A signal fires 100 times a day. A true positive earns $40, a false positive costs $25, and a missed opportunity costs nothing. The model's score is calibrated. Derive the probability threshold that maximises expected profit and compute it.

A signal fires 100 times a day. A true positive earns $40, a false positive costs $25, and a missed opportunity costs nothing. The model's score is calibrated. Derive the probability threshold that maximises expected profit and compute it.

Approach: Write the expected profit of acting on a signal with calibrated probability p, and find where it crosses zero.

0.385. Acting on a signal with calibrated probability p has expected profit 40p - 25(1 - p) = 65p - 25, which is positive when p > 25/65 = 5/13 = 0.3846. So the optimal rule takes every signal above 0.385 and nothing below it, and the threshold depends only on the ratio of the two payoffs, never on 0.5 and never on the class balance. The general form is threshold = C_fp / (C_fp + B_tp), where the numerator is the cost of a wrong action and the denominator is the total swing between being right and being wrong. If a missed opportunity carried a cost, that cost would be added to the numerator's counterpart and the threshold would fall. Two things break this in practice: the probability must actually be calibrated, since a threshold applied to an uncalibrated score selects the wrong set, and the payoffs must be net of costs, so the $40 has to be after spread, fees and slippage at the size traded. If the payoff varies per opportunity the correct rule is to act when p*B - (1 - p)*C is positive for that specific row, which means the model should be sized against expected profit rather than thresholded once for all rows.

Follow-up: If the win size is itself random with mean $40 and standard deviation $60, how does the optimal rule change under a growth-rate objective rather than expected profit?

Key concepts: expected profit, decision threshold, cost matrix, calibrated probability.