Risk decomposition (variance shares)
Splits a stock's or portfolio's return VARIANCE into shares attributable to market, sector, subsector, and residual. Shares sum to 100%. Answers "what drives the risk?" — not what drove realized return.
Glossary
Every term in the decomposition vocabulary, cross-linked. One distinction carries most of the weight: quantities on the risk axis (shares of variance) and quantities on the return axis (realized attribution) come from the same model but answer different questions — the glossary keeps them apart.
Two axes: risk vs return
Return attribution
Splits a REALIZED return over a window into contributions from the market, sector, and subsector legs plus the residual return. Answers "what drove the return?" — not what drives ongoing risk.
Residual risk share
The share of return VARIANCE that is stock-specific — what remains after the factor layers. A risk quantity, not a skill quantity: a high residual risk share means concentrated stock-specific risk, not good stock-picking. At the portfolio level it is diversification-credited: residuals are largely uncorrelated across names, so the portfolio's residual risk share is far smaller than the weighted average of its holdings'.
Residual return (stock selection)
The RETURN remaining after removing the market, sector, and subsector legs — the object manager evaluation looks at. Over a window it is the geometric residual leg: the compound gross return minus the compound factor-only return (the telescoping waterfall bar), which reduces to the arithmetic sum of daily ε only over short windows. A positive residual return is evidence relevant to stock-selection skill (not proof of it). Distinct from the residual risk share, which is a variance quantity.
Diversification credit
The reduction in a portfolio's residual risk share relative to the weighted average of its holdings' single-name shares. Position-level residuals are largely uncorrelated, so they cancel across names; factor exposures do not. A naive position-weighted aggregation ignores this and overstates stock-specific risk.
The three levels
L1 (Market Level)
The first level of our hierarchical model that captures broad market exposure through the S&P 500 (SPY). This is your beta to "the market".
L2 (Sector Level)
The second level that captures sector-specific exposure after removing market effects. This shows your stock's sensitivity to its sector ETF (e.g., XLK for Tech).
L3 (Subsector Level)
The third level that captures subsector exposure after removing market and sector effects. This shows granular industry-specific risk (e.g., semiconductors within Tech).
Hedge ratios
Hedge Ratio (HR)
The dollar amount of an ETF you should trade per $1 of stock position to neutralize factor exposure. Positive HR means go long the ETF; negative means short.
L1 Market HR
SPY hedge leg at L1 in dollar_ratio units (ETF notional per $1 stock). Closest to single-factor “market beta” intuition; still use the published HR for execution, not a hand-computed OLS slope.
L2 Market HR
Adjusted market hedge ratio at level 2, accounting for the market exposure embedded in your sector ETF positions.
L2 Sector HR
Hedge ratio for the sector ETF at level 2. Shows how much sector ETF to trade per dollar of stock.
L3 Market HR
Fully adjusted market hedge ratio at level 3, accounting for market exposure in both sector and subsector ETF positions.
L3 Sector HR
Adjusted sector hedge ratio at level 3, accounting for sector exposure embedded in your subsector ETF positions.
L3 Subsector HR
Hedge ratio for the subsector ETF at level 3. This is the raw beta to the subsector factor.
Explained risk
Explained Risk (ER)
The percentage of a stock's VARIANCE explained by factor exposures. ER = 1 - (residual variance / total variance). Higher ER means more systematic risk, less stock-specific. A risk quantity — says nothing about realized return.
L1 Market ER
Percentage of variance explained by market exposure alone. This is your R² from regressing the stock on SPY.
L2 Sector ER
Incremental variance explained by adding sector exposure beyond the market. This is the additional R² from the sector factor.
L3 Subsector ER
Incremental variance explained by adding subsector exposure beyond market and sector. This captures granular industry effects.
Residual ER (single-name residual risk share)
Percentage of a single stock's VARIANCE that is stock-specific — the single-name residual risk share. Diversifiable: in a portfolio, these largely cancel across names (see diversification credit). A risk quantity; the skill quantity is the residual return.
Core concepts
Link Beta
The beta relationship between ETFs at different levels (e.g., how much market exposure is in a sector ETF). This enables HR adjustments.
Residual (ε)
The return SERIES left after removing all factor legs. Two distinct readings derive from it and must not be conflated: its variance as a share of total variance is the residual risk share (a risk quantity); its compounded value over a window is the residual return (the stock-selection quantity).
Replication Equation
The mathematical identity showing stock returns decompose into hedge-ratio-weighted ETF returns plus residual. Reconciliation verified to within 0.1% at runtime as a data-integrity check.
Geometric Attribution
Multi-period return decomposition that compounds daily factor returns through the ERM3 hierarchy (L1 → L2 → L3), producing waterfall bars that sum exactly to the compound gross return.
Orthogonalization
The process of removing higher-level factor exposure from lower-level factors to ensure clean incremental betas at each level.
Beta (β)
The sensitivity coefficient showing how much a stock moves per unit move in a factor. Beta = Cov(stock, factor) / Var(factor).
Market Factor
Broad market exposure captured by the S&P 500 (SPY). This is the systematic risk common to all equities.
Sector Factor
Industry-level exposure captured by sector ETFs (e.g., XLK for Technology, XLF for Financials). One of 11 GICS sectors.
Subsector Factor
Granular industry exposure captured by subsector ETFs (e.g., SOXX for semiconductors, XBI for biotech).