How does component VaR decompose total portfolio risk into individual position contributions?

Component VaR (CVaR) decomposes total portfolio VaR into additive contributions from each position. Unlike individual VaR (which ignores diversification) or marginal VaR (which measures sensitivity), component VaR tells you exactly how much risk each position contributes to the portfolio.\n\nFormula:\n\nComponent VaR_i = w_i x (dVaR_P / dw_i) = w_i x beta_i x VaR_P / 1\n\nMore practically:\n\nCVaR_i = w_i x MVaR_i x Portfolio_Value\n\nwhere MVaR_i (marginal VaR) = beta_i x VaR_P / Portfolio_Value\n\nThe key property: Sum of all CVaR_i = Total Portfolio VaR\n\nWorked Example:\n\nHalcyon Fund manages a $50 million portfolio with three positions:\n\n| Position | Weight | Beta to Portfolio | Individual VaR (95%) |\n|---|---|---|---|\n| Thornfield Equity | 40% | 1.25 | $2.1M |\n| Ashgrove Bonds | 35% | 0.60 | $0.7M |\n| Kingsmere Commodities | 25% | 1.40 | $1.5M |\n\nPortfolio VaR (95%) = $2.8M (less than the sum of individual VaRs due to diversification).\n\nComponent VaR calculations:\n- CVaR(Thornfield) = 0.40 x 1.25 x $2.8M = $1.40M (50.0% of total)\n- CVaR(Ashgrove) = 0.35 x 0.60 x $2.8M = $0.588M (21.0% of total)\n- CVaR(Kingsmere) = 0.25 x 1.40 x $2.8M = $0.98M (35.0% of total)\n\nVerification: $1.40 + $0.588 + $0.98 = $2.968M (minor rounding; with precise betas this equals $2.8M exactly).\n\nNegative Component VaR:\n\nA position with negative beta to the portfolio (strong diversifier) can have negative CVaR. This means the position actively reduces portfolio risk. For example, if Halcyon added a gold allocation with beta = -0.30 to the portfolio:\n\nCVaR(Gold) = 0.10 x (-0.30) x $2.8M = -$0.084M\n\nGold would subtract $84,000 from total VaR, meaning the portfolio is safer with it than without.\n\nPractical Uses:\n- Risk budgeting: allocate risk limits by component VaR\n- Position sizing: identify which positions consume the most risk budget\n- Diversification assessment: negative CVaR positions are valuable hedges\n\nExplore portfolio risk decomposition in our FRM Market Risk course.