# Log return on short selling when the loss exceeds 100%

I'm working on a theoretical portfolio that includes both long and short positions. I already have the daily holding period returns, and I want to calculate both arithmetic and logarithmic returns for these positions (daily). However, I run into a problem with short positions when the return exceeds 100% (i.e., higher than 1). In such cases, I can't use the log return formula (LN(1+R)). Although these instances are rare, they do occur. Let's assume the return is 2, then for short selling it becomes LN(1-2).

I tried setting a cap, treating returns over -1 as -0.99, but this approach skews the portfolio because LN(1-0.99) equals -4.6051, which is an excessively negative value in logarithmic terms.

What would be a better solution to handle this issue?

Think about the following example. On day 1 the stock price $$P_1=50$$, on day 2 the stock price $$P_2 = 120$$ and on day three the stock price $$P_3 = 25$$.
The arithmetic return on a long position is $$\frac{P_{t+1}}{P_t} - 1$$ and the log return is $$\log \bigg ( \frac{P_{t+1}}{P_t} \bigg )$$.
• I fully agree on the note about the log return on a long positions being just the negative of the short. Another way people tend to represent returns on short positions is $(P_1-P_2)/P_1$ vs that of a long position that is the normal returns formula of $(P_2-P_1)/P_1$, which is why applying a negative sign on the log returns makes sense. Commented Aug 2 at 1:11
• KaiSqDist: That's interesting but also possibly confusing because, in the first equation, you're using an anchor of $P_{2}$ but then dividing by $P_{1}$. My brain can't handle that !!!!!!! phdstudent's method is more like back in the day when they told you that subtraction was like adding a negative. That works better for me. Commented Aug 2 at 3:58
• @markleeds not sure what you mean by anchor of $P_2$. The way I understand it is because $1$ comes before $2$ in terms of time, the denominator should always be $P_1$, so it makes perfect sense to me. The numerator itself is intuitive though. Commented Aug 2 at 6:31