AQR Capital is evaluating whether a short-term market signal should be trusted before increasing position size. The signal is noisy, so the risk team wants a Bayesian estimate of the probability that a true market downturn is underway after the signal fires.
Use Bayes' Theorem to update the probability of a market downturn given that the trading alert triggered. Then compare the expected loss of acting versus not acting.
| Quantity | Value | Notes |
|---|---|---|
| Prior probability of a true downturn on any trading day | 2.0% | Base rate of adverse market regime |
| Probability alert triggers when downturn is real | 85.0% | Sensitivity: |
| Probability alert triggers when no downturn is occurring | 8.0% | False positive rate: $P(A \mid |
| eg D)$ | ||
| Loss if firm does not hedge during a real downturn | $4.2M | One-day expected loss |
| Cost if firm does hedge and no downturn occurs | $0.35M | Opportunity cost / slippage |
| Residual loss if firm hedges during a real downturn | $0.9M | Hedge reduces but does not eliminate loss |
Let = true downturn and = alert triggered.