Abstract
In this article, I propose an extension of the Treynor–Black model to a case where the investor is not fully invested in the stock market at the outset and there is no need to explicitly specify securities’ expected returns. I derive explicit tangent portfolio weights based on a factor model of securities’ expected returns. The computational burden of the model is linear in the number of securities in the portfolio and does not involve any matrix inversion. I present an empirical application using the market model of Sharpe, the three-factor model of Fama and French and the four-factor model of Carhart with up to 30 industry portfolios between 1963 and 2012 and up to 1000 US stocks starting in 1992 until 2012. The portfolios perform well out-of-sample relative to the entire US value-weighted stock market portfolio with dividends reinvested from the Center for Research in Security Prices. The proposed framework can be extended in a straightforward way to time-varying factor models with multiple state variables affecting securities’ expected returns and factor loadings.
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Notes
Note that this is not strictly required in general. D can be a non-diagonal matrix if securities’ return innovations are expected to have non-zero covariances. A good example of this is securities of firms within the same industry as they are likely to be exposed to industry-specific shocks that will induce non-zero covariances between their return innovations. The general formulae that follow will hold for non-diagonal D as well but will then require the inversion of a large matrix when the tangent portfolio contains a large number of securities.
To see this, set x=D−1/2α and y=D−1/2β. Then (α′D−1α)(β″D−1β)−(α'D −1β)2=(x′x)(y′y)−(x′y)2⩾0.
The same argument applied previously to D applies here similarly to Ω.
This results in approximately 250 daily returns used for the estimation of α, β and D for the single-factor model and α, B and Ω for the multiple-factor model. Using an alternative estimation window of as few as 100 days or as many as 500 days yields qualitatively similar results. The findings of this robustness check are available from the author upon request.
The daily portfolio and factor return data as well as daily risk-free interest rates are obtained from Ken French’s online Data Library.
The RAR is given by the difference between the total cumulative return of the portfolio and the total cumulative return of the benchmark, that is, RAR=Πt=1t=T(1+r p, t )−Πt=1t=T(1+r b, t ), where rp, t is the total return of the portfolio in period t,rb, t is total return of the benchmark in period t and Π is the product operator.
The α reported is the abnormal return of the portfolio relative to the benchmark, that is,
, where 
and 
are the average returns of the portfolio and the benchmark, respectively, and β
p
is the β of the portfolio with respect to the benchmark.The findings regarding the performance of this algorithm using the FF3 and C4 models of securities’ returns is available from the author upon request.
, where 
and 
are the average returns of the portfolio and the benchmark, respectively, and β
p
is the β of the portfolio with respect to the benchmark.References
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Acknowledgements
The author would like to thank Turan Bali, Ron Bird, Evan Gatev, Berowne Hlavaty, Anthony Neuberger, Stephen Satchell (the editor), two anonymous referees, as well as participants in the 2013 J.P. Morgan quantitative conference in Sydney and seminar participants at the University of Adelaide for their valuable suggestions and comments. Any remaining errors are the author’s own.
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1obtained his PhD from the Olin School of Business at Washington University in St. Louis. He has worked at Koç University in Istanbul, Turkey before moving to the University of Adelaide. His research interests are in the fields of financial econometrics and applied asset pricing theory.
Appendix
Appendix
In order to derive equations (5) and (6) in sub-section ‘Single-factor model of returns’, we use the binomial inverse theorem for V−1 and perform the multiplication with μ as follows:
Note that the last two terms in equation (A.1) above can be combined as follows:
Combining equation (A.2) with the second term in equation (A.1) leads to:
Finally, combining equation (A.3) with the first term in equation (A.1) produces the stated result in equation (5) in sub-section ‘Single-factor model of returns’. To obtain equation (6), we substitute α=0 into equation (5) that leads to the following:
Note that the quantity in the large parentheses above is a scalar. Hence, we have that:
which, because of the diagonal nature of D−1, leads to equation (6) as stated in sub-section ‘Single-factor model of returns’.
In a similar spirit, we can obtain equations (14) and (15) in sub-section ‘Multiple-factor return models’ using the binomial inverse theorem for V−1 and multiplying through with μ:
The two terms in eqaution (A.7) can be collected as follows:
which leads to:
after collecting some terms. The final step involves collecting the second term in equation (A.6) with equation (A.5) and collecting terms:
producing the stated result in equation (14). The special case of α=0 in (15) obtains by substituting α=0 in equation (A.9) leading to:
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Glabadanidis, P. Tangent portfolio weights without explicitly specified expected returns. J Asset Manag 15, 177–190 (2014). https://doi.org/10.1057/jam.2014.22
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DOI: https://doi.org/10.1057/jam.2014.22