Trading of Stocks using Kelly Criterion
Team Latte
Sep 15, 2005
Let us say you have $100,000 to invest in a certain stock and you are faced with this question: you want to trade the stocks consistently over a period of one year but you don't know how much to put into the market at any one period, say, every month given the market's direction. As a day trader, you want to enter the market in the morning and exit at the close taking profits or losses with you. It is like a bet: you bet an amount in the market and at the end you either make money or lose money.
Kelly criterion can, with certain very simplistic assumptions, help you answer your question to a very great extent. Let us say that your investment horizon is three months and over the last three months the stock has had a mean monthly return of 0.2% and a monthly volatility of 7%, and that these numbers are calculated using monthly closing prices of the stock. You don't know which way the market will move, and you cannot short sell the stock, therefore, you would ideally want to bet a fraction of your total the total amount (that you hold) every month. You start with $100,000 on day zero and you want to bet a fraction q of that amount on day one of the first month. Now since markets follow a random walk the stock's closing price on the last trading day of the month would be a random outcome; let's denote this by a series of random numbers (one for each day) (z1,z2,z3,..............zN). Thus after the first month's bet (investment in the stock) you will have in your hand (at the end of the first month) an amount equal to:
On the first day of the second month, regardless of the outcome of the first day, you will bet the same fraction q of whatever is left from your investment from the previous month. Then, after the second month you will have in your hand an amount equal to:
After 12 months you will have the following amount:
Your problem is how do you choose the quantity q so that your strategy has the optimum payoff? In fact you are going to determine the quantity q such that it maximizes your expected long term growth rate. That growth rate is:
If we assume that the outcome of each month's result is independent of the previous day's result (which is to a very large extent true), i.e. that the random numbers are not correlated then the expected value of the above growth rate will be (ignoring the constant term in the above formula):
If we expand the argument of the above expected value in Taylor series around the mean and the standard deviation and then take the first mathematical derivative of that expression and equate it to zero (a condition for maxima) you will get the value of the fraction that you should invest in the stock in each month. In our example with a mean of 0.2% and volatility of 7% we will get the value for that fraction equal to 0.408. Therefore, you need to bet an amount equal to 0.40 of your holdings every month, i.e., starting with $100,000, you will invest roughly $40,800 in the first month, $16,600 in the second month and so on.
The most important constraint on the above framework is that the mean of the outcomes should be really small and the volatility should be high. If that is not true then the mathematical formulation will break down. Secondly, and equally importantly, in the stock market no trader or fund manager will bet (invest) a constant fraction of his holding every period regardless of the movement of the market. In fact, this dynamic nature of adjustment would render the whole exercise quite futile as in that event, when the quantity q is dynamically adjusted according to market's movement one cannot apply the above.
Still, Kelly criterion is can give an important insight into portfolio trading and asset allocation problem and can expanded greatly from the above simplistic model.
Disclaimer
"Risk Latte uses proprietary and non-proprietary mathematical and empirical models to measure the volatility and estimate the direction of the market. There is no guarantee of any particular outcome happening and readers must exercise caution while interpreting the conclusions of this article. Risk Latte Company is not a registered stock broker or an SFC registered entity and readers must take advise from their financial advisors, stock brokers, research analysts and bankers while making any buying or selling decisions. Risk Latte Company is not in the business of making stock or asset forecasts whether explicitly or implicitly and shall not be responsible for and/or liable for any losses arising out of any trading decisions based on the above article."
  
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