Sunday, January 29, 2012

The Meaning Of Edge.

In my previous post I gave you the answer to the ultimate question of life, the universe and everything. No, not 42, but .046! This number represents the "edge" that a trader needs to achieve provided that she risks 2% of her account per trade and is willing to accept a 1% risk of ruin. But what does this edge mean?

I have been searching all over the web for this and there are some misleading perceptions out there over the meaning of edge. First of all, my use of the word is strictly with respect to the risk of ruin formula:
PR = ((1-Edge)/(1+Edge))^RiskUnits
In this usage, we see that Edge is a number between 1 and 0. This is because PR is a probability number that is constrained to be between 1 and 0 and values of Edge outside of that create illegal probabilities. Edge is an advantage that creates the expectation of making money so if your edge is negative you need to fade that strategy.

Consider the meaning of Edge=0. This produces a PR of 1, certainty, no matter how small you are trading because 1 raised to any power is still 1. It is clear that Edge=0 is an even-money result that over time will eventually produce ruination. If we are playing an even-money game such as betting on coin-flips then if we play long enough then we will eventually encounter a string bad luck (not necessarily in consecutive flips) that reduces our capital to the point where we have nothing left and have to stop playing.

Now consider this infinite coin-flip contest: we win 1.02 for a heads result and lose 1.00 for a tails result on a fair coin. We risk 1 to make 1.02 but our average win/loss ratio is 1. One should say that we have an edge of  1% because we now have the expectation of a 2 cent gain on 1/2 of the coin flips. 

Suppose that the house in this coin-flip game wants to even things up by substituting an unfair coin. They wish to avoid detection by altering the coin by the smallest amount necessary. They only need make it so the win/loss ratio is 1/1.02. This is achieved by weighting the heads side so that the probability of a tails is .505. Thus in 1000 flips of that coin you would expect to see 495 heads paying almost 505 (504.9, to be precise) while the house collects 505 dollars on the tails outcomes. Thus #Wins/#Losses = Risk/Reward in an even-money game.

Thus edge can be mathematically defined as the win-rate times the difference between the Win/Loss ratio and the Risk/Reward ratio or:
Edge =  #Wins/#Trades  * ((#Wins/#Losses) - (Risk/Reward))
I find this particularly useful for my trading because I decree the Risk/Reward ratio by use of fixed stop and limit orders (or defined-risk option spreads.) Tracking the number of wins and losses is a simple matter of data collection.

Of course, average return per dollar of risk would also qualify as edge for the above risk of ruin calculation, too. This is because over a large number of trades one would come to expect this average in the future.

In one popular trading strategy (for which I am endlessly spammed) a limit order takes profit at 1.5 times the amount risked by a stop order. Yet, it is claimed that this wins about as many times as it loses. So I surmise from the claim that the edge of such a strategy is .167  (=.5 * (1-2/3)). We can then find out how many riskUnits we need  in order keep our risk of ruin low:
.01 = (1-.167)/(1.167)^RiskUnits
Plugging that into WolframAlpha because I am too lazy to solve for RiskUnits, tells me that I would need at least 14 times the amount risked, not including margin, in order to be able to trade that system safely.

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