11.2 Measuring and Managing Market Risk: VaR, CVaR & Stress Testing

11.2 Measuring and Managing Market Risk: VaR, CVaR & Stress Testing

Core Insight: Market risk is the potential for financial loss resulting from adverse movements in market prices, interest rates, exchange rates, and credit spreads. While traditional volatility ($\sigma$) measures two-sided dispersion, institutional risk management focuses on downside tail risk. Value at Risk (VaR), Conditional VaR (CVaR), and stress testing form the tripartite framework required to measure, monitor, and control portfolio market risk.

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1. Value at Risk (VaR): Formal Definition & Components

Value at Risk (VaR) is the minimum expected loss (in currency terms or as a percentage of portfolio value) over a specified time horizon $T$ at a given confidence level $(1 - \alpha)$ under normal market conditions.

$\text{Pr}(\text{Loss} \ge \text{VaR}) = \alpha \iff \text{Pr}(\text{Loss} \le \text{VaR}) = 1 - \alpha$

The Three Essential Elements of a VaR Statement

Every VaR declaration must specify three parameters:

1. Time Horizon ($T$): The holding period over which risk is measured (e.g., 1 day for trading desks, 10 days for Basel banking regulatory capital, 1 month for pension funds).
2. Confidence Level ($1 - \alpha$): Typically 95% ($\alpha = 5%$) or 99% ($\alpha = 1%$).
3. Loss Amount: Stated as a percentage ($VaR_{%}$) or currency value ($VaR_{$}$).

Example Interpretation: "A 1-day 95% VaR of $2.5 million means there is a 5% probability (or 1 trading day out of 20) that the portfolio will experience a loss of at least $2.5 million over a single day."

2. The Three Primary VaR Estimation Methods

VaR Estimation Methodologies
  ├── 1. Parametric (Variance-Covariance) -> Analytical, assumes normal distribution, fast
  ├── 2. Historical Simulation           -> Non-parametric, empirical price replays, fat tails
  └── 3. Monte Carlo Simulation          -> Stochastic paths, handles complex optionality, flexible

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Method 1: Parametric (Variance-Covariance) VaR

The Parametric VaR method (also known as the analytical or delta-normal method) assumes that asset returns follow a continuous probability distribution—typically the multivariate normal distribution.

Parametric VaR Formula

$VaR_{\%} = -(\mu_P - z_\alpha \sigma_P) = z_\alpha \sigma_P - \mu_P$

$VaR_{\$} = VaR_{\%} \times \text{Portfolio Value} = (z_\alpha \sigma_P - \mu_P) \times V_P$

Where:

• $\mu_P$ = Mean expected portfolio return over the horizon.
• $\sigma_P$ = Standard deviation of portfolio return over the horizon.
• $z_\alpha$ = Critical standard normal z-score corresponding to significance level $\alpha$.
  • 95% Confidence (5% tail): $z_{0.05} = 1.645$ (one-tailed)
  • 99% Confidence (1% tail): $z_{0.01} = 2.326$ (one-tailed)
  • 90% Confidence (10% tail): $z_{0.10} = 1.282$ (one-tailed)

Zero-Mean Assumption: For short time horizons (such as 1 day), the expected daily return $\mu_P$ is negligible and frequently set to zero ($\mu_P \approx 0$), simplifying the equation to $VaR = z_\alpha \sigma_P V_P$.

Time Horizon Scaling (Square Root Rule)

Under the assumption of independent and identically distributed (i.i.d.) normal returns, risk scales with the square root of time:

$\sigma_T = \sigma_1 \times \sqrt{T} \implies VaR_T = VaR_1 \times \sqrt{T}$

Example: If 1-day $VaR = $1,000,000$, the 10-day Basel regulatory $VaR = $1,000,000 \times \sqrt{10} = $3,162,277$.

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Method 2: Historical Simulation VaR

The Historical Simulation method applies the actual historical percentage price changes of all underlying risk factors over the past $N$ periods (e.g., past 500 or 1,000 trading days) to the current portfolio asset weights.

Step-by-Step Execution:

1. Collect historical daily return vectors for all portfolio assets over $N$ days.
2. Revalue the current portfolio holdings for each of the $N$ historical trading days to generate $N$ simulated portfolio returns: $R_{P,t} = \sum w_i R_{i,t}$.
3. Sort the $N$ simulated portfolio returns from lowest (most negative) to highest.
4. Identify the cutoff return corresponding to the $\alpha$ percentile. For $N = 1,000$ and a 95% confidence level ($\alpha = 5%$), the VaR is the 50th worst simulated return ($1,000 \times 0.05 = 50$).

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Method 3: Monte Carlo Simulation VaR

Monte Carlo Simulation generates thousands of hypothetical market trajectories based on user-specified stochastic processes, probability distributions, and parameter estimates (drift, volatility, correlations).

Step-by-Step Execution:

1. Specify stochastic models (e.g., Geometric Brownian Motion, jump-diffusion) and joint distributions for all risk factors.
2. Draw thousands of correlated random paths (e.g., 10,000 to 100,000 iterations).
3. Fully reprice all portfolio assets—including non-linear derivatives—at the horizon along each path.
4. Construct the empirical simulated P&L distribution and identify the $(1 - \alpha)$ percentile loss.

3. Comparative Evaluation of the Three VaR Methods

4. Conditional VaR (CVaR / Expected Shortfall)

The Fundamental Limitation of VaR

Value at Risk is inherently blind to the severity of losses beyond the VaR threshold. For instance, two portfolios may both have a 1-day 99% VaR of $10 million, but Portfolio A's maximum worst-case tail loss is $12 million, whereas Portfolio B (which wrote deep out-of-the-money options) faces a catastrophic tail loss of $100 million. VaR treats both portfolios as having identical risk.

Conditional VaR (CVaR) Definition

Conditional VaR (CVaR), also known as Expected Shortfall (ES) or Expected Tail Loss, measures the expected (average) loss given that the loss exceeds the VaR cutoff:

$CVaR_\alpha = E[L \mid L > VaR_\alpha] = \frac{1}{\alpha} \int_0^\alpha VaR_u \, du$

In a historical or Monte Carlo simulation with $M$ tail observations exceeding VaR, CVaR is simply the arithmetic mean of those $M$ tail losses:

$CVaR_\alpha = \frac{1}{M} \sum_{j=1}^M L_j \quad \text{for all } L_j > VaR_\alpha$

Coherence and the Subadditivity Property

A risk measure $\rho(\cdot)$ is defined as coherent (Artzner et al.) if it satisfies four mathematical axioms:

1. Monotonicity: If portfolio $X \le Y$ in all states, then $\rho(X) \ge \rho(Y)$.
2. Translation Invariance: $\rho(X + c) = \rho(X) - c$ for constant cash $c$.
3. Positive Homogeneity: $\rho(k X) = k \rho(X)$ for any scalar $k > 0$.
4. Subadditivity: $\rho(X + Y) \le \rho(X) + \rho(Y)$.

Critical Exam Distinction: VaR is NOT a coherent risk measure because it can violate subadditivity for non-normal, skewed, or fat-tailed distributions (e.g., portfolios with credit default risk or short option positions). Violating subadditivity means that merging two portfolios could result in a combined VaR that is greater than the sum of individual VaRs, falsely implying that diversification increases risk. In contrast, CVaR is mathematically coherent and strictly subadditive under all return distributions.

5. Sensitivity Measures, Stress Testing & Risk Limits

Sensitivity Risk Measures (First- and Second-Order)

While VaR provides a single aggregate loss number, risk managers require granular sensitivity metrics to diagnose specific market risk drivers:

Stress Testing and Scenario Analysis

Because VaR relies on statistical distributions calibrated during normal market conditions, it routinely fails during systemic market crises when correlations spike to 1.0 and liquidity vanishes. Stress testing complements VaR by analyzing severe, non-normal tail events:

1. Historical Scenarios: Replaying historical market shocks across the current portfolio (e.g., the 1987 Black Monday crash, 1998 Russian default/LTCM collapse, 2008 Lehman Brothers bankruptcy, and 2020 COVID-19 liquidity shock).
2. Hypothetical Scenarios: Modeling plausible forward-looking economic crises that have no historical precedent (e.g., a simultaneous 300 bps stagflation rate spike, 40% crude oil surge, and 200 bps corporate credit spread blowout).
3. Reverse Stress Testing: An institutional risk practice that starts with the catastrophic outcome—such as a loss that breaches the firm's regulatory capital buffer or causes insolvency—and works backward to identify the specific combination of market shocks that would trigger that failure.

Risk Limits Framework

Institutional capital allocation enforces multiple interlocking risk limits:

• Position Limits: Maximum dollar/nominal exposure to any single issuer or asset.
• Stop-Loss Limits: Mandatory risk reduction when cumulative losses breach predetermined thresholds.
• VaR Limits: Maximum allowable 1-day or 10-day VaR at the desk, division, and firm level.
• Scenario / Stress Limits: Maximum permitted loss under designated stress-test scenarios.

6. Worked Numerical Example: Parametric VaR, Historical VaR & CVaR

Part A: Parametric Portfolio VaR Calculation

An investment fund manages a $100 million portfolio split between two asset classes:

• Asset 1 (Equities): Weight $w_1 = 60%$ ($V_1 = $60\text{M}$), daily $\sigma_1 = 1.50%$.
• Asset 2 (Bonds): Weight $w_2 = 40%$ ($V_2 = $40\text{M}$), daily $\sigma_2 = 0.80%$.
• Correlation: $\rho_{12} = 0.20$. Daily expected returns $\mu_1 = \mu_2 = 0$.

Step 1: Calculate Daily Portfolio Volatility ($\sigma_P$)

$\sigma_P^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2$ $\sigma_P^2 = (0.60)^2 (0.015)^2 + (0.40)^2 (0.008)^2 + 2(0.60)(0.40)(0.20)(0.015)(0.008)$ $\sigma_P^2 = (0.36)(0.000225) + (0.16)(0.000064) + (0.096)(0.000120)$ $\sigma_P^2 = 0.00008100 + 0.00001024 + 0.00001152 = 0.00010276$ $\sigma_P = \sqrt{0.00010276} = 0.010137 \implies 1.0137\% \text{ per day}$

Step 2: Calculate 1-Day 95% and 99% Parametric VaR

• 1-Day 95% VaR ($z = 1.645$): $VaR_{1\text{d}, 95\%} = 1.645 \times 1.0137\% \times \$100\text{M} = \$1,667,537$
• 1-Day 99% VaR ($z = 2.326$): $VaR_{1\text{d}, 99\%} = 2.326 \times 1.0137\% \times \$100\text{M} = \$2,357,856$
• 10-Day 99% Regulatory VaR (scaled via $\sqrt{10}$): $VaR_{10\text{d}, 99\%} = \$2,357,856 \times \sqrt{10} = \$2,357,856 \times 3.162278 = \$7,456,214$

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Part B: Historical Simulation VaR and CVaR Calculation

A risk manager simulates 500 daily historical P&L returns for a $50 million fund. The 10 worst sorted simulated daily portfolio returns are:

\text{Rank 1: } -4.50\% & \text{Rank 2: } -3.80\% & \text{Rank 3: } -3.20\% & \text{Rank 4: } -2.90\% & \text{Rank 5: } -2.70\% \\ \text{Rank 6: } -2.50\% & \text{Rank 7: } -2.40\% & \text{Rank 8: } -2.30\% & \text{Rank 9: } -2.20\% & \text{Rank 10: } -2.10\% \end{array}$$ #### Step 1: Compute 1-Day 99% Historical VaR - Significance level $\alpha = 1 - 0.99 = 1.0\%$. - Tail cutoff rank: $500 \times 0.01 = 5\text{th worst observation}$. - Rank 5 return is $-2.70\%$. - **1-Day 99% Historical VaR:** $2.70\% \times \$50\text{M} = \mathbf{\$1,350,000}$. #### Step 2: Compute 1-Day 99% Conditional VaR (CVaR) - CVaR is the average of the losses in the tail beyond the VaR threshold (the 5 worst outcomes, Ranks 1 to 5): $$\text{Tail Losses} = 4.50\%, 3.80\%, 3.20\%, 2.90\%, 2.70\%$$ $$\text{Mean Tail Loss} = \frac{4.50\% + 3.80\% + 3.20\% + 2.90\% + 2.70\%}{5} = \frac{17.10\%}{5} = 3.42\%$$ - **1-Day 99% CVaR:** $3.42\% \times \$50\text{M} = \mathbf{\$1,710,000}$.