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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?

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  • $\begingroup$ 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. $\endgroup$
    – Richi Wa
    Commented Jun 23, 2015 at 10:31
  • $\begingroup$ @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]. $\endgroup$
    – epx
    Commented Jun 25, 2015 at 5:14

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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.

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  • $\begingroup$ Thanks and now I understand. It turns out that I have this problem because I did not understand the definition of covariance matrix. To calculate the covariance matrix, you need to use the exposure to the factors of both stocks. $\endgroup$
    – epx
    Commented Jun 25, 2015 at 5:16
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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.

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