## Factor attribution provides a deeper look into the driving forces of risk and return in a portfolio by showing the return series attributable to the various factor exposures of the portfolio.

Style factors are realizations of long-term risks and returns driven by logical groupings of underlying data. These style factors can be defined in a number of ways as directly measurable from stock specific data. Any number of style factors can be defined.

For example, Size typically looks at market value of equity and sometimes other data like total assets and total sales. As a factor, it is used to measure the difference in risks and returns between large companies and small companies.

There are two primary approaches to performing factor attribution: time-series regressions and cross-sectional regressions.

**Time-series regressions**(Fama-French) measure factor returns ex-ante and then estimate betas to those returns.

- Time-series regressions take a series of returns and estimate the exposure of a security or portfolio to each of those factors through a regression.
- Time-series regressions assume time-constant exposures to a factor within the regression period.
- This approach works okay (but not perfectly) with factors which are slow to change, like size, but particularly poorly for other factors that vary more with time.

**Cross-sectional regressions**(Barra) measure factor exposures and estimate the factor returns ex-post.- Cross-sectional regressions take a normalized measurement of the factor exposure and regress returns on each day for the stocks and those factor exposures.
- In this approach, factor exposures are allowed to vary with time and directly reflect the data from which they are measured.
- The factor returns, however, are often unintuitive and over-ascribe collinear returns to a factor.

Since both exposures and factor returns are measurable, we instead rely on regressions only to scale the exposures and returns appropriately with asset returns. This regression is computed daily on a per-stock basis on the series of factor returns and z-scores from the beginning of the backtest.

In this equation, 𝛿 represents the cross-sectional excess factor return (top 33% factor return - bottom 33% factor return), 𝑍 represents the cross-sectional z-score of the factor, and 𝛃 represents the regression coefficient to scale the product of z-scores and excess factor returns to stock returns. *This approach keeps the factor returns and factor exposures interpretable by avoiding holding time-varying measurements constant and avoiding over ascribing collinear returns to factors.*

**Factor Definitions:**

- Growth

- 28% Total Asset growth
- 26% sales growth
- 26% earnings growth
- 20% forecast of earnings growth

- Leverage

- 34% book leverage
- 33% market leverage
- 33% debt to total assets

- Market

- 1 year (252 day) trailing beta with benchmark

- Momentum

- 50% 12 Month return - 1 Month return
- 50% 6 Month return - 1 Month return

- Size

- 33% log market cap
- 33% log sales
- 34% log total assets

- Trading Activity

- 10% trailing 5 day volume
- 30% trailing 21 day volume
- 60% trailing 63 day volume

- Value

- 14% price to book
- 20% price to earnings
- 20% price to cash flow
- 5% EV/sales
- 20% EV/EBITDA
- 21% price to earnings forecast

- Volatility

- 1 year (252 day) trailing standard deviation of daily returns