How can I combine winrate, (risk/reward), and expectancy into a single formula?

winrate = wins / (wins + losses)

I fiddled around with it and found that:

requiredWinrate = risk / (reward + risk)

Expectancy formula

expectancy = (winProbability x averageWin) - (lossProbability x averageLoss)

So I reasoned that maybe

expectancy = (winrate x reward) - ((1-winrate) x risk)

Now I have two problems with this formula. One is I cannot be sure that it's correct and two is I cannot figure out how to solve for winrate or how to solve for (reward/risk).

In forex trade I may have a potential trade with given reward and risk values and given win rate in which case I would want to calculate expectancy to know if it has positive or negative expectancy. But I would also like to know what winrate I need in for given reward and risk values and positive expectancy in cases where I do not have a known winrate but can make a reasonable decision if I can calculate the required win rate.

Edit: I've confirmed that the formula is correct with a python script that simulates trades on a given winrate and compares a calculated expectancy and computed expectancy. https://pastebin.com/8FF1vAMm

• I think you need to use a measure of return instead of "winrate" - if you win \$1 100,000 times but lose \$100,000,000 once you're still winning according to your maths above. Feb 19, 2018 at 15:22
• I admit my understanding is merky but from your figures this is what my math seems to say winrate = 100,000 / 100,001 = 0.99999 and expectancy = (0.99999 * 1) - ((1-0.99999) * 100,000,000) = -999.00001 so I would never take such a trade according to that negative expectancy. But again I'm not sure if its right as I said. Feb 19, 2018 at 15:44