Forward Risk

Forward-Looking Risk Analysis

Core Concepts

Parametric (Variance-Covariance) VaR

Assumes returns are normally distributed. For a single asset or portfolio in dollar terms (assuming zero expected return over short horizons):

VaR = W * z_alpha * sigma_p

where:

• W = portfolio value
• z_alpha = z-score for confidence level (1.645 for 95%, 2.326 for 99%)
• sigma_p = portfolio volatility over the relevant horizon

More generally, including expected return:

VaR_alpha = mu - z_alpha * sigma

To convert from 1-day VaR to h-day VaR (assuming i.i.d. returns):

VaR_h = VaR_1 * sqrt(h)

Portfolio VaR (Multiple Assets)

For a portfolio with weight vector w and covariance matrix Sigma:

sigma_p = sqrt(w' * Sigma * w)
VaR_p   = W * z_alpha * sqrt(w' * Sigma * w)

The covariance matrix captures both individual volatilities and correlations between assets.

Monte Carlo VaR

Simulate a large number of portfolio return scenarios (e.g., 10,000+), then take the alpha-percentile of the simulated loss distribution.

Steps:

1. Estimate the return distribution parameters (mean vector, covariance matrix, or use a copula model).
2. Generate N random return scenarios (e.g., via Cholesky decomposition of the covariance matrix for multivariate normal).
3. Compute portfolio return for each scenario.
4. Sort results and identify the alpha-percentile loss.

Monte Carlo VaR can accommodate non-normal distributions, fat tails, path-dependent instruments, and nonlinear payoffs (e.g., options).

Conditional VaR (CVaR) / Expected Shortfall

CVaR answers: "Given that losses exceed VaR, what is the expected loss?"

ES_alpha = E[Loss | Loss > VaR_alpha]

For a normal distribution:

ES_alpha = mu + sigma * phi(z_alpha) / (1 - alpha)

where phi is the standard normal PDF.

CVaR is a coherent risk measure (unlike VaR) because it satisfies subadditivity: CVaR(A+B) <= CVaR(A) + CVaR(B). This means diversification always reduces or maintains CVaR, which is not guaranteed for VaR.

Component VaR

Decomposes total portfolio VaR into contributions from each position. Component VaRs sum to total VaR.

CVaR_i = w_i * beta_i * VaR_p

where beta_i = Cov(R_i, R_p) / Var(R_p) is the asset's beta to the portfolio.

Equivalently:

CVaR_i = w_i * (partial VaR / partial w_i)
sum(CVaR_i) = VaR_p

This decomposition identifies which positions are the largest contributors to portfolio risk.

Marginal VaR

Measures the rate of change of portfolio VaR with respect to a small increase in a position's weight.

MVaR_i = partial(VaR_p) / partial(w_i) = z_alpha * (Sigma * w)_i / sigma_p

Marginal VaR is used for position sizing: adding to a position with low marginal VaR reduces portfolio risk more efficiently.

Scenario Analysis

Apply specific historical or hypothetical market moves to the current portfolio to estimate P&L impact.

• Historical scenarios: Replay actual market events (e.g., 2008 GFC, 2020 COVID crash, 2022 rate hiking cycle) with current holdings.
• Hypothetical scenarios: Construct custom shocks (e.g., "equities -20%, rates +200bp, credit spreads +300bp, USD +10%").

Scenario P&L is computed by applying the scenario returns to current position exposures and revaluing.

Stress Testing

A structured framework for assessing portfolio resilience under extreme but plausible conditions.

Common stress scenarios:

• Equity crash: S&P 500 -30% to -40%
• Interest rate shock: +300bp parallel shift
• Credit crisis: investment-grade spreads +200bp, high-yield +800bp
• Liquidity freeze: bid-ask spreads widen 10x, forced selling at discount
• Currency shock: major currency pair moves 15-20%
• Stagflation: inflation +5%, GDP -3%, rates +200bp

Stress tests should include second-order effects: margin calls, liquidity demands, correlation spikes, counterparty risk.

Factor-Based Risk Decomposition

Separate total portfolio risk into systematic factor risk and idiosyncratic (security-specific) risk.

sigma^2_p = b' * Sigma_f * b + sum(w_i^2 * sigma^2_epsilon_i)

where:

• b = vector of portfolio factor exposures
• Sigma_f = factor covariance matrix
• sigma^2_epsilon_i = idiosyncratic variance of asset i

Common factor models: Fama-French (market, size, value, momentum), Barra risk models, PCA-based statistical factors.

Key Formulas

Worked Examples

Example 1: Parametric 95% VaR

Given: A $1,000,000 equity portfolio with an annualized volatility of 15%.

Calculate: 1-day 95% parametric VaR (assuming 252 trading days and zero expected daily return).

Solution:

Daily volatility:

sigma_daily = 0.15 / sqrt(252) = 0.15 / 15.875 = 0.00945

1-day 95% VaR:

VaR = $1,000,000 * 1.645 * 0.00945 = $15,545

Alternatively, computing directly from annual figures:

VaR_annual = $1,000,000 * 1.645 * 0.15 = $246,750
VaR_1day   = $246,750 / sqrt(252)       = $15,545

Interpretation: There is a 5% chance of losing more than $15,545 in a single day under normal market conditions.

Example 2: Monte Carlo VaR

Given: A two-asset portfolio (60% equities, 40% bonds). Equities: mu = 10%, sigma = 18%. Bonds: mu = 4%, sigma = 5%. Correlation rho = -0.2. Portfolio value = $1,000,000.

Calculate: 95% annual VaR via Monte Carlo simulation (conceptual steps).

Solution:

1. Construct covariance matrix:

Sigma = | 0.0324  -0.0018 |
        | -0.0018  0.0025 |

1. Cholesky decomposition of Sigma to get lower triangular matrix L.

1. Simulate 10,000 scenarios: For each simulation, draw z ~ N(0, I), compute r = mu + Lz, then portfolio return R_p = w' r.

1. Compute portfolio P&L for each scenario: P&L = $1,000,000 * R_p.

1. Sort P&L from worst to best. The 500th worst (5th percentile) is the 95% VaR.

For this portfolio, the analytical answer provides a benchmark:

sigma_p = sqrt(0.6^2 * 0.0324 + 0.4^2 * 0.0025 + 2 * 0.6 * 0.4 * (-0.0018))
        = sqrt(0.011664 + 0.0004 - 0.000864)
        = sqrt(0.0112)
        = 10.58%

VaR_95% = $1,000,000 * 1.645 * 0.1058 = $174,090

The Monte Carlo result should converge to approximately this value for a multivariate normal assumption.

Example 3: Expected Shortfall

Given: From the Monte Carlo simulation above, the losses exceeding VaR (the worst 500 out of 10,000 scenarios) have an average loss of $225,000.

Calculate: 95% CVaR.

Solution:

CVaR_95% = $225,000

Interpretation: When losses exceed the 95% VaR threshold, the average loss is $225,000. This is roughly 29% worse than the $174,090 VaR figure, highlighting the severity of tail events.

Common Pitfalls

• VaR says nothing about tail shape: VaR only identifies a threshold. Two portfolios with identical VaR can have vastly different tail losses. Always compute CVaR alongside VaR to understand tail severity.
• Parametric VaR assumes normality: Financial returns exhibit fat tails and skewness. Parametric VaR systematically underestimates tail risk. Use Monte Carlo with fat-tailed distributions or historical simulation for more realistic estimates.