# 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.

## The Model

Assume that the risk free rate is RF and the expected market return is RM 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:

${\displaystyle r_{i}=R_{F}+\beta _{i}(R_{M}-R_{F})+\alpha _{i}+\epsilon _{i}}$

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).

Then[1] it was shown by Treynor and Black that the active portfolio A is constructed using the weights

${\displaystyle w_{i}={\frac {\alpha _{i}/\sigma _{i}^{2}}{\sum _{j=1}^{N}\alpha _{j}/\sigma _{j}^{2}}}}$

(Note that if an alpha is negative the corresponding portfolio weight will also be negative, i.e. the active portfolio is in general a long-short portfolio).

The overall risky portfolio for the investor consists of a fraction wA invested in the active portfolio and the remainder (i.e. ${\displaystyle w_{M}=1-w_{A}}$) invested in the market portfolio. This fraction is found as follows:

${\displaystyle w_{A}={\frac {w_{0}}{1+(1-\beta _{A})w_{0}}}}$
${\displaystyle w_{0}={\frac {\alpha _{A}/\sigma _{A}^{2}}{(R_{M}-R_{F})/\sigma _{M}^{2}}}}$

where the alpha, beta and residual risk of A are found using the previously computed weights wi in the obvious way

${\displaystyle \alpha _{A}=\sum w_{i}\alpha _{i}}$
${\displaystyle \beta _{A}=\sum w_{i}\beta _{i}}$
${\displaystyle \sigma _{A}^{2}=\sum w_{i}^{2}\sigma _{i}^{2}}$

## References

1. ^ Kane et al.
• Treynor, J. L. and F. Black, 1973, How to Use Security Analysis to Improve Portfolio Selection, Journal of Business, January, pages 66–88. JSTOR 2351280
• Kane, Kim and White: Active Portfolio Management - The power of the Treynor–Black model, December 2003, [1]