A Simple Model of the Odds of Passing an Evaluation

[Fairman]

A Simple Model of the Odds of Passing an Evaluation

Model an evaluation as a walk where each trade wins or loses 1R, with a target of plus 8R and a failure at minus 10R. If the win probability is p, the chance of hitting the target first is given by a standard result. Let the ratio of loss to win probability be r equals q divided by p. The probability is (1 minus r to the power of 10) divided by (1 minus r to the power of 18).

At p equals 0.50, r is 1, and the formula reduces to 10 divided by 18, which is 55.6 percent.

At p equals 0.52, r is 0.923. Then r to the 10th is about 0.449, and r to the 18th is about 0.237. The probability is (1 minus 0.449) divided by (1 minus 0.237), which is 0.551 divided by 0.763, about 72 percent.

At p equals 0.55, r is 0.818. r to the 10th is about 0.134, and r to the 18th is about 0.027. The probability is 0.866 divided by 0.973, about 89 percent.

Notice how steeply the odds depend on a small edge. A coin-flip strategy passes just over half the time in this simplified model. A strategy winning 55 percent of 1R trades is far better.

What the model leaves out: time limits, daily loss rules, payoffs that are not symmetric, streak dependence, and costs. Each lowers the real pass rate.

Costs matter especially. If each trade costs 0.1R, a 55 percent win rate effectively becomes 50 percent.

Exercise: modify the target and the floor, and see how the pass rate changes at a fixed win probability.

[FXS HongKong Scalper]

A nice application of the gambler's ruin formula. It shows how much a small edge improves the odds. In reality trades are not exactly plus or minus 1R, but the formula gives a good intuition for how the target and failure levels interact.

[Fairman]

The formula and its values check out. At a 50 percent win rate, the pass probability is 10 divided by 18, about 55.6 percent. At 52 percent, it rises to about 72 percent, and at 55 percent to about 89 percent. The steepness is the striking feature: a 5 point improvement in win rate moves the pass rate from roughly a coin flip to nearly nine in ten. That's because the target and floor are many steps away, so a small edge compounds over the many trades needed to reach either one. The same logic explains why casinos with small edges win reliably over many bets.

[Amanhasnoname]

I played with the exercise on a spreadsheet and the floor mattered more than I expected. Keeping the target at plus 8R and a 52 percent win rate, but moving the failure level from minus 10R to minus 5R, drops the pass probability from about 72 percent to about 51 percent. A smaller target of plus 4R with the floor at minus 10R gives roughly 82 percent. So a tight drawdown rule hurts a lot.

The other number I like is expected length. For a fair coin the average number of trades until a boundary is hit equals the two distances multiplied together. That is 80 trades for 8 and 10, and only 40 for 8 and 5. The tighter version is faster, but you pay for it with a much lower pass rate.

This is why I do not just risk bigger to finish quickly. The model says the cheap improvement is a few more points of win rate.

[Fairman]

The cost adjustment is an important refinement. If each trade costs 0.1R, a win nets 0.9R and a loss costs 1.1R. At 55 percent, the expectancy becomes 0.55 times 0.9 minus 0.45 times 1.1, which is 0.495 minus 0.495, exactly zero. So in expectancy terms, the 55 percent strategy becomes a coin flip. The pass rate under the model would also fall sharply, toward the 50 percent case. That shows how costs can turn a strong edge into no edge at all, especially for strategies with symmetric payoffs where the margin is thin.

[Fairman]

The things the model leaves out each lower the real pass rate. Daily loss rules can end an evaluation during a bad day even if the total floor isn't reached. Time limits mean the walk might not reach either boundary before the deadline. Asymmetric payoffs, such as 2R wins and 1R losses, change the shape of the walk and need a different formula or a simulation. Streak dependence, where losses cluster because of market regimes, makes long losing runs more likely than the model assumes. A simple Monte Carlo simulation can handle all of these with a spreadsheet.

[Fairman]

Modifying the target and floor, as the exercise suggests, shows how the rules affect the odds. A target of plus 10R with a floor of minus 10R gives exactly 50 percent at a 50 percent win rate. A target of plus 5R with the same floor gives about 67 percent. Smaller targets relative to the floor improve the odds. Risking less per trade effectively moves both boundaries further away in R, which helps a trader with an edge and hurts one without. Has anyone here run this model for a real evaluation? I'd be interested in how the predicted odds compared with what happened.