Risky Portfolios and the CAPM

A Portfolio of Two Risky Assets

Suppose we have two risky assets \(A\) and \(B.\) The return of a portfolio \(P\) in which we invest \(1 - w\) in \(A\) and \(w\) in \(B\) is \[ r_{P} = (1 - w) r_{A} + w r_{B}. \tag{1}\] The expected return of the portfolio is \[ \mu_{P} = (1 - w) \mu_{A} + w \mu_{B}, \tag{2}\] whereas its variance can be computed as \[ \begin{aligned} \sigma_{P}^{2} & = \var((1 - w) r_{A} + w r_{B}) \\ & = (1 - w)^{2} \sigma_{A}^{2} + w^{2} \sigma_{B}^{2} + 2 w (1 - w) \sigma_{A, B}. \end{aligned} \tag{3}\]

Figure 1: The figure shows …

The Global Minimum Variance Portfolio

There is a portfolio that has the minimum variance among all portfolios between \(A\) and \(B.\) \[ \begin{aligned} \frac{d}{dw} \sigma_{P}^{2} & = \frac{d}{dw} (1 - w)^{2} \sigma_{A}^{2} + w^{2} \sigma_{B}^{2} + 2 w (1 - w) \sigma_{A, B} \\ & = - 2 (1 - w) \sigma_{A}^{2} + 2 w \sigma_{B}^{2} + 2 (1 - 2 w) \sigma_{A, B} = 0. \end{aligned} \tag{4}\]

\[ - (1 - w) \sigma_{A}^{2} + w \sigma_{B}^{2} + (1 - 2 w) \sigma_{A, B} = 0. \]

\[ w_{MV} = \frac{\sigma_{A}^{2} - \sigma_{A, B}}{\sigma_{A}^{2} + \sigma_{B}^{2} - 2 \sigma_{A, B}}. \]