Kelly Bet Sizing 2

[Leth]

More on Kelly Bet Sizing
by Michael Harris

APPLICATION OF THE KELLY FORMULA

An application of the Kelly formula for optimal position size determination requires the
actual values of the performance parameters of a trading system and specifically of the following
two: the success rate (profitability) and the ratio of average win to average loss. This is a
dramatic departure from other position sizing methods, such as the fixed risk percent method for
example, because of the information that must available in advance before one can determine
position size. Optimal position sizing based on the Kelly formula cannot be applied to newly
developed trading systems because the actual values for the two parameters that are used by the
formula are not available in advance. But even in the case of systems used in actual trading for
an extended period of time, there is no guarantee that these parameters will remain constant or
even within a certain range. This is especially true in the case of trend following systems, where
the average win to average loss ratio is a random variable because it depends on the magnitude
of unknown future trends and price volatility. Therefore, application of advanced optimal bet
sizing methods, like the Kelly formula, requires frequent evaluation of system performance
parameters in order to confirm that they maintain their minimum values.

Example: Stock Trading System

Trading capital: $100,000

Profitability: 65%

Ratio of average win to average loss: 1.25

In order to determine the optimal position size according to the Kelly formula, first we plug in
the numbers in equation 4.10 to calculate Kelly %, the optimal bet size:
Kelly % = 0.65 – (1 – 0.65)/1.25 = 0.37 or 37%

If the stop loss is 10% and the entry price is $85, then the number of shares according to the
optimal betting strategy of the Kelly formula is:
Number of shares N = (0.37 x $100,000)/(0.10 x $85) = 4,352 or 4,300 shares

If the stock drops 10% then the loss will be equal to: 4,300 x 85 x 0.10 = $36,550 close to the
allowable bet size due to the rounding of the number of shares.

However, the optimal bet size calculated above results in a number of shares that cannot be
purchased with the available capital of 100K. This is because 4,300 shares x $85 per share =
$365,500. In this case then, only $100,000/$85 = 1,176 or 1,100 shares can be purchased. The
actual risk if the stock drops 10% is 1,100 x 85 x 0.1/100,000 = 0.0935, or 9.35%, instead of the
37% risk calculated by the Kelly formula.
Therefore, the optimal bet size can be realized provided that the ratio of Kelly % to stop-loss
percent is less or equal to 1. In the above example this ratio is 37%/10% = 3.7. This means that
only 27.02% of the optimal bet size can be used, or in another sense, one could say that the
allowable solution is sub-optimal by a factor of about 3.7.
In comparison, a fixed fractional or fixed risk percent position sizing method for a system with
the parameters of the example presented and a risk percent per trade equal to 4% limits the
maximum number of shares to 465 only (just replace 0.37 above by 0.04 to get this result).

SUMMARY

The Kelly formula can often result in excessive risk per trade, in exchange for a geometric
growth of equity performance. The excessive risk is justified based on the assumption that the
profitability and ratio of average win to average loss estimates used in the formula correspond to
the figures of the system. It is up to the trader to decide which method to use given his personal
risk profile and confidence in his trading methods and systems. For beginners the fixed fractional
position sizing method with a maximum risk per trade of no more than 4% is recommended until
the performance of the method or system used is fully measured under actual market conditions.
All traders, especially new ones, should always keep in mind that trading involves substantial
financial risks and can result in total loss of capital.