Kelly Criterion - Position Sizing to Maximize Returns

Recently (since August 30th), I have started using the Kelly Criterion to decide the position size of my day trades. I have known about the Kelly Criterion for a long time, but have only had the confidence to use it now.

Why now? 'Cause, I have spent hours on backtests, and finally seem to have arrived at a trading system that gives consistent results over a few years, and gives me enough confidence to decide the p and b needed to calculate the position size according to the Kelly Criterion.

How confident? We will come to that later...

Kelly Criterion (Wikipedia link)

f = ( p * ( b + 1 ) -1 ) / b


where,
f = optimal fraction of capital to be risked to maximize long term gain
p = probability of winning (Win %)
b = expected reward per unit of risk (Risk to Reward Ratio)

As long as you know your p and b, risk f.

Kelly criterion - Why NOT?

 

Open any article about Kelly Criterion - not excluding the Wikipedia article - and you will find many reasons/arguments given to avoid Kelly Criterion. These include everything from the fact that you cannot predict the future, to the gambler's/trader's tendency to overestimate system performance, to reducing volatility of returns, to expected utility, to etc. etc. etc....

Also, to not incite novices and gamblers to risk ruin. (My risk disclaimer is at the bottom of this page)

From what I understand, I think that the Kelly Criterion was originally designed to determine the optimal bet sizes in scenarios where you know/assume the odds and probabilities. How do you apply it to markets that are dynamic? How do you determine the odds and probabilities? (The odds part may not be an issue for systems that have predetermined entries, stops, and targets)

Kelly criterion - Why not?!! 

 

Below, is an example that I had earlier posted in the Bakwaas Trading thread of the Traderji forum...

For example, if a system has a Win % of 50% and Risk to Reward Ratio of 1.6, then the optimal risk according to the Kelly Criterion is 18.75% of compounded capital on every trade.

- If we risk 19% per trade, then after 100 trades, the capital will be 15.42 times the initial capital (1442% return on capital)
- If we risk 18% per trade, then after 100 trades, the capital will be 15.36 times the initial capital
- If we risk 15% per trade, then after 100 trades, the capital will be 13.87 times the initial capital
- If we risk 10% per trade, then after 100 trades, the capital will be 8.61 times the initial capital
- If we risk 5% per trade, then after 100 trades, the capital will be 3.61 times the initial capital
- If we risk 2% per trade, then after 100 trades, the capital will be 1.76 times the initial capital (76% return on capital - not too bad)
- If we risk 1% per trade, then after 100 trades, the capital will be 1.34 times the initial capital (34% return on capital)

Oh, I forgot to mention what happens when we increase the risk beyond the optimal %.

For the same parameters as above,

- If we risk 20% per trade, then after 100 trades, the expectancy is that capital would be 15.25 times the initial capital
- If we risk 25% per trade, then after 100 trades, the expectancy is that capital would be 11.47 times the initial capital
- If we risk 30% per trade, then after 100 trades, the expectancy is that capital would be 5.86 times the initial capital
- If we risk 40% per trade, then after 100 trades, the expectancy is that capital would be 0.45 times the initial capital (55% loss of capital)
- If we risk 50% per trade, then after 100 trades, the expectancy is that capital would be 0.01 times the initial capital (99% loss of capital)
- Risk above 50%, then after 100 trades or lesser, and have the expectancy of losing 100% of your capital!!

 Would I not choose a 1442% return over 76% return? Why not?!!

Anti-Kelly reasons (excuses?) 

 

The problem is that we are unsure of the future win % and RR ratio of trading systems. I guess that is the reason most books recommend 1-2% risk.

The return with 19% risk looks grand... but there could be practical difficulties.

1. Will the Win % and RR ratio of the system hold into the future?
2. What happens when the system hits a continuous losing streak? According to calculations, we should still continue to risk the optimal risk %, if the system will get back to the Win % and RR ratio in future.
3. Margin requirements may limit position size, and not allow optimal risk.
4. The market/s may not have sufficient liquidity to allow compounding of position size.
5. With increased position size, slippages and fills may be affected. This will also impact the Win %, RR ratio, and % of Capital risked.

Points 4 and 5 only become relevant when the position size increases significantly. For a small trader, only question 1 to 3 are relevant, and the answer is that we should try to risk close to the optimal risk to get better return on capital. Notice that even if the system performance deviates slightly, the optimal fraction will only vary slightly.

p and b - The indispensable criterion to apply the Kelly Criterion

Read the formula again.

It says that for a given p and b, risk f. 
Risk f, only if you know p and b. 
If you do not know p and b, you cannot find f.

f = ( p * ( b + 1 ) -1 ) / b


where,
f = optimal fraction of capital to be risked to maximize long term gain
p = probability of winning (Win %)
b = expected reward per unit of risk (Risk to Reward Ratio)