Modeling Portfolio Risk Decomposition

When To Use

• Decomposing total portfolio variance into systematic (factor) and idiosyncratic (security-specific) components
• Attributing risk to individual factors (market, size, value, momentum, sector, rates, credit spread, etc.)
• Computing marginal contribution to risk (MCTR) and component contribution to risk (CCTR) for position-level decisions
• Evaluating whether portfolio risk budget aligns with intended factor tilts
• Preparing risk reports for investment committees, risk oversight, or investor communications
• Stress-testing factor exposure drift after rebalancing or new allocations

Inputs To Gather

• Position-level holdings: Weights (w_i) for each security or asset in the portfolio
• Factor exposure matrix (B): Security-level loadings on each factor (from regression, fundamental model, or vendor such as Barra/Axioma/Northfield)
• Factor covariance matrix (F): Covariance among systematic factors over the chosen estimation window
• Idiosyncratic variance matrix (D): Diagonal matrix of security-specific residual variances
• Return series (if estimating from scratch): Daily or monthly total returns for holdings and candidate factor proxies
• Benchmark (optional): Benchmark weights for active risk decomposition
• Lookback window and frequency: e.g., 252 trading days daily, 60 months monthly — confirm consistency with factor model vintage [VERIFY]

Workflow

1. Assemble the covariance structure

   • Total portfolio covariance: V = B F Bᵀ + D
   • Confirm dimensions match (n securities × k factors)
   • If using a vendor model, verify the factor covariance vintage date matches the holdings date [VERIFY]

2. Compute total portfolio variance

   • σ²_p = wᵀ V w
   • Report as annualized volatility: σ_p = √(σ²_p × annualization scalar)
   • Typical annualization: ×252 for daily, ×12 for monthly returns

3. Decompose into systematic and idiosyncratic

   • Systematic variance: σ²_sys = wᵀ (B F Bᵀ) w
   • Idiosyncratic variance: σ²_idio = wᵀ D w
   • Report the ratio σ²_sys / σ²_p — well-diversified equity portfolios typically show 85–95% systematic; significantly lower values signal concentrated idiosyncratic bets

4. Factor-level risk attribution

   • Portfolio factor exposures: β_p = Bᵀ w (k × 1 vector)
   • Factor contribution to variance for factor j: β_p,j × Σ_k (F_jk × β_p,k)
   • Sum of all factor contributions should equal σ²_sys (use as a reconciliation check)
   • Rank factors by absolute contribution to identify dominant risk drivers

5. Marginal contribution to risk (MCTR)

   • MCTR_i = (V w)_i / σ_p — measures how a small increase in weight_i changes portfolio volatility
   • Component contribution to risk: CCTR_i = w_i × MCTR_i
   • Verify: Σ CCTR_i = σ_p (Euler decomposition identity)

6. Active risk decomposition (if benchmark provided)

   • Active weights: w_a = w_portfolio − w_benchmark
   • Tracking error: TE = √(w_aᵀ V w_a × annualization)
   • Decompose TE into factor-active and stock-specific-active components using the same B F Bᵀ + D split
   • Report active factor bets and their contribution to tracking error

7. Sensitivity and stress analysis

   • Shock individual factor returns (e.g., +/−1σ, +/−2σ) and recompute portfolio P&L impact
   • Test covariance regime shifts: substitute a crisis-period F (e.g., 2008, 2020 March) and recompute σ_p
   • Evaluate concentration: flag any single factor contributing >40% of total variance or any single name contributing >10% of CCTR

Output

• Risk summary table: Total volatility, systematic vs. idiosyncratic split (absolute and percentage)
• Factor attribution table: Each factor's exposure, variance contribution, and percentage of total risk — sorted by magnitude
• MCTR / CCTR table: Per-position marginal and component contributions, highlighting top-10 risk contributors
• Active risk panel (if applicable): Tracking error, factor-active vs. stock-specific-active decomposition
• Stress scenario results: Portfolio impact under specified factor shocks and regime covariance matrices
• Reconciliation checks: Confirm variance decomposition sums match total; flag discrepancies >0.1%

Quality Checks

• Factor contributions sum to systematic variance within rounding tolerance
• CCTR values sum to total portfolio volatility (Euler identity holds)
• No negative idiosyncratic variances (indicates data or model error)
• Lookback window is consistent across factor model, covariance estimation, and holdings snapshot [VERIFY]
• Factor model covers ≥90% of portfolio NAV by weight; flag uncovered positions and their assumed treatment (benchmark proxy, peer mapping, or exclusion)
• Cross-check volatility estimate against realized rolling volatility — deviations >30% warrant investigation
• If using third-party risk model, confirm model version and release date match intended analysis period [VERIFY]