We introduce a generalization of the classical capital asset pricing model in which market uncertainty, market sentiment, and forms of idiosyncratic volatility and idiosyncratic skewness are priced in equilibrium. We derive two versions of the model, one based on a representative agent who cares about three criteria (risk, robustness, and expected returns), and the other with a microfoundation based on three types of investors (speculators, hedgers, and arbitrageurs). We apply the resulting capital asset pricing model with idiosyncratic risk (IR-CAPM) to provide a new theoretical account of the beta anomaly, one of the most fundamental and widely studied empirical limitations of the CAPM. We show that the IR-CAPM explains the main conditional relationships involving the beta anomaly in the literature including the time variation of the beta anomaly across optimistic and pessimistic periods and across high and low uncertainty periods, the relationship between the beta anomaly and the correlation between a stock\u27s beta and its idiosyncratic volatility, and the concentration of the beta anomaly among stocks with high idiosyncratic maximum returns
