Treynor-Black model
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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 priced efficiently, but who believes he has 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.
. There are N securities that have been analyzed and are thought to be mispriced, with expected returns given by:
where the random terms
are normally distributed with mean 0, standard deviation 
, and are mutually uncorrelated. (This is the so-called Diagonal Model of Stock Returns, or Single-index model due to Sharpe
).
Then it was shown by Treynor and Black that the active portfolio A is constructed using the weights
(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.
) invested in the market portfolio. This fraction is found as follows:
where the alpha, beta and residual risk of A are found using the previously computed weights wi in the obvious way
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
the Treynor–Black model is a mathematical model for security selection published by Fischer Black
Fischer Black
Fischer Sheffey Black was an American economist, best known as one of the authors of the famous Black–Scholes equation.-Background:...
and Jack Treynor in 1973. The model assumes an investor who considers that most securities are priced efficiently, but who believes he has information that can be used to predict the abnormal performance (Alpha
Alpha (investment)
Alpha is a risk-adjusted measure of the so-called active return on an investment. It is the return in excess of the compensation for the risk borne, and thus commonly used to assess active managers' performances...
) 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
Index fund
An index fund or index tracker is a collective investment scheme that aims to replicate the movements of an index of a specific financial market, or a set of rules of ownership that are held constant, regardless of market conditions.-Tracking:Tracking can be achieved by trying to hold all of the...
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. There are N securities that have been analyzed and are thought to be mispriced, with expected returns given by:where the random terms
are normally distributed with mean 0, standard deviation , and are mutually uncorrelated. (This is the so-called Diagonal Model of Stock Returns, or Single-index model due to SharpeWilliam Forsyth Sharpe
William Forsyth Sharpe is the STANCO 25 Professor of Finance, Emeritus at Stanford University's Graduate School of Business and the winner of the 1990 Nobel Memorial Prize in Economic Sciences....
).
Then it was shown by Treynor and Black that the active portfolio A is constructed using the weights
(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.
) invested in the market portfolio. This fraction is found as follows:where the alpha, beta and residual risk of A are found using the previously computed weights wi in the obvious way