14.2 Calculating and Applying VaR

Calculating and Applying VaR

Once VaR and ES are defined, the exam asks how desks compute them. VRM–2 contrasts portfolio linearity, three workhorse engines (historical simulation, delta-normal, Monte Carlo), and the practical failures of correlation assumptions and “VaR-only” thinking.

Linear Versus Nonlinear Portfolios

A portfolio is linear (in the risk-factor approximation) if P&L is well described by first-order (“delta”) exposures: stocks, FX spot, futures, and many cash bonds over small yield moves. For linear books, a covariance-based or historical return shock maps almost one-for-one into P&L.

A portfolio is nonlinear when value depends on convexity, optionality, or thresholds: equity options, bermudan callables, mortgage-backed securities with prepayment options, credit derivatives with discontinuous default. For these, a large factor move can produce P&L far from delta × shock. Risk engines respond with:

• Full revaluation under each scenario (historical or Monte Carlo),
• Delta-gamma (quadratic) analytic approximations,
• Or grid / least-squares interpolations between revaluation nodes.

Historical Simulation VaR and ES

Historical simulation (HS) applies past risk-factor moves (or past portfolio returns) to today’s positions and builds an empirical loss distribution.

Steps:

1. Choose a lookback window (e.g., 250 or 500 trading days) and horizon H (often 1 day).
2. For each historical day t, compute the simulated P&L of today’s book under the factor changes observed on day t (full revaluation or mapped returns).
3. Order the simulated losses from worst to best.
4. VaR_α is the empirical α-quantile (e.g., the 99th percentile loss).
5. ES_α is the average of losses strictly worse than (or at/beyond) that quantile, per your firm’s discrete convention.

Worked historical VaR/ES

Ten equally weighted historical one-day losses ($m), already sorted worst to best: 12, 9, 7, 5, 4, 3, 2, 1.5, 1, 0.5. For 90% VaR, the worst 10% is the single worst day → VaR_0.90 = 12. ES_0.90 = 12. For 80% VaR, the worst 20% covers {12, 9}; if VaR is defined as the milder edge of that tail band under a common discrete rule, desks often set VaR_0.80 = 9 and ES_0.80 = (12 + 9) / 2 = 10.5. Exact percentile interpolation varies by vendor; GARP cares that you know quantile versus tail average.

Strengths: no parametric distribution; naturally captures fat tails and some nonlinearities if you full-revalue; easy to explain to non-quants. Weaknesses: trapped in the sample (no worse crash than history); equal weight on old and new days unless you use weighted HS / age-weighted schemes; slow to reflect a new volatility regime unless the window is short (which then worsens quantile noise).

Delta-Normal VaR for Linear and Nonlinear Books

Delta-normal (variance-covariance) VaR assumes risk-factor returns are jointly normal (or elliptical) and maps positions through linear sensitivities.

For a linear portfolio, portfolio variance is σ_p² = wᵀΣw (or δᵀΣδ for dollar deltas). With zero mean over a short horizon,

VaR_α ≈ z_α × σ_p × V

where z_α is the standard normal quantile (1.65 for 95%, 2.33 for 99%).

Worked linear delta-normal

Two-asset book: dollar exposures x₁ = $40m, x₂ = $60m. Daily volatilities σ₁ = 1.2%, σ₂ = 0.8%, correlation ρ = 0.40.

σ_p² = (0.40)²(0.012)² + (0.60)²(0.008)² + 2(0.40)(0.60)(0.012)(0.008)(0.40)

Compute term-by-term:

• x₁²σ₁² = 0.16 × 0.000144 = 0.00002304
• x₂²σ₂² = 0.36 × 0.000064 = 0.00002304
• 2 x₁ x₂ σ₁ σ₂ ρ = 2 × 0.40 × 0.60 × 0.012 × 0.008 × 0.40 = 0.000018432

σ_p² = 0.000064512 → σ_p ≈ 0.00803 (0.803% of portfolio notional if weights are on $100m). One-day 99% VaR ≈ 2.33 × 0.00803 × 100m ≈ $1.87 million.

Nonlinear (delta-gamma) sketch

For an option-like position, a second-order Taylor expansion gives

ΔV ≈ δ ΔS + (1/2) Γ (ΔS)² + …

Under normal ΔS, the quadratic term makes ΔV non-normal (chi-square–like component). Cornish-Fisher or moment-matching adjustments shift the VaR quantile; alternatively, compute Var(ΔV) including gamma terms and still apply a normal z—faster but biased for large moves. Exam point: delta-normal alone misprices large moves in options; use delta-gamma carefully or switch to full revaluation.

Limits of delta-normal

• Assumes elliptical/normal factors → thin tails unless you fatten z.
• Linear (or local quadratic) mapping fails for barriers, digitals, and mortgage convexity spikes.
• Requires a full PSD covariance matrix; estimated correlations are noisy.
• Does not naturally produce ES without extra tail assumptions.
• Blind to liquidity jumps and gap risk overnight.

Monte Carlo VaR and ES

Monte Carlo (MC) draws risk-factor scenarios from a specified joint distribution (possibly with fat tails, stochastic vol, or copulas), revalues the book in each draw, and reads VaR/ES from the simulated loss histogram.

Pros:

• Handles nonlinear and path-dependent products.
• Can embed regime switches, jumps, and stressed correlations.
• Produces a full loss distribution → ES is immediate.

Cons:

• Model risk in the data-generating process dominates.
• Computationally expensive for full revaluation of complex books (hence proxy Greeks, adjoint AD, or least-squares MC).
• Simulation noise in far-tail ES needs many paths or importance sampling.
• Can create false precision if governance does not challenge inputs.

Correlation Breakdown and Crisis VaR

In calm markets, diversification “works”: average pairwise equity correlations might be 0.3–0.5. In crises, correlations often spike toward 1 as common factors dominate—correlation breakdown from the risk manager’s hopeful calibration, not from mathematics failing. A covariance matrix estimated in a quiet window then understates portfolio σ_p and VaR exactly when capital is most needed.

Mitigations tested on FRM items: use stressed VaR windows (2008-style), EWMA/GARCH correlations that react faster, factor models with a dominant market factor, floored correlations, and scenario add-ons that set ρ = 1 for key pairs.

Worst-Case Analysis Versus VaR

VaR answers a probabilistic question at a chosen α. Worst-case (or max-loss / scenario) analysis asks what happens under an adversarially chosen shock set—sometimes within a uncertainty set for factors (robust optimization), sometimes as named historical scenarios (1987, 1998, 2008, COVID crash).

Worst-case numbers are usually larger than VaR and are not probabilities. Regulators and GARP expect both: VaR/ES for routine statistical control, scenarios for “what the quantile missed.” Never treat a 99% VaR as a maximum loss.