Treynor–Black model

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:

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

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 alpha, beta and residual risk of the constructed active portfolio are found using the previously computed weights w_i:

${\displaystyle \alpha _{A}=\sum w_{i}\alpha _{i}}$ 

${\displaystyle \beta _{A}=\sum w_{i}\beta _{i}}$ 

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

The overall risky portfolio for the investor consists of a fraction w_A invested in the active portfolio and the remainder invested in the market portfolio. This active fraction is found as follows:

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

And corrected for the beta exposure of the active portfolio:

${\displaystyle w_{A}={\frac {w_{0}}{1+(1-\beta _{A})w_{0}}}}$ 

${\displaystyle w_{M}=1-w_{A}}$ 

The model is not bounded 0 ≤ w_A ≤ 1 and 0 ≤ w_M ≤ 1 i.e short positions in the market portfolio or active portfolio could be initiated to leverage a position in the other portfolio. This is often regarded as the major flaw of the model, as it often yields an unrealistic weight in the active portfolio. Imposing lower and upper bounds for w_A is a measure to counter this.

• 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]