Tangent Portfolio Weights without Explicitly Specified Expected Returns

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 models with multiple state variables affecting securities' expected returns and factor loadings