# How to perform risk factor calculation?

I am studying Arbitrage Pricing Theory (APT) and I have a question about calculating factor exposures.

Assume:

$$r = \beta_1r_1 + \beta_2r_2 + ... + \beta_kr_k + r_e$$

Where:

$\beta_i$ is the exposure of the asset to a factor

$r$ is the return attributable to a factor

I believe that beta will be the covariance of the factor with the underlying asset. Is this correct? Also how is the return attributable to a specific factor calculated? Is there a single way this is done or are there a variety of approaches?

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I believe that beta will be the covariance of the factor with the underlying asset. Is this correct?

Close, it's the covariance divided by the variance of the factor.

$$\beta_{f,a} = \frac{\sigma_{f,a}}{\sigma^2_f}$$

Also how is the return attributable to a specific factor calculated? Is there a single way this is done or are there a variety of approaches?

That depends on how you derive your factors. As mentioned in this earlier question, I once derived factors with cluster analysis. Thus, each factor was really a collection of highly correlated large-cap stocks. That meant the factor return was simply the cap-weighted average of all constituent stock returns, just like in a stock index.

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Does this imply that the return of a factor is equal to the value at the observed point minus the value at the beginning of the series? –  Steve Mar 23 '11 at 1:32
@Steve The return is (begin-end)/begin. That's true whether it's an asset or a factor. Again, this is for a selection process that allowed a dollar price in the time series, since the factor is a cap-weighted selection of stocks. –  chrisaycock Mar 23 '11 at 2:03
This is only the case for simple linear regression with a single factor (i.e. CAPM), and not for multiple linear regression. If computing a multiple linear regression were that simple, there wouldn't be the vast mountain of literature on the topic (do a google search for 'solving normal equations' ) –  shabbychef Mar 23 '11 at 4:40
@shabbychef I reviewed my notes from when I did this. The process then was to select a single "best" factor through cluster analysis, then subtract its cap-weighted returns from the universe before iteratively re-running the cluster analysis and selecting another "best" factor. I found a comment that all selected factors at the end were independent because of the subtractions. Is that logic valid? –  chrisaycock Mar 23 '11 at 14:08
if you perform e.g. a en.wikipedia.org/wiki/Gram_schmidt normalization before the regression, you will have an orthogonal design matrix. However, it is difficult to interpret the resultant regression coefficients. –  shabbychef Mar 23 '11 at 16:34

I don't have much to add, but wanted to address the "price of risk" question.

APT is kind of "economics"-free and tries to price assets without the utility maximization required in CAPM/ICAPM. Ross's APT observes that groups of assets move together (e.g., tech stocks) and that is the risk you're bearing because the idiosyncratic risk, like the firing of HP's CEO, can be diversified away. Because this risk is easily diversifiable, the market won't pay you to take it. So in your APT model these factors are returns to asset classes, industries, etc.

Although the model looks the same, in Merton's ICAPM, the factors are state variables (e.g., industrial production, inflation). These are purely academic points -- in practice you run a multivariate regression with return on the LHS and whatever factors you think are priced on the RHS. OLS and GMM are common. So you'll estimate $$E ~ \left[ ~r_i~ \right] = \alpha_i + \beta_i^1 f_1 + \beta_i^2 f_2 + \ldots + \beta_i^k f_k$$

Also how is the return attributable to a specific factor calculated?

Now you regress the returns back on the betas. $$E ~ \left[ ~r_i~ \right] = \sum_{j \in K} \lambda_j \beta_i^j$$

Where $\lambda_j$ is the return to factor $j$. Typically the Fama-MacBeth approach is used here. If you've done it correct and found something, $\lambda > 0$ (i.e., the market is paying you to take this risk).

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Note that APT assumes the the alpha term is zero... and that the cross-product of exposures and factor returns + the risk-free rate result in the E[Ri]. So you need to think a little bit about whether to include the intercept or not depending on the application (asset-pricing test would include the intercept, for example). –  Quant Guy May 29 '12 at 11:50
I think all of the theories have zero intercepts (i.e., only one risk-free rate)? Empirically you include the intercept to avoid forcing $\alpha_i = 0$ so that you can test if there is a return not correlated with the risk factors. –  Richard Herron May 30 '12 at 13:32

If you have a series of observations of the return as a vector, $\mathbf{r}$ with corresponding observations of the factor returns in matrix $Z$, then the least squares estimate of the vector of betas is $$\hat{\beta} = \left(X'X\right)^{-1} X'\mathbf{r},$$ where $X$ is the matrix with $Z$ and a column of all ones (for the intercept term). The last value of $\hat{\beta}$ will be the estimate of the 'idiosyncratic' return. In general, the estimate of the $j$th coefficient, $\hat{\beta_j}$ will not be correlation of the return to the return of the $j$th factor, nor will it be that value adjusted for the volatility of the factor.

If you have only one factor (in which case it is CAPM, not APT), then the computation does simplify. Also, if the sample returns of the different factors are independent vectors (highly unlikely to happen by accident), you will get the simplification.

See wikipedia for more on multiple linear regression.

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