The Many Faces of Trading Expectancy

Some authors define trading profitability using the expected value or expectation, a.k.a expectancy. However, it would only suffice for them to say that a trading system, trading method, or even investment program, is profitable if the net profit/loss is positive, i.e. if it makes money. I will actually prove in this post that expectation has many faces that often they make more sense than a fancy mathematical definition. 

The mathematical expectation of a trading system, or method in general, is usually referred to as expected gain, expectancy or expected profit per trade by various authors. It is defined mathematically as the expected value of the random variable T, the P/L of trades, as follows:

E[T] = w × avgW – (1-w) × avgL                   (1)

where w is the win rate, avgW is the average winning trade and avgL is the average losing trade (in absolute value).

If E[x] > 0 then the trading system, or method, is profitable:

 w × avgW -(1-w) × avgL  > 0      <=>  Profitability                 (2)

meaning that a positive expectancy is a necessary and sufficient condition for profitability. 

I will demonstrate here that the notion of positive expectation, as defined by equation (2), is mathematically equivalent to the following claims:

(A) The Profit Factor is greater that 1
(B) The Net Profit/Loss is greater than zero
(C) The Average Trade is greater than zero

Actually, I will show that since there is a mathematical equivalence between the notion of positive expectancy and A, B and C, there is no advantage in using any of these notions in declaring the necessary and sufficient condition for profitability, meaning that

w × avgW -(1-w) × avgL  > 0 <=>  PF > 1 <=> NPL > 0 <=> AvgT > 0 <=> Profitability                

where PF is the profit factor, NPL is the net profit and AvgT is the average trade. (Note: some authors define expectancy as the expectation divided by the average loser. In this post I use expectation and expectancy for the expected trade value.)

A. Positive expectancy is equvalent to a positive profit factor 

The profit factor PF is the ratio of the sum of winning trades to the sum of losing trades. The equivalence is demonstrated using equation (1) as follows:

E[T] > 0 => w × avgW – (1-w) × avgL > o => w × ∑W/Nw – (1-w) × ∑ L/NL > 0 => (Nw/N) × ∑W/Nw – (NL/N) × ∑ L/NL > 0 => ∑W – ∑L > 0  => ∑W / ∑L > 1  (2)

(requires ∑L > 0)

where Nw and NL are the number of winning and losing trades, respectively, and ∑W and ∑L the sum of winning and losing trades respectively. Recall that w = Nw/N and 1-w = NL/N.

B. Positive expectancy is equivalent to a positive net profit/loss

Equation (2) can be re-written as follows:

∑W – ∑L > 0 or NPL = ∑W – ∑L > 0  (3)

Equation (3) simply states that the net profit/loss NPL is greater than zero. Thus, Equation (1) is mathematically equivalent to equation (3). This can be also proved if one starts with equation (3):

∑W – ∑L > 0  (3) => avgW × Nw – avgL × NL > 0     (4)

(because avgW = ∑W/Nw  and avgL =  ∑L/NL)

Now, recall that Nw + NL = N, or that winners and losers sum to the total trades. Then equation (4) becomes:

avgW × Nw – avgL × (N – Nw) > 0        (5)

Now, assuming that N > 0, divide equation (5) through by N:

avgW × Nw/N – avgL × (N – Nw)/N > 0  =>

=> avgW × Nw/N – avgL × (1 – Nw/N) > 0       (7)

Recall that w = Nw/N, i.e. the win rate, is equal to the number of winning trades divided by the number of total trades. Then equation (7) can be written as follows:

avgW × w – avgL × (1 – w) > 0       (8)

which is identical to equation (1). This means that, mathematically, equations (1) and (3) mean the same thing.

C. Positive expectancy is equivalent to a positive average trade

Next, it is fairly easy to show that equation (2) is equivalent to the statement that the average trade is greater that zero. The average trade is given by:

AvgT = (∑W – ∑L)/ N > 0 => ∑W/N – ∑L/N > 0     (9)

Now, multiply through by N >0 and we get equation (3)

∑W – ∑L > 0  (10)

We can go about proving the equivalence as before or we can try a different way. After multiplying the first term of equation (10) by (Nw/Nw) and the second by (NL/NL) and re-arranging, we obtain

(Nw/N) × ∑W/Nw + (NL/N) × ∑L/NL > 0     (11)

We now make use of the fact that Nw + NL = N => NL = N – Nw and (11) becomes

(Nw/N) × ∑W/Nw + ((N – Nw)/N) × ∑L/NL > 0     => (Nw/N) × ∑W/Nw + (1 – Nw/N) × ∑L/NL > 0  (12)

Recall that avgW = ∑W/Nw, avgL =  ∑L/NL and  w = Nw/N, then equation (12) becomes identical to equation (8)

avgW × w – avgL × (1 – w) > 0       (13)

which is identical to equation (1). This means that mathematically, equations (1) and (10) mean the same thing.

Conclusion

I have proved above that the definition of a trading expectancy for profitable trading systems, or trading methods in general, given by

E[x] = w × avgW -(1-w) × avgL  > 0 

is equivalent to the (common sense) statements that

(A) The Profit Factor is greater that 1
(B) The Net Profit/Loss is greater than zero
(C) The Average Trade is greater than zero

There is no advantage in using any of the above when defining profitability. The only advantage may arise from specific uses, like for example, equation (1) is useful in determining the optimal bet size for geometric equity growth. The optimal bet size has been shown to be equal to the expected gain divided by the average winner. This is the well-know Kelly ratio discussed here in more detail.

It is true that some authors insist on defining profitability based on the mathematical definition of expected value given by equation (1). However, it would only suffice for them to say that a trading system, trading method, or investment program, is profitable if the net profit is positive, i.e. if it makes money. Using definitions that entail certain conditions to be valid is both confusing and also an overkill. I will just mention a simple case where definition (B) holds but equation (1) does not.

Consider a long-term trading system that has produced only 3 trades, one profitable and two non-profitable but due to successful trend-following it made money and thus the net profit/loss is positive. In this case, definition (1) cannot be used because the notion of expected value applies only to sufficient samples [Ref. 1]. Although the average trade is positive, one cannot calculate the expectancy (mean) based on just 3 observations from an unknown distribution of returns. The average trade will only be equal to the expectancy in the presence of a sufficient sample. In other words, the expectancy is not known and one cannot talk about positive expectancy based on just 3 observations, yet the average trade is known and positive and the system is profitable.

Ref[1] Papoulis, A., Probability, Random Variables and Stochastic Processes, 1965, McGraw-Hill Inc.

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