Correct calculation of returns from a pairs trade

Question

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

Answer 1

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)