Why you should add to winners and never add to losers

I recently got an email from Kevin Davey of KJ Trading, aka @kevinkdog, regarding his latest blog post entitled “Peel Off Trading”. Kevin is a highly respected member of nexusfi.com (formerly BMT) who has done several webinars, has his own “ Ask me Anything Thread", has written a book “ Building Winning Algorithmic Trading Systems, A Trader's Journey From Data Mining to Monte Carlo Simulation to Live Trading” and is also I believe the current coordinator of “ The Battle of the Bots".

The basic concept of “Peel off Trading” is

In its simplest form, the trader starts out with 2 contracts, exits the first at a small profit (the “peel off”), and then holds the second as a “runner,” going for big profits. In some cases when the first contract is exited, the second contract is then modified to have a breakeven stop.

Like almost all of Kevin’s work it’s a very interesting piece based upon supporting analysis. Kevin actually back tests the exit rules upon three different strategies but the results are inclusive

Results of the out-of-sample analysis are mixed. The ES is the same, the GC strategy is better with the “peel off” method, and the JY strategy is better with the baseline (always trading 2 contracts).

While reading it I was actually reminded of a different blogpost “ Does [AUTOLINK]Averaging in[/AUTOLINK] Work?” by Ernie Chan. While I don’t believe Ernie posts here he has done several webinars. He has also written two books “ Quantitative Trading: How to Build Your Own Algorithmic Trading Business” and “ Algorithmic Trading: Winning Strategies and Their Rationale”.

While Kevin’s blog was looking at exit scenario’s, Ernie’s blog was looking at entry rules. It was also more theoretical in nature, but his conclusion was more decisive.

Ron Schoenberg and Al Corwin recently did some interesting research on the trading technique of "averaging-in". For e.g.: Let's say you have $4 to invest. If a future's price recently drops to $2, though you expect it to eventually revert to $3. Should you
A) buy 1 contract at $2, and wait for the price to possibly drop to $1 and then buy 2 more contracts (i.e. averaging-in); or
B) buy 2 contracts at $2 each; or
C) wait to possibly buy 4 contracts at $1 each?
Let's assume that the probability of the price dropping to $1 once you have reached $2 is p. It is easy to see that the average profits of the 3 options are the following:
A) p*(1*$1+2*$2) + (1-p)*(1*$1)=1+4p;
B) 2; and
C) p4*$2=8p.
Profit A is lower than C when p > 1/4, and profit A is lower than profit C when p > 1/4. Hence, whatever p is, either option B or C is more profitable than averaging in, and thus averaging-in can never be optimal.

As I started thinking about Chan’s analysis I didn’t like the way he considered the one situation “that the probability of the price dropping to $1 once you have reached $2 is p” and excluded other scenario’s. In a moment of boredom I decided to expand Chan’s analysis to a binomial tree where we have a probability p of going down $1, but also a probability (1-p) of going up $1.

Hence at time T+1 we have probability p of trading $1, and a probability (1-p) of trading $3. Then at time T+2 we have probability p*p of trading $0, a probability 2p(1-p) of trading $2 and a probability of (1-p)*(1-p) of trading $4.

For those of you were not aware this is what we would call a binomial tree which is used to derive the Binomial Distribution ( Wikipedia, Wolfram Mathworld) and there are many option and derivative pricing models based upon binomial trees and even more based upon other type of trees.

If we consider as an example scenario A) where we buy 1 contract at $2, and wait for the price to possibly drop to $1 and then buy 2 more contracts.

In case i) where the market goes up at T+1, and T+2 to $4 we never buy the additional contracts, hence we buy a single contract at $2, that has a $2 profit. Since this has a probability of (1-p)*(1-p) the expected payout of case i) is 2*(1-p)*(1-p)

Case ii) the market goes up at T+1, and then down at T+2 to $2 we never buy the additional contracts, and we have no profit. Hence the expected payout of case ii) is 0.

Case iii) the market drops at T+1, and then rallies back to $2 at T+2. In that case our initial contract purchased at $2 is breakeven, but when prices dropped to $1 at T+1 we added two more contracts. These two contracts now have a total profit of $2. Hence the expected payout of case iii) is 2*p*(1-p).

Finally in case iv) the prices drops at T+1 and T+2 and ends at $0. In this case we lose $2 on our initial one contract purchase and lose an additional $2 on our two extra purchases at $1. Hence we have a total loss of $4 and an expected payout of -4*p*p

Adding this all together our expected payout for scenario A is
2*(1-p)*(1-p) + 0 + 2*p*(1-p) - 4*p*p = -4p^2 -2p +2
It follows that the other expected payouts are B (-8p +4) and C (-8p^2 +4p)

If we plot these functions we can see that while A) and C) might outperform B) (well actually lose less) when p>0.5 (ie market is more likely to go down) but they underperform even more when p<0.5 and hence the expected returns of A) and C) are both negatively skewed in relation to B).


While this doesn’t completely agree with Chan’s analysis that’s it’s never better to scale in, it does support his analysis in that from a risk reward perspective it’s never better to scale down into a position.
I then added a fourth scenario D) where we Buy 1 at $2, and if prices go up we buy 2/3rds more at $3. I know fractions make it messy but it’s the only way to keep it an equal investment size.
The expected payout of D) is +1.33p^2 - 6p +2.67

This is the complete opposite of A) and C). It underperforms 2) when p>0.5 but outperforms by a greater margin when p<0.5, hence it has a positive expectancy skew!

This would imply that not only should you not “average down” but that adding to winning positions has a better risk reward.



While D has the best risk/reward it does have a lower reward than the base scenario B). Hence to achieve the same desired results, you can/should scale the positions appropriately. This is actually a great way to visualize what we are discussing as you can see below. You will see that while 3 of the 4 have similar positive profiles, scenario D has the least worse profile when the market goes against the position.



I then expanded the analysis significantly, going out to T+5, and including additional scale in scenarios. The conclusions were the same though. Averaging down has a worse risk reward profile than buying your entire position at entry, while adding to winners has a better risk reward profile.

Disclaimers/Notes/Flaws/Weaknesses:
A couple of the obvious flaws of this analysis are
- it uses a fixed price movement of $1 as opposed to a percentage move - while this is actually easy to fix for the purposes of this example it would add undue complication since +1-1=-1+2 but +1%-1%<>-1%+1%
- it assumes a binomial distribution when in reality a log normal distribution would be a better fit.
- Price (changes) in reality are not log normally distributed. As we all know tails are a lot lot fatter.
- Stops could potentially completely change the analysis.

I guess my questions is: for those of you who add to the existing trade, how do you do that? On retracements, at fixed intervals, any other way?