Annualized Returns When a Trading Balance Turns Negative

Summary

The document considers how to interpret compound annualized return when a strategy’s ending balance is negative. The standard calculation raises the ratio of final to initial balance to a fractional power based on the length of the test. If the final balance is below zero, that operation can produce a complex result, making the usual annualized return unsuitable as a performance measure.

The response notes that sustained negative account equity would generally trigger a demand for additional collateral, so the account balance alone may not represent the capital supporting the strategy. It suggests including margin in the balance used to assess returns. The discussion does not provide a general replacement formula or compare alternative performance metrics, and it cautions that negative equity creates broader problems for rate-of-return calculations. The central lesson is to define the capital base consistently with the collateral and financing required by the strategy before annualizing its performance.

Key ideas

• The usual compound annualized return formula is not meaningful when the ending balance is negative and the exponent is fractional.
• Negative account equity can make standard rate-of-return measures fail.
• Margin requirements mean the trading balance may not capture all capital supporting a strategy.
• Including margin in the return base may provide a more relevant calculation.
• The response offers no universal alternative formula for annualizing returns from negative balances.

Tags

# Calculating Compound Annualized Rate of Return (Negative Mantissa)


# Calculating Compound Annualized Rate of Return (Negative Mantissa)












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?

## Answer by Tim Wilding (score 1)

https://quant.stackexchange.com/a/42381

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.

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.