Perfect Position Sizing

There is a perfect position sizing formula, which I derived many years ago (see derivation) and dub the Sanden Criterion. It is as follows:

Where:

  • f = Optimum fraction (percentage) of portfolio to invest on this opportunity
  • W = Winning percentage (chance to earn G percent profit)
  • G = Gains on win (percentage)
  • L = Loss on lose (percentage)

The key is that average profit and loss over the long term depend on how much you invest in each trade. If you invest too little, or if you invest too much, you lose. After you’ve made about 100 manual trades, analyze your trade statistics and determine your personal best trading size.

Let’s look a bit deeper and understand this, because it’s important.

A 5% loss followed by a 5% gain always loses money. So does a 5% gain followed by a 5% loss.

Now imagine a coin flip carnival game. You can bet any amount of money, and flip a coin. If you bet $10, and get Heads, then you get $10 profit. If you get Tails, you lose the $10 that you bet. Simple right?

Now let’s make it more interesting. You’ve been watching this carnival game for the past few weeks, and been taking statistics. The coin is actually unfair — it doesn’t come up 50% Heads and 50% Tails — but it’s actually unfair in your favor. In other words, it’s coming up 55% Heads and only 45% Tails. You start to get pretty excited.

You walk up to the table with $1000 and grab the coin. The carny smiles. Now here’s the question: How much do you bet?

This is the problem. Even though the coin flip is going to land in your favor 55% of the time, you’ll still lose all your money if you bet too much. For example, pretend we invest 25% each time:

That is, start with $1000; gain 25% 55 times and lose 25% 45 times. You end up with $510. Again start with $1000; gain 25% 550 times and lose 25% 450 times. You end up with $1. We’re losing money even though we have the advantage!

Investing 25% of our capital in each coin flip will ruin us in the long run. Interestingly, however, instead investing 10% each time maximizes profits:

Now we’re making money. It’s all the same system (double-or-nothing with 55% chance to win), just with a different percentage of the total portfolio invested each time (10% vs 25%).

Anyone can see that the underlying system inherently represents a good deal. Furthermore, investing 10% shows it makes money. The problem is that a neophyte might naturally feel that if you are making money by putting in 10%, then you can only make more by putting in 25%. But we see that this is not the case. Even in a double-or-nothing bet game in which you win 90% of the time, invest 100% of your capital every time and in 10 trades you’ll still likely go broke.

So, where did the 10% come from in this example? This is the maximum-profit investment-percentage, given by the Sanden Criterion shown earlier:

W = 55% chance to win
G = 100% gain on win
L = 100% loss on lose

In real trading, you should assume that the optimum investment percentage is 1%, until you have sufficient results data from your own trading to raise or lower that number.

Finally, an astute reader may note that the Sanden Criterion is similar to the Kelly Criterion, but they’re not quite the same. The Sanden Criterion accounts for the possibility of leverage and gives the optimal investment size. The Kelly Criterion never accounts for leverage and gives the optimal risk size instead. The above coin flip game with a 1% gain/loss (instead of 100%) demonstrates the difference.

Derivation:

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