How do you use marginal VaR to evaluate the risk impact of adding a new position to an existing portfolio?

Marginal VaR (MVaR) measures the rate of change of portfolio VaR with respect to a small change in a position's weight. It is precisely the partial derivative of VaR with respect to the position weight, and it provides the most efficient tool for evaluating whether a new allocation increases or decreases total risk.\n\nFormula:\n\nMVaR_i = dVaR_P / dw_i = beta_i x VaR_P\n\nwhere beta_i is the beta of asset i with respect to the portfolio (Cov(r_i, r_P) / Var(r_P)).\n\nFor a parametric (normal) VaR:\n\nMVaR_i = z_alpha x (Sigma x w)_i / sigma_P\n\nwhere Sigma is the covariance matrix, w is the weight vector, and sigma_P is portfolio volatility.\n\n`mermaid\ngraph TD\n A[\"Current Portfolio
VaR_P = $3.2M\"] --> B{\"Consider New Asset\"}\n B --> C[\"Calculate beta_new
to portfolio\"]\n C --> D[\"MVaR_new = beta_new x VaR_P\"]\n D --> E{\"Compare MVaR_new
to average MVaR\"}\n E -->|\"MVaR_new < Avg MVaR\"| F[\"Adding reduces
risk per dollar\"]\n E -->|\"MVaR_new > Avg MVaR\"| G[\"Adding increases
risk per dollar\"]\n E -->|\"MVaR_new < 0\"| H[\"Strong diversifier
reduces total VaR\"]\n`\n\nWorked Example:\n\nStonecrest Advisors manages a $120M equity portfolio with VaR(99%) = $4.8M. They consider adding Meridian Biotech with the following data:\n\n- Portfolio annual volatility: 16%\n- Meridian annual volatility: 28%\n- Correlation(Meridian, Portfolio): 0.35\n\nBeta of Meridian to portfolio:\nbeta_M = rho x sigma_M / sigma_P = 0.35 x 0.28 / 0.16 = 0.6125\n\nMVaR(Meridian) = 0.6125 x $4.8M = $2.94M per 100% weight\n\nPer $1M invested: $2.94M / $120M x $1M = $24,500\n\nAverage MVaR per $1M currently: $4.8M / $120M x $1M = $40,000\n\nSince Meridian's MVaR per dollar ($24,500) is below the portfolio average ($40,000), adding it would reduce the portfolio's VaR-to-capital ratio. The low correlation (0.35) provides diversification that makes the position risk-efficient.\n\nDecision Rule:\n- If MVaR_new < average MVaR per dollar, the new position improves risk efficiency\n- If MVaR_new > average MVaR per dollar, the new position worsens risk efficiency\n- The optimal portfolio has equal MVaR per dollar across all positions (risk parity condition)\n\nLimitations:\n- MVaR is a local measure valid only for small weight changes\n- For large allocations, incremental VaR provides a better estimate\n- Assumes constant covariance structure (breaks down in stressed markets)\n\nDive into portfolio risk analytics in our FRM course.