imply?

Stationarity and mean reversion require $\alpha+\beta<1$ (with $\omega>0$). If $\alpha+\beta$ is close to 1, volatility is highly persistent and reverts very slowly; if $\alpha+\beta=1$ (EWMA/IGARCH), there is no mean reversion and shocks persist indefinitely.

How does the EWMA model relate to GARCH(1,1)?

EWMA is a special case of GARCH(1,1) with $\omega=0$, $\alpha=1-\lambda$, and $\beta=\lambda$, giving $$\sigma_{t}^{2}=\lambda\sigma_{t-1}^{2}+(1-\lambda)u_{t-1}^{2}.$$ Because $\alpha+\beta=1$, EWMA has no long-run variance and exhibits no mean reversion. RiskMetrics uses $\lambda=0.94$ for daily data.

What is the volatility term structure, and how does a mean-reverting model shape it?

The volatility term structure shows how forecasted (average) volatility varies with forecast horizon. With mean reversion, if current volatility is above $V_{L}$ the term structure slopes downward toward the long-run level, and if below $V_{L}$ it slopes upward; volatility-of-volatility decreases with horizon.

Distinguish implied volatility from realized (historical) volatility.

Implied volatility is the forward-looking volatility backed out of observed option prices via a pricing model (e.g., Black-Scholes); it reflects market expectations. Realized volatility is the backward-looking volatility computed from actual historical returns. Implied often exceeds realized, reflecting a volatility risk premium.

What empirical patterns are observed when comparing implied and realized volatility?

Implied volatility tends to exceed subsequently realized volatility on average (volatility risk premium), is a biased but informative predictor of future realized volatility, and exhibits the volatility smile/skew across strikes. Both display clustering and mean reversion, but implied vol reacts faster to market stress.

Define correlation risk and give an example.

Correlation risk is the risk that the correlation between assets or risk factors changes adversely, affecting portfolio value or hedge effectiveness. Example: a diversified portfolio loses its diversification benefit when correlations rise toward 1 during a crisis, or a basket option's value changes as pairwise correlations shift.

What is 'correlation breakdown' and why is it dangerous in stress periods?

Correlation breakdown is the tendency for correlations to change sharply (usually increase toward 1) during market stress, so that assets that were weakly correlated move together. It is dangerous because diversification benefits evaporate exactly when they are most needed, increasing tail losses beyond what normal-period correlations predict.

What is the empirical relationship between correlations and market volatility/direction?

Correlations are not constant; they tend to be higher during periods of high volatility and during market downturns (correlations rise in bear markets more than in bull markets — an asymmetry). This means equicorrelation assumptions understate joint tail risk.

List several stylized empirical properties of asset return correlations.

(1) Correlations are time-varying and exhibit clustering; (2) they tend to mean-revert toward a long-run level; (3) they rise during crises (correlation breakdown); (4) they show asymmetry, increasing more in down markets; (5) correlation estimates are noisier and less stable than volatility estimates.

Why can relying on a single historical correlation matrix understate portfolio risk?

Because correlations are unstable and rise during stress, a correlation matrix estimated over calm or mixed periods understates the co-movement that occurs in a crisis. This leads to underestimated portfolio VaR and overstated diversification benefits, motivating stress correlations and conditional (regime-dependent) models.

Compare the Gaussian copula and the use of a Student-$t$ copula for modeling dependence in risk management.

Both copulas join marginals into a joint distribution, but the Gaussian copula has zero tail dependence, so extreme joint losses are underestimated. The Student-$t$ copula has symmetric, positive tail dependence (controlled by degrees of freedom), better capturing the tendency for assets to crash together, making it more suitable for stress and tail-risk modeling.