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According to Kaufman (Trading Systems and Methods, 2013), the compound annualized rate of return is calculated as follows:

$$\mathrm{AROR}_\mathrm{compound} = \left[ \left( \frac{\mathrm{Final Balance}}{\mathrm{Initial Balance}} \right)^ {\frac{252}{\mathrm{length of testing period}}} \right]- 1$$

where the selection of 252 is based on an American trading calendar.

My question is: If the mantissa is negative, i.e. we incurred a negative balance at the end and we raise it to a fraction (which would happen with testing periods more than a year), then the result of this calculation would be a complex number. How do we calculate $\mathrm{AROR_{compound}}$ in this case?

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  • $\begingroup$ You cannot compute a rate of return if you have a negative final balance. The rate of return mathematically does not exist in this case. Usually you start trading with a certain amount of money, once the balance goes to zero you have to stop. $\endgroup$
    – nbbo2
    Oct 25, 2018 at 17:52
  • $\begingroup$ The negative returns result from allowing short positions with no limit (assuming infinite credit). When resolving these positions by the end of testing, you could sometimes easily lose all your current balance and end up in debt (hence the negative returns). How would you suggest I handle such positions? $\endgroup$
    – Ismail Mohamed
    Nov 3, 2018 at 23:54

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If a balance can go negative, then using any rate of return is likely to cause problems (see Appropriate method for calculating negative returns on a trading strategy? ).

In practice, I doubt anybody you deal with would let you hold a negative balance for a great length of time without you posting further collateral (margin). Hence, I would suggest that, if your balance can go negative, then it might be worth considering calculating the rate of return on the balance including the margin, rather than using only the balance.

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  • $\begingroup$ The negative returns arise from holding short positions that are still active by the end of the testing period. How would I better handle them? I resolve them by the end of the testing period using the last available close, and that would sometimes result in your balance dipping into the negative side. $\endgroup$
    – Ismail Mohamed
    Nov 3, 2018 at 23:52

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