What is approximate revaluation of VaR?

Approximate revaluation of VaR replaces the computationally expensive full revaluation of each position (repricing every instrument from scratch under each risk scenario) with a mathematical approximation using sensitivities. The most common approximations: (1) Delta approximation — linear sensitivity to each risk factor; (2) Delta-gamma approximation — adds the second-order (convexity) term from a Taylor series expansion; (3) Key rate duration revaluation — for fixed income, applies duration sensitivities at specific tenor points rather than repricing the full yield curve. The trade-off: approximate methods are 10-100x faster than full revaluation but introduce approximation error, especially for positions with significant convexity (options, MBS, callable bonds).

When does the delta-gamma approximation break down?

Delta-gamma approximation breaks down in four situations: (1) Large risk factor moves — the Taylor series is a local approximation; for a 1-day VaR at the 99th percentile, the moves are usually small enough for delta-gamma to be accurate, but for stressed VaR over longer horizons, errors increase. (2) High-gamma positions — deep in-the-money or near-the-money options have significant gamma; the approximation captures gamma but misses higher-order terms (speed). (3) Vega exposure — delta-gamma only captures price sensitivity, not the volatility sensitivity. For options portfolios with significant vega, a separate vega approximation is needed. (4) Path-dependent instruments — barriers, lookbacks, and mortgage-backed securities with prepayment optionality have payoffs that depend on the path of rates, not just the endpoint. Delta-gamma assumes payoff depends only on the terminal risk factor value.

What is key rate duration revaluation for fixed income VaR?

Key rate duration (KRD) revaluation decomposes a bond portfolio's interest rate sensitivity into sensitivities at specific maturity buckets (typically 9-12 key rates: 3M, 6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 20Y, 30Y). For each scenario, the yield change at each key rate point is multiplied by the portfolio's KRD at that point, and the sum across all key rates gives the approximate P&L: ΔP ≈ -Σ(KRD_i × Δy_i × P), where i indexes the key rate tenors. This is more accurate than using a single modified duration because it captures the non-parallel yield curve shifts that drive most real-world interest rate risk.

What is the regulatory treatment of approximate revaluation in VaR models?

Regulatory guidance (Basel II/III, SR 11-7) requires banks to validate their VaR models, including the adequacy of revaluation methods. The SR 11-7 model risk management guidance requires that approximation error be documented, quantified, and incorporated into model uncertainty adjustments. The FRTB (Basel IV) Internal Models Approach requires backtesting of P&L attribution — the gap between the model's approximate P&L and the actual P&L (the P&L explain) is a key validation metric. If the P&L explain ratio (modeled P&L / actual P&L) consistently falls outside the 80-120% range, the approximation is likely inadequate for that instrument class.

Approximate Revaluation of VaR — Delta-Gamma & Duration

How quant risk teams use approximate revaluation for fast VaR calculation: delta approximation, delta-gamma Taylor series for options books, key rate duration revaluation for fixed income portfolios, and P&L attribution validation under FRTB. Claude AI prompts.

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Full Revaluation vs. Approximate Revaluation

Value at Risk requires repricing a portfolio under thousands of scenarios — historical simulation typically uses 250-500 scenarios; Monte Carlo uses thousands. Full revaluation reprices every instrument under every scenario from scratch using the full pricing model. For a portfolio with 10,000 positions repriced under 1,000 scenarios, full revaluation requires 10 million pricing calls. Approximate revaluation computes sensitivities once (delta, gamma, vega, key rate durations) and applies them across all scenarios using Taylor series expansion — 10 million multiplications instead of 10 million pricing calls, running orders of magnitude faster.

The choice between full and approximate revaluation is instrument-by-instrument and scenario-by-scenario. Linear instruments (bonds, futures, forwards) have negligible approximation error even with just delta. Options and structured products require at minimum delta-gamma; path-dependent instruments ideally get full revaluation. Hybrid approaches are common: full revaluation for nonlinear books, delta-gamma for linear books.

Delta Approximation (Linear VaR)

The simplest approximation: portfolio P&L is linear in the risk factor changes.

"Delta-only P&L approximation for a bond portfolio: I have 45 bonds with modified durations and market values. For each of the following 5 yield scenarios (parallel shifts of -200bps, -100bps, 0, +100bps, +200bps), compute the approximate P&L for each bond using: ΔP ≈ -MD × ΔY × P, where MD is modified duration, ΔY is the yield change in decimal, and P is market value. Sum across all bonds to get portfolio P&L under each scenario. Also compute: (1) The 1-day 99th percentile VaR, assuming yield changes are normally distributed with daily standard deviation 8bps; (2) The portfolio's dollar duration (DV01): how much does the portfolio gain/lose per 1bp parallel yield shift? (3) Identify the 5 bonds contributing most to yield sensitivity."
"Delta approximation for an equity portfolio using factor sensitivities: my portfolio has 80 equity positions. I have estimated each position's beta to 4 factors: Market (SPX), Size (IWM-SPX), Value (RPV-RPG), and Momentum (MTUM). Daily factor return standard deviations: Market 0.85%, Size 0.40%, Value 0.35%, Momentum 0.60%. Factor correlation matrix: [provided]. Using the delta approximation, compute the 1-day 99th percentile VaR using the variance-covariance method: Portfolio variance = β' Σ β, where β is the vector of portfolio factor exposures and Σ is the factor covariance matrix. The 99th VaR = 2.326 × √(Portfolio variance × Portfolio value²). Show: factor contribution to VaR for each of the 4 factors."

Delta-Gamma Approximation (Taylor Series)

Adding the second-order term captures convexity — crucial for options and callable bonds.

"Delta-gamma P&L approximation for an options book: I have a portfolio of 20 equity option positions (calls and puts, various strikes and expiries on SPX). For each position: delta, gamma, vega, and notional. Risk factor: SPX daily move. (1) Compute the portfolio delta: sum of (position delta × notional) — this is the first-order P&L sensitivity. (2) Compute the portfolio gamma: sum of (position gamma × notional) — this is the second-order term. (3) For a SPX move of +2% and -2%, compute approximate portfolio P&L using: ΔP ≈ δ × ΔS + ½ × Γ × ΔS², where ΔS is the SPX move in dollar terms. (4) Compute the 1-day 99th percentile VaR using the delta-gamma-normal method: the distribution of P&L under the delta-gamma approximation is non-normal (it's a quadratic function of a normal variable). Use the Cornish-Fisher expansion to adjust the 99th percentile quantile for skewness and excess kurtosis: adjusted z = z_99 + (z_99² - 1)/6 × γ₁ + (z_99³ - 3z_99)/24 × γ₂ - (2z_99³ - 5z_99)/36 × γ₁², where γ₁ is skewness and γ₂ is excess kurtosis of the P&L distribution."
"Comparing full revaluation vs. delta-gamma approximation accuracy: I have 6 option positions (including 2 near-the-money short-dated options with significant gamma). For 10 SPX scenarios ranging from -5% to +5%: (1) Compute the exact option value under each scenario using Black-Scholes (inputs: current S, K, T, r, σ for each option); (2) Compute the delta-gamma approximation under each scenario; (3) Compare: for each scenario and each option, compute the approximation error (exact - approximate) in dollars and as a percentage of full revaluation P&L. (4) Identify which options show the largest approximation error and at what scenario size the error becomes material. This establishes whether delta-gamma is adequate for this book or whether full revaluation is needed."
"Delta-gamma-vega approximation for an options portfolio with volatility risk: in addition to delta and gamma, my portfolio has significant vega (sensitivity to implied volatility). The daily implied vol move has standard deviation 0.8 vol points. Portfolio vega: $42,000 per vol point. For a 2D scenario grid (SPX moves from -3% to +3% in 1% steps; IV moves from -2 to +2 vol points in 1-point steps): compute the approximate P&L at each grid point as: ΔP ≈ δ × ΔS + ½ × Γ × ΔS² + ν × Δσ. Show the P&L matrix as a heatmap description (most profitable and most loss-making scenarios). This is the standard risk report for options trading desks — the 'greeks P&L grid'."

Key Rate Duration Revaluation for Fixed Income VaR

"Key rate duration VaR for a bond portfolio: my portfolio holds 35 bonds (government, IG corporate, HY, MBS). I have computed key rate durations (KRDs) at 9 tenors: 0.25Y, 0.5Y, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 30Y. Portfolio total market value: $280M. The 9×9 yield curve factor covariance matrix (based on 2-year historical daily yield changes at each tenor) is provided. (1) Compute portfolio-level KRD vector: for each tenor bucket, sum the dollar duration contribution (KRD_i × MV_i) across all bonds; (2) Compute 1-day yield change standard deviations at each tenor from the covariance matrix; (3) Compute 1-day VaR using the variance-covariance method: Portfolio P&L variance = KRD_vector' × Cov_matrix × KRD_vector; VaR(99%) = 2.326 × √(variance); (4) Compute the KRD contribution to VaR at each tenor (marginal contribution); (5) Compute the diversification benefit: undiversified VaR (sum of individual tenor VaRs) vs. diversified VaR — the difference shows how much yield curve correlation reduces the portfolio risk."
"Historical simulation with KRD revaluation: I want to compute 1-day 99% VaR using historical simulation over 260 daily scenarios (1 year of history). I have: (1) portfolio KRDs at 9 tenor buckets; (2) 260 historical daily yield changes at each tenor. Rather than repricing each bond under each of the 260 scenarios (260 × 35 = 9,100 pricing calls), use KRD revaluation: for each scenario date d, approximate P&L = -Σ(KRD_i × Δy_{i,d} × Portfolio_MV). This gives 260 daily P&L estimates. Sort the 260 values and identify the 3rd worst loss (99th percentile of the 260 observations). Also: (1) Identify the top 5 worst historical scenarios and the date they occurred — what market event drove each? (2) Compute the realized standard deviation of the 260 daily P&Ls — what is the implied annual volatility of the portfolio?"
"Stressed VaR using KRD approximation: Regulators require Stressed VaR using a 12-month historical window that includes a period of significant financial stress. I need to identify the optimal stress window from daily yield curve data from 2007-2025. For each possible 260-day window, compute the portfolio Stressed VaR using KRD revaluation: (1) For each 260-day window, compute portfolio daily P&L under each scenario using KRDs; (2) Take the 99th percentile of the 260 P&Ls as the Stressed VaR for that window; (3) Identify the 260-day window with the highest Stressed VaR — this is the regulatory worst-case period. For a bond portfolio with 7-year duration, the 2013 taper tantrum or 2022 rate shock are likely candidates. Show the 5 highest-VaR windows and the dates."

P&L Attribution: Validating Approximation Quality

Under FRTB, approximate revaluation must be validated against actual P&L. The P&L attribution test measures how well the model explains real daily P&L changes.

"P&L attribution analysis for model validation: over the last 60 trading days, I have two P&L time series: (1) Hypothetical P&L (HPL): the portfolio P&L computed by full revaluation using yesterday's portfolio and today's market prices; (2) Risk-theoretical P&L (RTPL): the approximate P&L computed by the VaR model using delta-gamma revaluation. For each day, compute: HPL - RTPL = unexplained P&L (model error). Analyze: (1) Mean and standard deviation of unexplained P&L; (2) Unexplained P&L as a percentage of |HPL|; (3) Count days where unexplained P&L exceeds 20% of |HPL| — under FRTB, exceeding this threshold on more than 4 out of 60 days triggers the P&L attribution test failure; (4) Identify which risk factor exposures (delta? gamma? vega?) are driving the largest unexplained P&L days — likely the positions with the highest convexity."

Approximate Revaluation of VaR: Delta-Gamma and Key Rate Duration Methods with AI (2026)

Full Revaluation vs. Approximate Revaluation

Delta Approximation (Linear VaR)

Delta-Gamma Approximation (Taylor Series)

Key Rate Duration Revaluation for Fixed Income VaR

P&L Attribution: Validating Approximation Quality

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Approximate Revaluation of VaR: Delta-Gamma and Key Rate Duration Methods with AI (2026)

Full Revaluation vs. Approximate Revaluation

Delta Approximation (Linear VaR)

Delta-Gamma Approximation (Taylor Series)

Key Rate Duration Revaluation for Fixed Income VaR

P&L Attribution: Validating Approximation Quality

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