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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 [[Efficient-market hypothesis|priced efficiently]], but who believes he has information that can be used to predict the abnormal performance ([[Alpha (investment)|Alpha]]) of a few of them; the model finds the optimum portfolio to hold under such conditions.
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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 ''R<sub>F</sub>'' and the expected market return is ''R<sub>M</sub>'' with standard deviation <math>\sigma_M</math>. There are ''N'' securities that have been analyzed and are thought to be mispriced, with expected returns given by:
 
:<math>r_i = R_F +\beta_i(R_M-R_F)+\alpha_i+\epsilon_i</math>
 
where the random terms <math>\epsilon_i</math> are normally distributed with mean 0, standard deviation <math>\sigma_i</math>, and are mutually uncorrelated. (This is the so-called Diagonal Model of Stock Returns, or [[Single-index model]] due to [[William Forsyth Sharpe|Sharpe]]).
 
Then<ref>Kane et al.</ref> it was shown by Treynor and Black that the active portfolio ''A'' is constructed using the weights
 
:<math>w_i = \frac{\alpha_i/\sigma_i^2}{\sum_{j=1}^N \alpha_j/\sigma_j^2}</math>
 
(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 ''w<sub>A</sub>'' invested in the active portfolio and the remainder (i.e. <math>w_M=1-w_A</math>) invested in the market portfolio. This fraction is found as follows:
 
:<math>w_A = \frac{w_0}{1+(1-\beta_A)w_0}</math>
:<math>w_0 = \frac{\alpha_A/\sigma_A^2}{(R_M-R_F)/\sigma_M^2}</math>
 
where the alpha, beta and residual risk of ''A'' are found using the previously computed weights ''w<sub>i</sub>'' in the obvious way
 
:<math>\alpha_A=\sum w_i \alpha_i</math>
:<math>\beta_A=\sum w_i \beta_i</math>
:<math>\sigma_A^2=\sum w_i^2 \sigma_i^2</math>
 
== References ==
 
{{reflist}}
 
* 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, [http://weber.ucsd.edu/~mbacci/white/pub_files/hwcv-112.pdf]
 
{{DEFAULTSORT:Treynor-Black model}}
[[Category:Mathematical finance]]

Latest revision as of 07:44, 25 November 2014

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