# Detrending market data to calculate expected return (ER)

I'm a complete newbie so please be kind.

Evidence-Based Technical Analysis: Applying the Scientific Method and Statistical Inference to Trading Signals by David Aronson

And I'm struggling to understand the following excerpt:

ER = [p (long) × avg. daily return] − [p (short) × avg. daily return]

For example, if the position biases were 60 percent long and 40 percent short, the expected return is zero.

0 = [0.60) × 0] − [0.40 × 0] Position Bias: 60 percent long, 40 percent short

If, on the other hand, a rule does have predictive power, its expected return on detrended data will be greater than zero. This positive return reflects the fact that the rule’s long and short positions are intelligent rather than random.

I fail to see how its possible for this equation to ever produce a result greater than zero if the detrended market data mandates the avg. daily return to be 0.

• The point is simply that a trading system selects some days to be long and some days to be short, and if you are lucky or good it selects days that will have a positive return even after removing the drift or upward bias during the overall period. If you try trading rules at random on average they will have zero detrended ret (you are right about this), but some will have pos and some negative detrended return. Jul 5, 2020 at 17:06
• Suppose S&P starts the year 2020 at 2500 and ends the year at 3000. That is a 500 point trend. The detrended S&P starts at 2500 and also ends at 2500, but with a lot of waves up and down. On this detrended series you can still make P&L if you "catch the waves" correctly. Jul 5, 2020 at 17:10
• @noob2 Ah. I see, I think its maybe the wording that has confused me. So in other words when he says "its expected return on detrended data" he is referring to the actual observed return on detrended data? not the ER formula itself Jul 6, 2020 at 11:06
• Yes, you got it. There are 2 ways to do it. Compute P&L on detrended data (easy), or use the formula above, which can be used to adjust the actual P&L per day to detrended P&L by subtracting ER. For example there are 250 trading days in the period so **avg. daily return**=500/250 =2. Suppose a strategy was long for 100 days and short for 2 days. Then **ER**= (100/250)*2-(2/250)*1= 0.792 per day. If your strategy had actual P&L of more than this per day it would have positive adjusted P&L. Makes sense? (I don't have Aronson's book any more but this what I remember). Jul 6, 2020 at 15:21
• Slight errata: ER = (100/250)*2 - (2/250)*2 = 0.784. The average daily change (ADC in formula on page 26 of book) should be the same for both the probability of a long and the probability of a short. Jul 7, 2020 at 13:09

The point is simply that a trading system selects some days to be long and some days to be short, and if you are lucky or good it selects days that will have a positive return even after removing the drift or upward bias during the overall period. If you try trading rules at random on average they will have zero detrended ret (you are right about this), but some will have pos and some negative detrended return.

Suppose S&P starts the year 2020 at 2500 and ends the year at 3000. That is a 500 point trend. The detrended S&P starts at 2500 and also ends at 2500, but with a lot of waves up and down. On this detrended series you can still make P&L if you "catch the waves" correctly.

Ah. I see, I think its maybe the wording that has confused me. So in other words when he says "its expected return on detrended data" he is referring to the actual observed return on detrended data? not the ER formula itself.

Yes, you got it. There are 2 ways to do it. Compute P&L on detrended data (easy), or use the formula above, which can be used to adjust the actual P&L per day to detrended P&L by subtracting ER. For example there are 250 trading days in the period so avg. daily return=500/250 =2. Suppose a strategy was long for 100 days and short for 2 days. Then ER= (100/250)*2-(2/250)*1= 0.792 per day. If your strategy had actual P&L of more than this per day it would have positive adjusted P&L. Makes sense? (I don't have Aronson's book any more but this what I remember).

Slight errata: ER = (100/250)*2 - (2/250)*2 = 0.784. The average daily change (ADC in formula on page 26 of book) should be the same for both the probability of a long and the probability of a short.

• Note: I concocted this answer from comments of a year ago (see above) since they seemed to hit the spot. Thanks also to Babelproofreader for his contribution. Apr 2, 2021 at 13:38

by definition once you remove the trend and other volatility clustering patterns, you are left with white noise, without correlation whatsoever