What's the difference between common factor variance and specific variance?

Question

AcadiFi · Accepted Answer

In a multi-factor model Rᵢ = αᵢ + β₁ᵢF₁ + β₂ᵢF₂ + ... + βₖᵢFₖ + εᵢ, total return variance decomposes as:

Var(Rᵢ) = Common Variance + Specific Variance

Common Variance = βᵢ' Σ_F βᵢ (sum of factor exposures squared times factor covariance matrix)

Specific Variance = σ²(εᵢ)

Common variance is the portion explained by systematic factors — it's shared with other assets via similar factor exposures. Not diversifiable in cross-section.

Specific variance is the idiosyncratic residual — unique to that asset, uncorrelated across assets. Diversifies away in large portfolios.

Example: 2-factor model, factor 1 is market (variance 0.04), factor 2 is value (variance 0.01), factors uncorrelated. Stock A has β₁ = 1.2, β₂ = 0.3, residual variance 0.008.

- Common variance = (1.2)² × 0.04 + (0.3)² × 0.01 = 0.0576 + 0.0009 = 0.0585
- Specific variance = 0.008
- Total variance = 0.0665
- Common share = 0.0585/0.0665 = 88%
- R² = 88% of the stock's variance is explained by factors.

Portfolio context: N equally-weighted stocks, each with specific variance σ²_ε. Portfolio specific variance = σ²_ε / N → 0 as N → ∞. Portfolio common variance persists because all stocks share factor exposures.

This is why diversification reduces specific risk but not systematic risk — a core CAPM/APT insight.

Practical use: risk attribution tables in Barra-style systems split portfolio tracking error into factor-contribution (common) and stock-specific parts. Managers typically underweight specific risk (via diversification) while taking factor bets.

What's the difference between common factor variance and specific variance?

Master Level II with our CFA Course

Related Questions

Practice Questions