# The Surprising Effect of Small Edges

Imagine you have a trading method which trades the GLOBEX west Texas crude oil contract (ticker:CL).  This is a futures contract that represents 1000 barrels of crude oil.  At current oil prices, the contract has a value of about \$100,000.  Price for CL is specified in dollars per barrel, and the tick is one cent per barrel, or \$10 for the contract.  Now, let’s say this method has very simple trade management – once a trade is entered, a stop loss order is placed 10 ticks away from the entry price, and a limit order to exit is placed 10 ticks the other direction.  These orders are set up as a “one cancels all” or OCA group, meaning that if one of the two exit orders executes, the other is canceled.

This is a very simple type of trade setup, and is common for very short term trades.  In this example once you enter a trade (assume a 1 contract position) you should get one of two resuls:

1. You win \$100 minus fees
2. You lose \$100 plus fees + slippage

Fees in this example are simply your transaction fees.  Slippage is the cost associated with stop orders executing at a price worse than the trigger price, which they can do since they trigger market orders.  For now, let’s assume that the round trip transaction fees for 1 contract of CL are \$4 (my broker charges \$4.01 for some strange reason) and that slippage is zero.  The slippage assumption is unrealistic and I’ll address it in a subsequent post, but for now I want to get at a different aspect of the situation.  I want to know:

How often do you have to place your trade in the right direction in order to make good money here?

Luckily, the math for this is easy.  Assign a probability P that your trade is a winner.  For readability sake I’ll write these in percent – 50% would mean that half the trades hit the profit target and half stopped out for a loss.  But for arithmetic purposes we need to represent that as P=0.5 since probabilities range 0 to 1.  Now, what I want to do is calculate our expectation for the trade as a function of P.  Expectation is simply the result you expect to get for one trade if you did a large number and averaged the outcomes.  Some trading literature will term it “expectancy” instead – same thing.  The formula for expectation is

The sum across all possible outcomes of P(outcome)*result in dollars

P(outcome) is simply the probability that you get a given outcome.  All the P(outcomes) should add up to one since every trade should produce some result.  Now, I’ll re-name P from earlier to P(win) – the probability that you win 10 ticks minus fees.  And I’ll coin P(loss) = 1 – P(win) as the probability that you lose ten ticks plus fees.  Since the trade management for this trade is so simple, there are only two possible outcomes and the relationship between them is easy to understand.  Now, the formula for our expectation is as folows:

Expectation = P(win)*(\$96) + P(loss)*(-\$104)

note: P(loss) = 1 – P(win)

Now I’ve gone and computed a table of some expectations for various values of P(win):

 Win percentage P(win) Expectation on \$100 bet, \$4 fees Results from 250 trades per year Simple Annual Return on \$10,000 Performance Bond Approximate years to double your money assuming full compounding 45% -\$14.00 -\$3,500.00 -35% N/A 50% -\$4.00 -\$1,000.00 -10% N/A 52% \$0.00 \$0.00 0% N/A 54% \$4.00 \$1,000.00 10% 7.2 56% \$8.00 \$2,000.00 20% 3.6 58% \$12.00 \$3,000.00 30% 2.4 60% \$16.00 \$4,000.00 40% 1.8 62% \$20.00 \$5,000.00 50% 1.4 64% \$24.00 \$6,000.00 60% 1.2 66% \$28.00 \$7,000.00 70% 1.0

The results from the third column are more interesting in the context of a \$10,000 capital requirement.   Earning \$2,000 on \$100,000 isn’t very attractive but earning \$2,000 on \$10,000 is pretty good.  That’s what the fourth column represents in the form of annual return.