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Evaluating Strategies with Negative Cumulative PnL

Article Quant Q&A · Author: Maxime Willemet
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Summary

The document asks how to form a return series for strategies whose cumulative profit and loss can become negative, in the context of statistical strategy comparisons such as White's Bootstrap Reality Check. The issue is that conventional log returns cannot be computed when the value used as the denominator or base is negative, which can distort or obstruct analysis.

The response suggests using absolute PnL changes as the series, illustrated by taking the differences between successive cumulative PnL observations. It asserts that reality checks and permutation tests can use such a series, just as they can use log returns. However, it does not discuss capital allocation, risk exposure, or how to normalize PnL across strategies, so the suggested series may not represent economically comparable investment returns. The answer is concise and provides no validation or further methodological qualifications.

Key ideas

  • Log returns are undefined when their underlying value becomes negative.
  • Successive changes in cumulative PnL can provide an absolute PnL series for analysis.
  • The response claims that White's Reality Check and permutation tests can use absolute PnL changes.
  • Absolute PnL differences do not normalize for capital or risk, limiting comparisons across strategies.

Tags

StatisticsBacktestingOptionsForex
Full text
# How to calculate daily returns when the cumulative PnL can become negative?


# How to calculate daily returns when the cumulative PnL can become negative?












I would like to know a way to compute returns when the total PnL of a strategy can become negative. For example: Total PnL day 1: 100 Total PnL on day 2: -20 Total PnL on day 3: 30

I have that kind of PnL when implementing systematic selling of straddle/strangle on FX options. As I would like to perform a statistical analysis on the return series such as the Bootstrap Reality Checkof White, I need to be able to compute daily returns for each strategies. However, the fact that the PnL can become negative distort the analysis.

Any ideas ? Thanks :)

## Answer by babelproofreader (score 2)

https://quant.stackexchange.com/a/71553

Why don't you use an absolute return series, e.g. 100, -120 and 50, based on your given figures? White's reality check, permutation checks etc. will work with this type of return series just as well as with log returns (which I presume is the thrust of your question, i.e. you can't take the log of a negative number).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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