How is delta-normal VaR calculated for a multi-asset portfolio?

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AcadiFi · Accepted Answer

Delta-normal VaR linearizes positions using their deltas (first-order sensitivities) to underlying risk factors, then treats the risk factor returns as jointly normal. Steps: build the sensitivity vector delta (dollar change in portfolio per unit move in each factor), build the factor covariance matrix Sigma from historical returns, compute portfolio variance as delta' * Sigma * delta, take the square root to get portfolio volatility sigma_p, and apply VaR = z * sigma_p where z is the confidence quantile. Example: two positions, $10M in S&P with daily vol 1.1% and $5M in 10Y Treasury with daily vol 0.6%, correlation -0.3. Portfolio variance = (10 * 0.011)^2 + (5 * 0.006)^2 + 2 * 10 * 0.011 * 5 * 0.006 * (-0.3) = 0.01210 + 0.00090 - 0.00198 = 0.01102 so sigma_p = $0.105M. 99% daily VaR = 2.326 * $105K = $244K. Strengths: fast, analytical, intuitive decomposition into marginal and component VaR. Weaknesses: fails for nonlinear payoffs (options, mortgages with prepayment), ignores fat tails, assumes stable correlations. Banks still use delta-normal for daily risk reports on linear books, while equity and rates desks with optionality use full-revaluation Monte Carlo or historical simulation. FRM commonly asks for component VaR, which equals (delta_i * (Sigma * delta)_i) / sigma_p, and for interpreting negative component VaR as a diversifier.

How is delta-normal VaR calculated for a multi-asset portfolio?

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