Proof That Detrending Is Equivalent to Benchmarking Based on Position Bias
In order to compensate for market bias interacting with long-short bias, one should compute the expected return of a random system with the same long-short bias, and subtract this bias-related return from the return of the candidate system.
The expected value of a single raw return drawn from the population of historical raw returns is, by definition, the mean of these returns.
Let Pi signify the position of a trading system at opportunity i. This will be long, short, or neutral. The expected return of a trading system in which positions are random is:
The total return of the candidate system is the sum of the raw returns at those times when the candidate system is long, minus the sum of the raw returns when the system is short:
The total return of the candidate system (Equation (A.3)) is corrected for long/short prejudice by subtracting the expectation of a similarly prejudiced random system (Equation (A.2)). Subtraction distributes over the summation, giving:
Notice that the corrected return of Equation ...