Forget Kelly, forget fractional sizing. Where is the general theory?

Question

I am struggling to find a general theory of position sizing. Help!

The literature is all about fractional position sizing, but that's just one of the innumerable strategies. What about all the other sizing strategies?

The problem:

• Suppose I have a daily trading strategy that generates a daily return $r_i \sim N[\mu,\sigma^2]$
• I start with capital $x_0$ and run my strategy once a day, for 1 year
• each day I begin with the previous day's capital $x_{i-1}$ and I risk an amount $y_i=f(x_{i-1})$
• at the end of the year, I end up with a capital $x_n \ge 0$

Let's look at common metrics:

• $R=(x_n-x_0)/x_0$ the yearly return on investment
• $\mu_R=E[R] =$ the yearly return expectation
• $\sigma_R^2=\text{VAR}[R] =$ the yearly return variance
• $Ϛ_R=\frac {\mu_R} {\sigma_R} = $ the Sharpe Ratio

Questions:

• what utility function $U(R)$ would a risk-averse investor optimise?
• what is the best trade sizing function $f(x_i)$ that optimises $U(R)$ ?

An example:

• let's pick the Sharpe Ratio as utility function, so $U(R)=Ϛ_R$
• let's explore how different trade sizing functions yield different utility

• so function "B" in the chart corresponds to fractional trade sizing, while function "D" corresponds to constant trade sizing. The others two functions are less common, I picked them arbitrarily.

• besides being different functions, each function has a tuning parameter $\theta$. As an example, for function "B", $\theta$ is the % of my current capital I risk at each trade. That is $y_i=f(x_{i-1})=\theta x_{i-1}$

• I simulated 1M runs of each trade sizing function, also varying the tuning parameter $\theta$ between $0$ and $1$

• the various trade sizing functions yield a very different Sharpe Ratio, and -somewhat surprisingly- fractional trade sizing is the worst! (see red curve below)

• in the chart below are the results for daily return $\mu=1, \sigma=20$. But the differences in performance remain similar when I change $\mu$ and $\sigma$.

More questions:

• why is everyone talking about fractional trade sizing, if it shows such a bad Sharpe Ratio?

• did anyone study the problem more generally, instead of just trying arbitrary trade sizing functions, like I did in my empirical study?

Your opinion is welcome, thank you!

Answer 1

I've recently had to do quite a bit of work on position sizing.

Leonard C MacLean, Edward O Thorp, and William T Ziemba have written an incredible amount of literature on this. The following text book encompasses an incredibly deep study of the topic on position sizing, different utility functions and so on.

From what I can tell the two broad branches of position sizing split into capital growth theory and variants of mean-variance. In the context of Kelly, the Sharpe ratio may not be the best metric since it is trying to maximize growth rather than stable growth as measured by the Sharpe ratio.

Much of this is highlighted in the book above. I think you will find it a noteworthy read.

It has been shown that if you know the probability of success and the payout then there exists no other algorithm that outperforms the Kelly criterion over an infinitely long horizon (in terms of maximizing growth, not Sharpe). Much of the problem is that we, of course, don't have the probability of success upfront and often what this results in is us moving from Kelly in a two outcomes discrete setting to Kelly in continuous time which can be also be used in a portfolio management setting.

For games such as 21 Black Jack where we can determine the payout and probability of success, Kelly works fantastically well and has been coined Fortunes Formula.