Chapter 22: Value at Risk and Expected Shortfall

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VaR and ES Defined

Value at Risk (VaR)

The loss level over a certain time horizon that is exceeded with a given probability.

$days\text{VaR}_{(X, T)} = \text{loss that has a } X\% \text{ chance of being exceeded over } T \text{ days}$

Example: A 1-day 99% VaR of $10 mi l l i o nm e an s t h er e i s a 1$ 10 million in a single day.

Expected Shortfall (ES, CVaR, C-VaR)

The expected loss conditional on the loss being greater than VaR:

$ES=E[L∣L>VaR]\text{ES} = E[L \mid L > \text{VaR}]$

ES captures tail risk better than VaR. Regulators (Basel Committee) now favor ES over VaR — Basel IV uses 97.5% ES for market risk capital.

VaR vs. ES

Measure	Advantages	Disadvantages
VaR	Easy to understand, widely used	Does not capture tail severity, not sub-additive
ES	Captures tail risk, sub-additive (coherent risk measure)	Harder to backtest, more sensitive to model assumptions

Historical Simulation

Use actual historical daily changes in market variables to construct scenarios. For a 1-day 99% VaR with 500 days of data:

Compute the change in portfolio value for each of the 500 historical scenarios
Sort the changes from worst to best
The 5th worst loss (1% of 500) is the VaR estimate

Advantages

Non-parametric (no distribution assumptions)
Captures correlations and fat tails naturally
Easy to explain to management

Disadvantages

Assumes the past will repeat
Limited by the length of historical data
Equal weighting of all historical observations (can be addressed with exponential weighting)

Model-Building Approach

The Linear Model

Assumes the change in portfolio value is a linear function of changes in market variables:

$ΔP=∑i=1nδiΔxi\Delta P = \sum_{i=1}^{n} \delta_i \Delta x_i$

where $δi\delta_i$ is the sensitivity (delta) to variable $i$ .

For a portfolio with normally distributed returns, the 1-day $X%X\%$ VaR is:

$VaR=N−1(X)⋅σP⋅V\text{VaR} = N^{-1}(X) \cdot \sigma_P \cdot V$

where $σP\sigma_P$ is the daily portfolio standard deviation and $V$ is portfolio value.

For converting to $T$ -day VaR (assuming i.i.d. returns):

$VaR(T)=VaR(1-day)⋅T\text{VaR}(T) = \text{VaR}(1\text{-day}) \cdot \sqrt{T}$

The Quadratic Model

For portfolios with significant gamma (options), the change is quadratic:

$ΔP=∑δiΔxi+12∑∑γijΔxiΔxj\Delta P = \sum \delta_i \Delta x_i + \frac{1}{2}\sum \sum \gamma_{ij} \Delta x_i \Delta x_j$

Monte Carlo Simulation for VaR

Assume a multivariate distribution for market variables
Sample from this distribution to generate scenarios
Revalue the portfolio under each scenario
Compute VaR and ES from the distribution of outcomes

This approach is the most flexible but also the most computationally intensive.

Back Testing

Compare actual daily losses with VaR estimates. If the 99% VaR is correct, exceptions (days when loss exceeds VaR) should occur approximately 1% of the time.

The Kupiec test (proportion of failures test) checks whether the observed exception rate is consistent with the VaR confidence level. Regulators use traffic light systems:

Green zone (0-4 exceptions): Acceptable
Yellow zone (5-9 exceptions): Requires investigation
Red zone (10+ exceptions): Model rejected, higher capital requirements

Principal Components Analysis (PCA)

PCA reduces the dimensionality of interest rate risk. Instead of modeling each maturity point separately, PCA identifies the main factors driving yield curve movements:

PC1 (Shift): Parallel shift of the yield curve (accounts for ~80-90% of variance)
PC2 (Twist): Steepening/flattening (changes slope) (5-10%)
PC3 (Butterfly): Curvature changes (1-5%)

Using these 3 factors captures 95-99% of yield curve movements, dramatically simplifying VaR computation for fixed-income portfolios.