# 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?

• You write "the factor model is" and then two numbers. This is not a model. Do the 2 stocks "have" 2 different models? This is very unclear. Please format the question and insert the formulas that describe the models. – Richard Jun 23 '15 at 10:31
• @Richard. The factor model for the two stocks is a two-factor model, so there are two numbers which are the value of the factor. The two stocks do have different factor model, for stock A, it is [0.5, 0.75], for stock B, it is [0.75, 0.5]. – epx Jun 25 '15 at 5:14

$$r_A = (0.5, 0.75) (r_F^1, r_F^2) + \epsilon_A$$ 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.