Risk Decomposition

Risk Decomposition: Systematic vs. Idiosyncratic

CAPM decomposes total asset risk into two components:

$\sigma_i^2 = \underbrace{\beta_i^2 \sigma_m^2}_{\text{systematic}} + \underbrace{\sigma_{\varepsilon_i}^2}_{\text{idiosyncratic}}$

Systematic Risk (Market Risk)

$\sigma_{\text{sys}}^2 = \beta_i^2 \cdot \sigma_m^2$

This risk is driven by broad market movements and cannot be diversified away. It is the only risk that commands a return premium under CAPM.

Idiosyncratic Risk (Firm-Specific Risk)

$\sigma_{\varepsilon}^2 = \sigma_i^2 - \beta_i^2 \cdot \sigma_m^2$

This risk is specific to the individual asset and can be eliminated through diversification across many assets.

R-Squared

The fraction of total variance explained by systematic risk is R²:

$R^2 = \frac{\beta_i^2 \sigma_m^2}{\sigma_i^2}$

Your Task

Implement:

• systematic_risk(beta, var_market) — returns the systematic variance component: β² × σ_m²
• idiosyncratic_risk(total_var, beta, var_market) — returns the idiosyncratic variance component