# Correct calculation of returns from a pairs trade

I am trying to understand how pairs trading works but I am confused about how to go about calculating the return on the pairs trade when I reverse my positions. I have been reading 'Pairs Trading - Quantitative methods and analysis' by Ganapathy Vidyamurthy. My understanding is limited at this time, although the whole point of my venture is to understand better so please bare with me if this seems a naive question or I am mistaken in any assumptions. I am omitting any hedge ratio for academic purposes to simplify this question.

I am calculating the spread as:

log(price_a) - log(price_b)

my question is, if I long stock A and short stock B at time t then is it correct that I can get the return at t+1 as:

spreadt+1 - spreadt ?

This is a new concept to me and my general thinking would be that in fact I can get the overall return as the combination of both individual returns:

return on stock A: (price_At+1 - price_At) / price_At

return on stock B: (price_Bt+1 - price_Bt) / price_Bt

total return on pairs trade: return on stock A + return on stock B

Some clarification on the above would be helpful.

Thank you

If you use logreturns it becomes simpler:

logreturn on stock A: log(price_At+1/price_At)

logreturn on stock B: log(price_Bt+1/price_Bt)

then

total logreturn on pairs trade: logreturn on stock A + logreturn on stock B =

=log((price_At+1*price_Bt)/(price_At*price_Bt+1))=

=spreadt+1 - spreadt

Now it is all consistent, thanks to the property that $$\log(x y)=\log(x)+\log(y)$$

(Of course no one stops you from calculating simple returns as well, your program can print both. simplereturn = -1.0 + exp(logreturn) )

(Also, I assumed hedge ratio of 1, like you did)

• There seems to be an error here: total logreturn on pairs trade: logreturn on stock A + logreturn on stock B = log((price_At+1*price_Bt+1)/(price_At*price_Bt)) Therefore this is not equal to spreadt+1 - spreadt If I'm correct then how are return of pairs trade computed? Could you please clarify this. Thanks in advance. Commented Sep 7, 2022 at 23:29
• The change in spread is not the return on the trade, but it is an approximation because it is the difference of two logs each of which is approximation of the returns of the stock positions. I don't have the Vidyamurthy book, so I am not sure if this is what he had in mind. The return is generally computed as the dollar profit/loss divided by the equity or capital at time t. Commented Sep 8, 2022 at 6:27
• Thank you for the clarification. Perhaps you could include that in your response. Thanks again. Commented Sep 8, 2022 at 6:34