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Tuesday, September 19th, 2006

How much of my total trading account should I risk? (Risk, part IV)

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Last week we looked at how much to risk on each trade. But unlike gamblers, who play one hand at a time, traders have more to consider. An account usually consists of many individual stocks. So we also need to work out how much exposure we should have to the market across all positions.

The Kelly Criterion is one method of determining optimal bet sizes in gambling that can also be useful for investors. Legg Mason fund manager Bill Miller said that “the Kelly Criterion is integral to the way we manage money.” The story of how the Kelly Criterion came about is told in William Poundstone’s “Fortune’s Formula, the Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street”.

Texas-born Kelly worked at Bell Laboratories, where he was regarded as one of the smartest scientists. He became interested in optimal gambling strategies after the success of a game show led to betting scams. Some shrewd operator got the result of the game show when it was broadcast in New York and then bet on the West Coast where the show was aired three hours later.

Kelly reasoned that an optimal gambling strategy in this situation lay between betting everything on one hand and not betting enough to make winning worthwhile. With inside information, or an ‘edge’, what would be the optimal amount to gamble? Kelly realized that work by a colleague at Bell Labs on information theory could help find the answer of how much to bet to produce the greatest compound return on capital.

The formula he came up with was edge/odds = optimal bet size. The edge is how much you expect to win on average. The odds are the profit if you win, which are the public odds.

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KELLY ADAPTED TO THE STOCK MARKET

Working out the optimal amount of exposure for your account using the Kelly Criterion is also easy with a bit of number crunching. Fortunately for investors, mathematician and hedge fund trader Edward O. Thorp adapted Kelly’s work to blackjack and the stock market. We’re left with a simple formula to help us determine our optimal exposure to the market: Kelly % = W – (1-W)/R. Where Kelly% is the percent of your capital to risk in the market, W is the historical winning percentage of the trading system and R is the historical win/loss ratio.

For example, if your system has a winning percentage of 50 per cent and profits on average are twice as big as losers, W = 0.5 and R = 2. Slotting those numbers into the equation yields 0.25. So to maximize your returns you’d bet 25 per cent of your account.

That doesn’t mean you risk 25 per cent on each stock, but that should be your total risk across all your holdings. For example, if we’re risking 2 per cent on each stock and our optimal exposure based on the Kelly equation is 25 per cent, then we would be able to hold 12 positions. Similarly if you were risking 1 per cent on each individual stock you’d have 25 positions.

Most people will ask: how do I find out my win rate and win/loss ratio? If you have detailed records of past transactions it should not be that difficult. If you don’t, start a simple spreadsheet database of all your investments. To determine W, just divide the number of winning investments by the number of losing investments. To get R, divide the average amount you made on the winning investments by the average amount you lost on the losing investments.

MODIFIED KELLY

The Kelly formula is an aggressive form of betting and is designed to maximize long-term returns. Because of that it will create significant volatility. Therefore, some recommend using between 50 per cent (‘half Kelly’) and 80 per cent of the Kelly equation. Perhaps your optimal exposure based on the Kelly Criterion is 20 per cent, but you want to cut the volatility of your account. Halving the optimal Kelly would give you 10 per cent total market risk. Ultimately it will depend on your investment objectives and tolerance of volatility, and will require you to play around with different risk levels to find what suits you.

There are alternative ways to determine optimal portfolio exposure. One is the inverted Sharpe ratio method used by Kenneth L Grant, which I will take a look at later this week. Calculating this ratio will require more detail, including statistics on account volatility, but on the whole it’s quite simple and effective. Most will find the inverted Sharpe ratio and Kelly Criterion give similar results.

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Word Count: 763. This entry was posted on Tuesday, September 19th, 2006 at 7:19 pm and is filed under Risk management. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.