Calculating Stock Covariance from Shared Factor Exposures
Summary
The discussion explains how a two-factor model can estimate covariance between two stocks using their exposures to common factor returns. Each stock’s return is represented as a weighted combination of the same factor returns plus an idiosyncratic error. If those errors are uncorrelated with the factors and with each other, the cross-stock covariance comes from the shared factors alone: multiply the first stock’s exposure vector by the factor-return covariance matrix and then by the second stock’s exposure vector.
The factor covariance matrix is common to both stocks because it describes the covariance of the underlying factors, not a separate matrix belonging to each stock. The example supplies exposures and a matrix, but the main contribution is the model logic rather than empirical validation. The result depends on the assumptions about residual correlations and common factors; correlated stock-specific errors would add another covariance term. The explanation also does not cover estimation error or how to choose the factors.
Key ideas
- A factor model represents each stock return as factor exposures plus an idiosyncratic residual.
- The same factor-return covariance matrix applies to stocks exposed to the same factors.
- Cross-stock covariance is obtained by combining each stock’s exposure vector with the shared factor covariance matrix.
- The simple expression assumes residual errors are uncorrelated with factors and with each other.
- Correlated residuals would contribute an additional term to stock covariance.
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# Covariance between two stocks in a two-factor model
# Covariance between two stocks in a two-factor model
I am studying the Arbitrage Pricing Theory using Pairs Trading: Quantitative Methods and Analysis.In page 44 the author gives an example on how to calculate the covariance between two stocks. I will tell how the author do it first.
There are two stocks using two factor model, for stock A, the two factor model is (0.5, 0.75) and the factor covariance matrix is [ 0.625 0.0225,0.0225, 0.1024]. And for stock B, the two factor model is (0.75, 0.5). Then the author says we can calculate the covariance between stocks as [0.5, 0.75][0.625 0.0225,0.0225, 0.1024][0.75,0.5].
What I do not understand is that in calculating the covariance between stocks, the mid-term is the factor covariance matrix for stock A, we do not know the factor covariance matrix for stock B, so is it right to calculate the covariance between the stocks as the author says?
## Answer by Richi Wa (score 6, accepted)
https://quant.stackexchange.com/a/18490
Your questions is unclear but I guess you mean that for the return of stock A you find a model
$$ r_A = (0.5, 0.75) (r_F^1, r_F^2) + \epsilon_A $$ where $r_F^i$ are the factor returns and $\epsilon_A $ is an uncorrelated error. Let us denote $e_A = (0.5, 0.75)$, the exposure of stock $A$ to the factors. For $B$ you have $$ r_B = (0.75, 0.5) (r_F^1, r_F^2) + \epsilon_B. $$
Furthermore the covariance matrix of the factor returns is given by $$ \Sigma_F:= \left( \begin{array}{ccc} 0.625 & 0.0225 \\ 0.0225 & 0.1024 \end{array} \right). $$ Then the covariance of $r_A$ and $r_B$ can be calculated as follows \begin{align} cov(r_A,r_B) &= cov(e_A(r_F^1,r_F^2)+\epsilon_A,e_B(r_F^1,r_F^2)+\epsilon_B ) \\ &= cov(e_A(r_F^1,r_F^2),e_B (r_F^1,r_F^2) ) \\ &= e_A \Sigma_F e_B, \end{align} where we have used the assumption that the errors are uncorrelated from all other random variables and some matrix algebra to arrive at the vector times matrix times vector expression that you have up there.
## Answer by vanguard2k (score 2)
https://quant.stackexchange.com/a/18488
The factors are the same for both stocks, so there is just one factor covariance matrix for both A and B.
Factor models are a way to reduce the dimension of a problem. If every stock had its own set of factors, this would increase the problem dimension.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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