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# Minimum Risk:Reward

Discussion in 'Strategy Building' started by tireg, Aug 26, 2006.

1. ### tireg

Kind of an old subject but I thought I'd touch up on it.

We've often heard the idea that a system can be profitable even though its number of winners is relatively less frequent than the number of losers. Common sense would dictate that in these situations, risk-management stops help keep the losers much smaller than the winners. A good example of this is most typical trend-following systems, which often have win rates of 30%-40%, yet are still profitable. Van-Tharp even had a whole piece about how a random-entry system could make money because of money management. But how much of this is attributed to an Edge, and how much to money management? More importantly, what is the minimum risk:reward one needs to break even (and then make money) with a given win rate?

Remember the formula for expectancy:
<span style="font-weight: bold;">Expectancy = (Probability of Win * Avg. Win Size) - (Probability of Loss * Avg. Loss Size)</span>

Now, let's simplify and see what happens with 0 expectancy, meaning breakeven system:

0 = (Prob. win * avg win size) - ((1 - Prob. win) * avg loss size)

<div style="text-align: center;"> </div><table str="" style="border-collapse: collapse; width: 234pt;" border="0" cellpadding="0" cellspacing="0" width="312"></table><div style="text-align: center;"></div><table str="" style="border-collapse: collapse; width: 234pt;" border="0" cellpadding="0" cellspacing="0" width="312"> <tbody><tr style="height: 12.75pt;" height="17"> <td class="xl26" style="height: 12.75pt; width: 49pt; text-align: center;" height="17" width="65">P(W)</td> <td class="xl27" style="width: 63pt; text-align: center;" width="84">Avg. Win</td> <td class="xl26" style="width: 66pt; text-align: center;" width="88">P(L) = 1-P(W)</td> <td class="xl27" style="width: 56pt; text-align: center;" width="75">Avg. Loss</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.01" align="right" height="17">1%</td> <td class="xl25" num="99" fmla="=C2/A2" align="right">99.00 </td> <td class="xl24" num="0.99" fmla="=1-A2" align="right">99%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.05" align="right" height="17">5%</td> <td class="xl25" num="19" fmla="=C3/A3" align="right">19.00 </td> <td class="xl24" num="0.95" fmla="=1-A3" align="right">95%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.1" align="right" height="17">10%</td> <td class="xl25" num="9" fmla="=C4/A4" align="right">9.00 </td> <td class="xl24" num="0.9" fmla="=1-A4" align="right">90%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.15" align="right" height="17">15%</td> <td class="xl25" num="5.666666666666667" fmla="=C5/A5" align="right">5.67 </td> <td class="xl24" num="0.85" fmla="=1-A5" align="right">85%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.2" align="right" height="17">20%</td> <td class="xl25" num="4" fmla="=C6/A6" align="right">4.00 </td> <td class="xl24" num="0.8" fmla="=1-A6" align="right">80%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.25" align="right" height="17">25%</td> <td class="xl25" num="3" fmla="=C7/A7" align="right">3.00 </td> <td class="xl24" num="0.75" fmla="=1-A7" align="right">75%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.3" align="right" height="17">30%</td> <td class="xl25" num="2.3333333333333335" fmla="=C8/A8" align="right">2.33 </td> <td class="xl24" num="0.7" fmla="=1-A8" align="right">70%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.35" align="right" height="17">35%</td> <td class="xl25" num="1.8571428571428574" fmla="=C9/A9" align="right">1.86 </td> <td class="xl24" num="0.65" fmla="=1-A9" align="right">65%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.4" align="right" height="17">40%</td> <td class="xl25" num="1.5" fmla="=C10/A10" align="right">1.50 </td> <td class="xl24" num="0.6" fmla="=1-A10" align="right">60%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.45" align="right" height="17">45%</td> <td class="xl25" num="1.2222222222222223" fmla="=C11/A11" align="right">1.22 </td> <td class="xl24" num="0.55" fmla="=1-A11" align="right">55%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.5" align="right" height="17">50%</td> <td class="xl25" num="1" fmla="=C12/A12" align="right">1.00 </td> <td class="xl24" num="0.5" fmla="=1-A12" align="right">50%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.6" align="right" height="17">60%</td> <td class="xl25" num="0.66666666666666674" fmla="=C13/A13" align="right">0.67 </td> <td class="xl24" num="0.4" fmla="=1-A13" align="right">40%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.7" align="right" height="17">70%</td> <td class="xl25" num="0.42857142857142866" fmla="=C14/A14" align="right">0.43 </td> <td class="xl24" num="0.3" fmla="=1-A14" align="right">30%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.8" align="right" height="17">80%</td> <td class="xl25" num="0.25" fmla="=C15/A15" align="right">0.25 </td> <td class="xl24" num="0.2" fmla="=1-A15" align="right">20%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="0.9" align="right" height="17">90%</td> <td class="xl25" num="0.11111111111111108" fmla="=C16/A16" align="right">0.11 </td> <td class="xl24" num="0.1" fmla="=1-A16" align="right">10%</td> <td class="xl25" num="1" align="right">1.00 </td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl24" style="height: 12.75pt;" num="1" align="right" height="17">100%</td> <td class="xl25" num="0" fmla="=C17/A17" align="right">0.00 </td> <td class="xl24" num="0" fmla="=1-A17" align="right">0%</td> <td class="xl25" num="1" align="right">1.00
</td> </tr> </tbody></table>
Notice how much larger win sizes have to be in relation to losses in the smaller % win rows, while having a higher win-rate means you can have smaller gains relative to losses. Keep in mind, this table is for break-even, so anything above the figures here imply positive expectancy.

For a 30% win-rate system, one must have at least a 2.33 unit gain for every 1 unit risk. Many people use a 3:1 R:R as a general rule of thumb, implying at least a 25% win rate.

Another way of interpreting these results is Profit Factor, which is Avg. Win / Avg. Loss.

2. ### tireg

Also read a very interesting study of stops and exits by Dr. Koch. Many of the conclusions drawn are worth exploring.

One of the results of his findings that pertain to the above post:

"Assymetric stops don't change expectancy

asymetric stop loss/profit target: Changes Win/Loss and AvgLoss/AvgWin accordingly, no change to expectancy and profit. If your stops are too wide, the probability of getting hit may be low. Perhaps too low to happen within you historical data interval."

This would imply that arbitrarily setting high R:R or low R:R without an edge will result in no change to expectancy. Furthermore, even with an edge, tightening stops or raising targets arbitrarily (as in, possibly an attempt to optimize) will also result in the same.

The original study is here.

EDIT: This goes to show that no amount of money management will make a nonprofitable system profitable; yet effective money management will enhance the results of an already profitable system.

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