Accuracy versus Profitability

Accuracy treats every correct call as one and every incorrect call as zero. Trading PnL instead depends on purchase price, payout, size, and costs. They answer different questions.

Definitions #

For $n$ trades,

$$Accuracy=\frac1n\sum_{i=1}^n\mathbf 1(Y_i=1), \qquad Total\ PnL=\sum_{i=1}^n(Y_i-p_i)N_i-costs_i.$$

$Y_i$ is settlement, $p_i$ is execution price, and $N_i$ is shares. Accuracy discards three of those four economic inputs.

Below 50% accuracy and profitable #

Buy 100 YES shares across independent markets at $p=0.20$. If only 30 win, accuracy is 30%, but

$$PnL=30-100(0.20)=+10.$$

Each rare win earns 0.80 while each common loss costs 0.20.

Above 50% accuracy and unprofitable #

Buy 100 shares at $p=0.90$. If 80 win, accuracy is 80%, but

$$PnL=80-90=-10.$$

Frequent ten-cent wins do not cover twenty ninety-cent losses.

Life example #

A delivery company can complete 99 ordinary parcels profitably and lose more on one uninsured high-value parcel than it earned on all 99. Transaction count is not monetary outcome.

Application to no-hedge #

No-hedge evaluation must report net PnL and probability calibration on filled orders, not only win rate. High accuracy can arise from repeatedly buying expensive favorites; low accuracy can be healthy when discounted longshots are correctly valued.