How do you decompose portfolio VaR by risk factor using a multi-factor model?

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AcadiFi · Accepted Answer

Factor-based VaR decomposition attributes portfolio risk to underlying systematic risk factors rather than individual positions. This approach reveals the true sources of risk and prevents double-counting when multiple positions share the same factor exposures. **Multi-Factor VaR Framework:** Portfolio return is modeled as: r_P = sum(beta_k x f_k) + epsilon where beta_k is the portfolio's sensitivity to factor k, f_k is the factor return, and epsilon is idiosyncratic risk. The systematic VaR is: VaR_P = z_alpha x sqrt(B^T x Sigma_F x B + sigma_epsilon^2) where B is the factor loading vector and Sigma_F is the factor covariance matrix. ```mermaid graph TD A["Portfolio VaR"] --> B["Factor Decomposition"] B --> C["Equity Market Factor
beta_1 x sigma_equity"] B --> D["Interest Rate Factor
beta_2 x sigma_rate"] B --> E["Credit Spread Factor
beta_3 x sigma_credit"] B --> F["FX Factor
beta_4 x sigma_fx"] B --> G["Idiosyncratic Risk
sigma_epsilon"] C --> H["Factor CVaR_1"] D --> I["Factor CVaR_2"] E --> J["Factor CVaR_3"] F --> K["Factor CVaR_4"] G --> L["Specific Risk VaR"] H --> M["Sum = Total VaR"] I --> M J --> M K --> M L --> M ``` **Worked Example:** Bramblewood Asset Management runs a $200M multi-asset portfolio. A three-factor model (equity, rates, credit) yields: | Factor | Portfolio Beta | Factor Volatility (daily) | |---|---|---| | Equity | 0.85 | 1.10% | | Rates | -3.20 (duration) | 0.05% (yield change) | | Credit | 2.10 | 0.08% | Factor correlation matrix: | | Equity | Rates | Credit | |---|---|---|---| | Equity | 1.00 | -0.25 | 0.55 | | Rates | -0.25 | 1.00 | -0.30 | | Credit | 0.55 | -0.30 | 1.00 | Factor contributions to portfolio variance (after covariance computation): - Equity contribution: 62.4% of total variance - Rates contribution: 18.7% - Credit contribution: 14.2% - Cross-factor (already allocated): 4.7% (idiosyncratic) If total portfolio VaR(99%) = $6.4M: - Equity factor VaR = 0.624 x $6.4M = **$3.99M** - Rates factor VaR = 0.187 x $6.4M = **$1.20M** - Credit factor VaR = 0.142 x $6.4M = **$0.91M** - Idiosyncratic VaR = 0.047 x $6.4M = **$0.30M** **Advantages over Position Decomposition:** - Identifies hidden concentrations (e.g., equity and credit might both load heavily on economic growth) - Supports hedging decisions (hedge the factor, not individual positions) - Works naturally with scenario analysis (stress one factor at a time) Practice factor decomposition problems in our FRM course.

How do you decompose portfolio VaR by risk factor using a multi-factor model?

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