# Correct return calculation if short position involved

I wanted to calculate the daily return for a long-short position. Say one has 1 unit of stock A (long) and alpha units of stock B short. How can one now calculate correctly the daily percentage return? The literature puts different formulas forward, but they reach different values. Two of them are

1. $$\log(P^{A}_t / P^{A}_{t-1}) - alpha * \log(P^{B}_t / P^{B}_{t-1})$$, or

2. $$(P^{A}_t - alpha * P^{B}_{t}) / (P^{A}_{t-1} + alpha * P^{B}_{t-1}) -1$$

Are they both correct?

• You could either mean the return on the net position, or on the net capital at risk, which of these you use depends on how you want to think about the risk of the position. Usually log returns are used over simple returns. Dec 18, 2021 at 23:16
• The denominator of 2. shows that they are computing returns on "gross exposure" (i.e. Long+|Short|) while 1. seems to be return on "net exposure" (i.e. on Long-|Short|). In addition 1. is logarithmic return. (BTW is there a typo in 2. ? Something doesn't quite look right. Are they missing a "-1" at the end?). Dec 20, 2021 at 21:40
• Thank you @ noob2. See it. Yes you are right, there should be -1 in that formula. Jan 5, 2022 at 19:07