# Options For Determining How Much To Risk On A Trade (An Intro To The Kelly Criterion)

#### mojoyoyo

##### Recruit
Options For Determining How Much To Risk On A Trade (An Introduction To The Kelly Criterion)

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By Shon Shampain, fx@zencowsgomu.com

© 2010, All Rights Reserved. No part of this article may be reproduced, in whole or in part, without the specific written permission of the author.

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Disclaimer: Trading is risky in nature and not suitable for all persons. It can never be proven (only disproven) that past results are indicative of future performance, and no claim to that effect is being made either explicitly or implicitly. Substantial losses are possible.

Notation: In this article, 1 pip is a move in the 4th decimal place for non-JPY pairs, and and move in the 2nd decimal place for JPY pairs. So a move from 1.2345 to 1.2346 in EUR/USD is a move of 1 pip (likewise a move from 91.23 to 91.24 in USD/JPY). When quoting 5 decimal places (or 3 for JPY pairs), a move in the 5th decimal place (or 3rd for JPY pairs) is referred to as a tenth of a pip. Therefore, a move from 1.23456 to 1.23457 in EUR/USD is a move of one tenth of a pip, as is a move from 91.234 to 91.235 in USD/JPY.

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Conventional trading wisdom says that you should never risk much of your account on any one trade. Opinions vary, but most advice recommends you stick with a figure of about 2% or 2.5% of your account. Here is how risk sizing would look in this case:

Account balance: \$10,000
Stop loss level: 30 pips
Trading mini lots on EUR/USD (1 pip = \$1)
Risk level: 2.5%

\$10,000 x 2.5% = \$250
\$250 / 30 = 8.3

In this manner, if we are stopped on the trade, we only lose 30 pips on 8 mini lots, which is just under 2.5% of our account balance.
The Kelly Criterion says that this is not the mathematically optimum amount of money to risk on the trade if your goal is to maximize the amount of money you want to earn, while minimizing the amount of time in which you earn it. In order to calculate the amount of money to risk using a Kelly system, you need to keep track of your trades over time.

Here is an example of how Kelly sizing would work.

(These are the metrics that need to be maintained over time for your trading system.)

Number of winners: 179
Number of losers: 79
Total pips won: 4296
Total pips lost: 2638

(Now perform the Kelly calculations.)

Win percent = 179 / (179+79) = 69.38%
Average winning trade = 4296 / 179 = 24.0 pips
Average losing trade = 2638 / 79 = 33.4 pips
b = average winning trade / average losing trade = 24.0 / 33.4 = 0.7186
f = ((b x win percent) – (100.0% – win percent)) / b
f = ((0.7186 x 69.38%) - (100.0% - 69.38%)) / 0.7186 = (0.4986 – 0.3062) / 0.7186 = 0.2677

The Kelly calculations are saying that the mathematically optimum amount to risk on each trade is 26.77% of your bankroll. This might seem somewhat outrageous, and in many ways it is, but before we pass judgment, let's look at some of the assumptions we would be making.

First, it would be assumed that your bankroll and ability size your wagers is infinitely divisible. If you start with a million dollars, this might be true. If you start with \$100, this may not be true.

Second, it would be further assumed that the metrics you are using to calculate your Kelly parameters are a true indication of how your trading system will perform in the future. This is a rather dicey proposition even without discussing it further, as it implies that past results are a true indicator of future results (which cannot be proved, only disproved). Complicating the matter is the number of observations you have made. More observations over more time (and a broader range of conditions) is better.

Third, and psychologically very important, is the fact that even if you can reconcile the assumptions put forth up to this point, trading at the Kelly level is intensely volatile. The swings in your account balance would be positively mad. However, if you would look at simulations of Kelly systems, what you would find is that there is a type of fracticality to the sequence.

Suppose you started with \$10,000. For the first period of time your balance would oscillate around this value, perhaps dropping down to a crazy amount like \$500, and perhaps climbing up to \$30,000. And then repeating in a maddening sequence of fits and starts. But then a sequence of trades would occur that will take your balance up to the next level, suppose its \$50, 000.

Now, looking backwards, the volatility around the \$10,000 level that you just left look tame. On a graph, its hardly a blip. But they are exactly the same kind of variations you will experience as your balance runs up from \$50K to \$90K and down to \$7K, in a stomach-churning sequence that seems to have neither rhyme nor reason. But then a sequence of trades will occur that will elevate you to the next level, and looking backwards the volatility at the \$50K level will seem pedestrian.

This sequence will repeat itself over and over, with the highs getting progressively higher and the downswings getting more intense, until the investor finally has had enough and takes all the money into T-bills and plans a nice retirement in the South of Spain, or unfortunately discovers, as his account balance mercifully reaches zero, that an initial assumption had turned out to be incorrect.

Take a look at this chart (http://www.zencowsgomu.com/Kelly1.gif) and look at the very beginning. The very mild variation that seemed to be occurring around the \$10K level was just as wild and extreme as the oscillations at the end when the account balance was around \$250K.

Taking a percentage of the Kelly value, whether its one half or one quarter, is often viewed as a good compromise. Not only does it remove you from the bleeding edge of volatility, it protects you in case you have overestimated your winning percent (it should be explicitly noted that even if your past results are indicative of your future results, and again this can never be proved, if you wager more than the Kelly percent, your account will go to zero, eventually).

In closing it might be noted that the conservative 2.5% risk from the first example would have been about a one eighth Kelly system as described by the second example. How to manage risk is a decision that every investor must make personally, and it is always a good idea to be aware of the range of possibilities.

#### L. Jay Smith

##### Recruit
Another limitation to using the Kelly Criterion is allowing for short term deviation from our long term Win/Loss ratio.
While over the long term, say the last 100 trades, we might average a good 3 out of 4 trades profitable, but looking at any short term run in that series, and we can easily find places that we lose 4 trades out of 4. (And worse)
The Kelly formula also does not concern itself with little details like leverage or a trade sitting temporarily at a loss (with leverage). This is where over-trading will stop you out at a big loss even though eventually your prediction could have been correct.