Volatility Decomposition
$$ \Omega_r = B\, \Omega_f\, B^T + \Omega_\epsilon $$ This equation tells us what causes the ups and downs in a group of asset returns, and how those sources mix together. If you want to understand risk, build portfolios, or analyze any group of assets, this formula is a fundamental “recipe” for how returns vary together.
Start from the recipe for asset returns: $$ r = Bf + \epsilon $$ The covariance (how things wiggle together) of \(r\) is: $$ \Omega_r = \mathrm{Var}(r) $$ Plug in our recipe: $$ = \mathrm{Var}(Bf + \epsilon) $$ If errors and factors are independent, the variance of a sum is just the sum of variances: $$ = \mathrm{Var}(Bf) + \mathrm{Var}(\epsilon) $$ When you scale factors by B, variances get mapped like so: $$ \mathrm{Var}(Bf) = B\, \mathrm{Var}(f)\, B^T $$ $$ \Omega_r = B\, \Omega_f\, B^T + \Omega_\epsilon $$
Socratic Q\&A
Q: What does the first term \(B \Omega_f B^T\) mean in words? A: It shows how group movements (factors) flow through to each asset and spread out to influence asset risk.
Q: Why do we need \(B\) and \(B^T\), not just \(\Omega_f\)? A: Because \(B\) tailors the factor wiggles to each asset, and $B^T $connects back to all assets, creating a full map of shared risk.
Q: What does \(\Omega_\epsilon\) add to the mix? A: It captures each asset’s personal quirks—randomness that factors cannot explain.
Q: Why is the sum of two covariances—the shared and the unique? A: Because return variations come from big market shocks (factors) and small asset-specific surprises (errors).
Q: What’s the practical importance of splitting up risks this way? A: It lets investors see what risks can be diversified (factor risks) and what is stubborn (idiosyncratic risk).