Business Context

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.

Problem Statement

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.

Given Data

Let $D$ = true downturn and $A$ = alert triggered.

Requirements

1. Compute the posterior probability $P(D \mid A)$ using Bayes' Theorem.
2. Compute the probability that the alert is a false alarm, $P( eg D \mid A)$.
3. Calculate the expected loss if the firm hedges after an alert.
4. Calculate the expected loss if the firm ignores the alert.
5. Recommend the lower-loss action after an alert.
6. Briefly explain why Bayes' Theorem is useful in trading and risk problems.

Assumptions

• The prior downturn rate is stable over the period analyzed.
• Sensitivity and false positive rate were estimated from historical backtests and are applicable going forward.
• The decision is made only after an alert has triggered.
• Expected losses are one-day conditional averages.