Interpreting the Intercept in Factor Asset Pricing Models

# What's the meaning of the intercept in asset pricing model?


# What's the meaning of the intercept in asset pricing model?












I would like to understand the role of alpha (intercept) in the regression-based asset pricing model or $n$-factor models; one of the most famous of those one is the Fama-French 3-factor model.

About this kind of model particularly, what's the meaning of the intercept?

I know that, technically speaking, from an econometric point of view, it should be the value assumed by the dependent variable on average, given the independent variables of the model set to be equal to 0. But, how can you interpret that from an economic point of view?

Does it have to be necessarily significant and equal to zero in order that the model can model properly the asset prices?

I looked some answers for the internet, but I found only contradictory opinions. Thanks for helping.

## Answer by Quantopik (score 3, accepted)

https://quant.stackexchange.com/a/17608

The factor models are based on the following linear regression model:

$(R_t - R_f)$ = $\alpha$ + $\beta_{mkt}$*$(R_{mkt} - R_f)$ + $\sum\limits_{i=1}^n {x_{k,t}}$ + $\epsilon_t$

$\alpha$ is the regression model intercept and indicates the portfolio performance in excess to the market excess return and the other factor; It has to be strictly positive and significant, in order to be able to measure properly the portfolio performance and the risk-adjusted portfolio returns; look at this [answer] for the joint hypothesis problem1.

$x_{k,t}$ represents the set of variables that can be (or that were been added over time) added following other kind of factor models (see, for instance, the Carhart's 4-factor model (1997)).

## Answer by Alexander Didenko (score 8)

https://quant.stackexchange.com/a/10054

Glad you've asked :)

Technically speaking, in factor model $\alpha$ stays for return or risk premia, which asset pays when all factor returns are zero.

Then, to answer question in more details, we have to specify, are we dealing in our model with return ($R_i$ for asset $i$) or with risk premia over risk free ($R_i-R_f$).

In the first case, interpretation of $\alpha$ is straightforward: most probably, it's $R_f$. As for latter case, this is one of white spot in my understanding of modern finance. I don't know correct answer. As far as I understand, in efficient market it should be equal to zero. If not - market is inefficient. Or there is still some risk factor which is priced, but not reflected in the model.

Or may be something else. In fact, for several months after discovering http://quant.stackexchange.com I was going to ask that question myself someday :) So, let me humbly join the questioning crowd.

## Answer by Yang Bai (score 2)

https://quant.stackexchange.com/a/51484

The traditional cross-sectional asset pricing focuses on the factors implied by the theory of stochastic discounting factor. Specifically, the existence of stochastic discounting factor leads to $1=\mathbb{E}(mR)$, where $m$ is the discounting factor. We can further rewrite the expression as \begin{align} 1&=cov(m,R)+\mathbb{E}(m)\mathbb{E}(R)\\\\ \frac{1}{\mathbb{E}(m)}&=\frac{cov(m,R)}{\mathbb{E}(m)}+\mathbb{E}(R)\\\\ \mathbb{E}(R)-\frac{1}{\mathbb{E}(m)}&=-\frac{cov(m,R)}{\mathbb{E}(m)}. \end{align} In theory, $\frac{1}{\mathbb{E}(m)}=R_f$, where $R_f$ is the risk free rate of return. Then, we have \begin{align} \mathbb{E}(R)-R_f&=-\frac{cov(m,R)}{\mathbb{E}(m)}.\\\\ \mathbb{E}(R)-R_f&=\frac{cov(m,R)}{var(m)}\cdot (-\frac{var(m)}{\mathbb{E}(m)}). \end{align} We then further define $\lambda_m=-\frac{var(m)}{\mathbb{E}(m)}$ as the risk premium. Note that $\frac{cov(m,R)}{var(m)}$ is the coefficient of the linear regression regressing $R$ on $m$. Therefore, we have \begin{align} \mathbb{E}(R)-R_f&=\beta \cdot \lambda_m. \end{align} This expression is interpreted as the risk premium of an asset equals its exposure on the underlying risk factor multiplied by the compensation(risk premium) on the risk factor. Note that we do not have this interpretation if $\lambda_m$ is not return.

This implies many factor models in the form of linear regression $R_i-R_f=\alpha_i + \sum\beta_{ij}f_{j} + \varepsilon_i$ or $\mathbb{E}(R_i-R_f)=\alpha_i+\sum\beta_{ij}f_{j}$. Note that the above theory implies no extra terms beyond the factor and the risk premium. Therefore, both the intercept and the regression error term are "errors". In general, asset pricers call $\alpha_i$ the cross-sectional pricing error, while $\varepsilon_i$ is called idiosyncratic risk/error.