The purpose with this note is to rescue a simple procedure presented by Black (1972), Merton (1973) and later by Levy and Samat (1982), Elton and Gruber (1995) and Benninga (1997). They just propose that the optimal portfolio can be found maximizing the slope of the line that joins the point of riskfree return and the efficient frontier. When this maximum tangent is reached, that line is the capital market line (CML) (it is tangent to the efficient frontier).
This is a simple procedure that does not require one to calculate the efficient frontier and is an easy task with Excel Solver. It is just one point of the efficient frontier.
In this article are provided simple methods of constructing known results. At the core of the methods is the identification of a simple concise basis that spans the Capital Market Line (CML). It is shown that a portfolio whose risky assets weights are the product of the inverse variance-covariance matrix of (nonredundant) security rates of return times the vector of the excess expected rates of return over the riskfree rate is a CML portfolio. This portfolio and the riskfree security span the CML. In addition, with this basis, there is immediate construction of the efficient frontier of risky assets (the “hyperbola”), “tangency” portfolios, “reflection” portfolios, and a CAPM relationship. Used method is quick and simple. It is easy to derive, teach, implement, interpret, and remember.
In a well developed financial market with liquid short term fixed income trading, the volatility of short term fixed income securities forms a continuous spectrum that converges to zero, the volatility of riskless asset. This means that the attainable combinations of risky assets contain the whole region under the capital market line and the capital market line is the efficient frontier for the risky assets.
According to the CAPM theory, a portfolio that lies in the efficient frontier and combined with certain proportion of risk free investment, and given a desired risk level, maximizes the return of the combined portfolio. This definition is valid even if the desired risk level is less than the minimum defined by the efficient frontier.
That optimal risky portfolio is just the point of tangency between the Capital Market Line and the efficient frontier. As this optimal portfolio has to lie along the efficient frontier, then the point of tangency is located at the line with the maximum tangent between that line and the horizontal line. This solution is very good because it is not easy to determine the indifference curves for each decision maker.
According to the CAPM theory, the investor will prefer a position in the “market portfolio” either levered or unlevered. Then, the optimal portfolio is given by this optimization problem.