Prop Firm Risk-to-Reward & Mathematical Expectancy Calculator for 100k Challenges

The Mathematical Expectancy Equation in Trading

Mathematical expectancy defines the average dollar amount you expect to win or lose for every dollar placed at risk over a large sample of trades:

# Mathematical Expectancy Formula
E = (Win_Probability * Average_Win_R) - (Loss_Probability * 1.0)

For example, a strategy with a modest 40% win rate executing trades at an asymmetrical 1:3.0 Risk-to-Reward ratio yields:

E = (0.40 * 3.0) - (0.60 * 1.0) = 1.20 - 0.60 = +0.60 R per trade

Over a 50-trade sequence on a $100,000 account risking $500 (0.50%) per trade, a +0.60R expectancy produces an expected net return of +$15,000 (+15.0%)—comfortably exceeding both Phase 1 (8%) and Phase 2 (5%) challenge profit targets.

Expectancy Matrix: Win Rate vs. Risk-to-Reward Ratio

Python Monte Carlo Expectancy Simulator

import numpy as np

def simulate_prop_challenge(equity: float = 100000, risk_pct: float = 0.5, win_rate: float = 0.40, rr: float = 2.5, trades: int = 60, iterations: int = 1000):
    risk_amount = equity * (risk_pct / 100.0)
    successes = 0
    breaches = 0
    
    for _ in range(iterations):
        eq = equity
        max_daily_loss = eq * 0.05
        day_start_eq = eq
        failed = False
        
        for t in range(trades):
            is_win = np.random.rand() < win_rate
            if is_win:
                eq += risk_amount * rr
            else:
                eq -= risk_amount
                
            # Check 10% max total drawdown
            if eq <= equity * 0.90:
                breaches += 1
                failed = True
                break
                
            # Check 8% profit target
            if eq >= equity * 1.08 and not failed:
                successes += 1
                break
                
    pass_rate = (successes / iterations) * 100
    print(f"Simulation of {iterations} runs: Pass Rate = {pass_rate:.1f}%")

if __name__ == "__main__":
    simulate_prop_challenge()