Calculating Log Returns for a Long-Short Pairs Trade
Summary
The document explains how the change in a log-price spread relates to the return from a pairs position with a hedge ratio of one. For a spread defined as the log price of asset A minus the log price of asset B, its change equals A’s log return minus B’s log return. That corresponds to being long A and short B: the short leg contributes the negative of B’s return. Thus the spread change can represent the combined log-return exposure under that position convention.
The answer emphasizes that signs matter when combining the legs and that simple returns can also be calculated, with a conversion from a combined log return by exponentiating and subtracting one. The discussion assumes a unit hedge ratio and omits fees, financing, borrow costs, rebalancing, and capital or notional conventions. It is an algebraic explanation, not evidence that the spread itself is a complete measure of realized portfolio performance.
Key ideas
- The change in log price A minus log price B equals A’s log return minus B’s log return.
- A long A and short B position gives the second leg a negative return contribution.
- The relationship assumes a hedge ratio of one.
- A combined log return can be converted to a simple return by exponentiating and subtracting one.
- The calculation omits trading costs, financing, and capital-allocation conventions.
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# Correct calculation of returns from a pairs trade # 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 ## Answer by nbbo2 (score 0, accepted) https://quant.stackexchange.com/a/54063 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)
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