In a previous article, we talked about CAPM and its applications. As a quick review, CAPM is a model used to calculate the expected return on a security, based on its market risk, commonly denoted as β (Beta). The model says that a security that is risky relative to the market should generate a higher return, and vice versa. Expected return can be calculated with the following equation:

*E(r _{i}) = r_{F} + β_{i} * (E(r_{M}) – r_{F}) *

One glaring weakness of the CAPM is that it only considers one variable, market risk, as a predictor of return. In practice, applying CAPM to a set of stocks will result in a relatively low R Squared value, or a large unexplained variance. These attempts at building models with more accurate predictions are collectively known as Multi Factor Models (MFMs).

### Multivariate Regressions

Most people are familiar with linear regressions, or at least the intuition behind linear regressions. By minimizing the squared error for the dependent variable, we can draw a line of best fit through a series of x and y values in a scatter plot. The line of best fit will tell us if and how the dependent variable relates to y, described by this equation:

*y = mx + b or y = β_{0}_{ }+ β_{i} * x_{i}*

where *m* or *β _{i} * is the slope or the coefficient that describes how x relates to y, and

*b*or

*β*describes the y intercept. This is essentially what the CAPM is, a line of best fit that relates expected return to market risk.

_{0}To improve on the CAPM, we must add additional factors into the relationship, leveraging multivariate regression. Multivariate regression allows us to add additional independent variables into the equation, to better predict the dependent variable. The equation looks like this:

*y = β _{0}_{ }+ β_{i} * x_{i }+ β_{j} * x_{j} … + β_{n} * x_{n}*

Each independent variable will have its own coefficient that describes its relationship with the dependent variable, in the context of the other relationships. This is the basis for Fama French and other multi factor models. You do not need to know the theory to run the regression, Excel or most other stats packages handles it for you. Some practical notes about multivariate regressions are:

- R Squared (explained variation) always increases as you add variables (or factors), use adjusted R Squared to test whether the new variable increases predictive power
- Each coefficient will have its own p-value, which describes how significant the associated variable is
- To test if the model can be generalized (applied to other data sets), use a holdout set to test your model

### Fama French 3-Factor Model

Fama and French identified 2 other factors on top of market risk as predictors of expected return, the size of the stock and the value of the stock. In 1993 Fama and French proposed to add these two factors into the CAPM, denoted by SMB (Small Minus Big) and HML (High Minus Low).

#### Size Effect

The size effect states that smaller companies (measured by market capitalization, or the total value of their shares) perform better than expected relative to larger companies. To rationalize this, consider stocks with positive alpha (risk adjusted returns): to generate alpha they must be priced lower than their intrinsic value. Therefore stocks with lower prices and market capitalization are more likely to outperform. Alternatively, smaller stocks are more likely to be overlooked by institutional investors, and hence more likely to be mispriced.

#### Value Effect

The value effect states that value companies (measured by book equity to market equity ratio) perform better than expected. The same rationale works here as well: stocks that generate alpha must be priced lower than their intrinsic value. The book value of equity is used as a proxy for intrinsic value – a high intrinsic value relative to market value will result in a low book to market ratio.

The Fama French 3-Factor Model looks like this:

*E(r _{i}) = β_{0}_{ }+ β_{i} * (E(r_{M}) – r_{F})_{ }+ si * E(SMB) + h_{i} * E(HML)*

### Fama French Carhart 4-Factor Model

In 1996, Carhart proposed an additional factor in the Fama French Model, one that accounts for momentum. This was after he noticed that stocks that did well continue to do well and vice versa. He named the factor MOM for momentum:

*E(r _{i}) = β_{0}_{ }+ β_{i} * (E(r_{M}) – r_{F})_{ }+ si * E(SMB) + h_{i} * E(HML) + m_{i} * E(MOM)*

### Fama French 5-Factor Model

In 2015, Fama and French once again extended their model to include two additional factors: profitability and asset growth. Profitability, denoted by RMW (Robust Minus Weak), describes how profitable the firm is. Asset growth is denoted by CMA (Conservative Minus Aggressive).

*E(r _{i}) = β_{0}_{ }+ β_{i} * (E(r_{M}) – r_{F})_{ }+ si * E(SMB) + h_{i} * E(HML) + r_{i} * E(RMW) + c_{i} * E(CMA)*

### Multi Factor Models

There are many more variations of the CAPM, which include many more factors than those described. In general, they all have factors that describe some sort of risk, which are associated with additional expected returns per unit of risk. Both MFM and CAPM are used to calculate a risk-adjusted return for an asset with a particular risk profile. In practice, MFMs seem to work better than CAPM, but is not grounded in solid economic theory.

Like CAPM, MFMs can be used to calculate cost of equity, which is then used to calculate Weighted Average Cost of Capital (WACC). MFMs can also be used to calculate alpha, which in some circumstances may be better than CAPM alpha for fund performance evaluation.

Statistics 101 | · · · |
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