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A Trading Math Question with Nearly 75% Wrong Answers

Nearly 75% of those who voted  in a poll about a basic trading math question got it wrong. Here is the question and the solution.

Question: The profit factor is 2, the win rate is 50% and the average loss is 0.5. What is the expectation assuming sufficient samples?

Below is the tweet with the poll:

Solution

The average trade AvgT is given by the following equation:

AvgT=w  × avgW – (1-w) × avgL  (1)

where w is the win fraction, avgW is the average win and avgL is the average loss.

The average trade AvgT, converges to the mean of the distribution at the limit of sufficient samples, also know as the expectation E:

E = w  × avgW – (1-w) × avgL  (2)

The problem statement referred to sufficient samples and we can use equation 2. We know the win fraction (0.50) and AvgL (0.5). We need to determine avgW to calculate E.

The avgW can be determined after applying the Profitability Rule, a formula found in Chapter 4, page 51, equation 4-18, of my book Profitability and Systematic Trading.  A free PDF of the book can be found here. 

Profitability Rule:         w = PF/(PF+R)  (3)

where PF is the profit factor and R is the payoff ratio = avgW/AvgL. (Note that in the book the win fraction is denoted by P.)

Solving (3) for R yields the following:

R = PF ×  (1-w)/w     (4)

Since PF = 2 and w is 0.5, then R is 2. Then, from the definition of R we get that avgW is 2 × 0.5 = 1. Now we have all the values required to calculate E in equation 2.

E = 0.5 × 1 – 0.5 × 0.5 = 0.25

and this is the correct answer that received nearly 25% of the votes.

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