Treynor–Black model

In Finance the Treynor–Black model is a mathematical model for security selection published by Fischer Black and Jack Treynor in 1973. The model assumes an investor who considers that most securities are priced efficiently, but who believes they have information that can be used to predict the abnormal performance (Alpha) of a few of them; the model finds the optimum portfolio to hold under such conditions.

In essence the optimal portfolio consists of two parts: a passively invested index fund containing all securities in proportion to their market value and an 'active portfolio' containing the securities for which the investor has made a prediction about alpha. In the active portfolio the weight of each stock is proportional to the alpha value divided by the variance of the residual risk.

Assume that the risk free rate is R_F and the expected market return is R_M with standard deviation ${\displaystyle \sigma _{M}}$. There are N securities that have been analyzed and are thought to be mispriced, with expected returns given by:

where the random terms ${\displaystyle \epsilon _{i}}$ are normally distributed with mean 0, standard deviation ${\displaystyle \sigma _{i}}$, and are mutually uncorrelated. (This is the so-called Diagonal Model of Stock Returns, or Single-index model due to William F. Sharpe).

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