# How to optimally allocate capital among trading strategies?

I'm trying to find an optimal way to allocate capital among trading strategies.

"Quantitative Trading" by Ernie Chan claims on page 97 that the optimal fraction of capital to allocate to a given trading strategy can be calculated by the following formula.

f = μ/σ2, where μ is the mean and σ2 is the variance of the return of the trading strategy.

This assumes that the trading strategies are statistically independent and their returns have a normal distribution.

• The formula looks deceptively simple. Does it actually work?
• Do professionals use it at all?
• The author calls this the Kelly formula. Is this correct? I thought the Kelly formula was $\frac{p(b + 1) - 1}b$?
• The formula "f" is only optimal with respect to a specific level of risk-aversion assuming a mean-variance utility function – Ram Ahluwalia Oct 7 '11 at 3:23
• Keep in mind also that finding trading strategies with returns that are independent (and stationary) with normally distributed returns is unlikely. – strimp099 Oct 7 '11 at 3:39

To respond to your questions in order:

• The formula looks deceptively simple. Does it actually work?

That depends on what you mean by "work". Chan spends the rest of the chapter discussing the pitfalls of investing at "full Kelly".

• Do professionals use it at all?

Professionals may maximize geometric growth, but I don't know anyone who does so with such a simple and unconstrained objective function.

• The author calls this the Kelly formula. Is this correct? I thought the Kelly formula was (p(b + 1) - 1)/b?

The Kelly formula you cite is based on discrete events with known probabilities and outcomes. As Chan says several times in that chapter, the formula from his book is an approximation of the Kelly formula for continuous outcomes.

First of all a very warm welcome to Quantitative Finance Stack Exchange :-)

Concerning your question there are some basic points that seem to be unclear. In general "Quantitative Trading" by Ernie Chan is a good starting point for learning about quantitative trading strategies. The problem is of course that in this small book there are many concepts whose interrelation may not always be completely clear.

I think it is always helpful to separate a trading system into three general levels:

1. Forecast
3. Risk management

Concerning the first point: You must have some idea about the market. If you thought everything was efficient you would not trade (even "buy-and-hold" reveals some forecast, be it the exact asset allocation, be it that you believe in a rising market in the long run).

Concerning the second point: When you think that you found some structure in the market the question is how to exploit it. The latter does not necessarily follow from the former (I won't go deeper into this matter here).

When talking about the "Kelly rule" this is normally referring to the third point, so it is about position or money management. But you must first have some edge (forecast) to use it!

To understand some of the intricacies of the Kelly formula I would recommend another excellent book (which is a real pageturner by the way):

Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone

If you have K strategies and each strategy has an expected return, a variance, and you can measure the covariance of your strategies performance then a mean-variance optimization would answer how to optimally allocate capital amongst your strategies. Key to this approach is accurate estimation of your the input parameters identified above.

The formula looks deceptively simple. Does it actually work? The formula is a correct approximation if the asset in question is not too volatile. It works good if a) you exactly know the parameters mu and sigma b) if you can commit a lot of trade In practice the parameters are often very roughly estimated. The rule of thumb is to reduce mu and increase sigma to avoid overbetting.

I have developed a formula for multivariate portfolios with many assets: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2259133 In this paper I implicitly assume that the assets are sold or bought simultaneously.

You may also have a look at my book "Knowledge rather than Hope", in which I discuss a practical application of Kelly Criterion to a portfolio with non-simultaneous buys and sells. The book is available on Amazon, the R-scripts can be downloaded from my webpage http://www.yetanotherquant.com

Regards Vasily