Handle Short Position Returns Without Invalid Logarithms

Summary

The document addresses a return-calculation problem for portfolios with short positions: applying the usual logarithm to a short’s arithmetic return can become undefined when the loss exceeds the investor’s capital under the stated return convention. The questioner considers capping extreme returns, but worries that this distorts portfolio results.

The accepted response advises separating the underlying asset’s return calculation from the position direction. Compute the asset’s arithmetic and logarithmic returns from its price change, then reverse the sign for the short position. A sequence of stock prices is offered to illustrate the sign convention. This framing avoids applying a logarithm to a short-position return that has been formed by subtracting the asset return from one. The brief answer does not discuss margin, leverage, financing, liquidation, or the path-dependent wealth consequences of a short position, so it is a return-sign convention rather than a complete model of short-sale P&L.

Key ideas

• Calculate the underlying asset’s log return from its price ratio.
• Represent the short position’s return as the opposite sign of the corresponding long return under the stated convention.
• Capping returns to force a logarithm into its domain can distort portfolio calculations.
• The sign-flip explanation omits margin, financing, leverage, and other short-sale mechanics.

Tags

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


# 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?

## Answer by phdstudent (score 4, accepted)

https://quant.stackexchange.com/a/80175

The easiest way is to think for a return on a short position as being the negative of the return on the long position. In that way you never get confused.

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 $\log \bigg ( \frac{P_{t+1}}{P_t} \bigg )$.

The short return is just the same as above with the opposite sign.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.