MTH 4500: Introductory Financial Mathematics

Portfolio Theory

Minimum variance portfolio

Assume that we are given \(n\) risky securities and that the expected returns of the securities are respectively \(\mu_1\), \(\mu_2\), \(\dots\), \(\mu_n\). We can write these expected returns in a form of a vector \( m=\left[\begin{array}{cccc}\mu_1& \mu_2&\dots &\mu_n\end{array}\right]\). Let \(C\) denote the covariance matrix between the returns. Assume that \(C\) is invertible. Let us denote by \(u\) the vector whose \(n\) components are equal to \(1\), i.e. \(u=\left[\begin{array}{cccc}1&1&\dots&1\end{array}\right]\).


Among all portfolios that consist of the given \(n\) securities, the minimal variance has the portfolio with the weights \[w=\frac{u C^{-1}}{uC^{-1}u^T}.\]


Among all portfolios that consist of the given \(n\) securities and that have the expected return \(\mu\), the minimal variance has the portfolio with the weights \[w=\left[\begin{array}{cc}\mu&1\end{array}\right]M^{-1}\left[\begin{array}{c}mC^{-1}\newline uC^{-1}\end{array}\right],\quad\mbox{where } M=\left[\begin{array}{cc} mC^{-1}m^T& mC^{-1}u^T\newline uC^{-1}m^T& uC^{-1}u^T\end{array}\right].\]

Notice that \(M\) and \(M^{-1}\) are \(2\times 2\) matrices and that \(mC^{-1}\) is a product of \(1\times n\) matrix \(m\) and \(n\times n\) matrix \(C^{-1}\). Thus \(mC^{-1}\) is \(1\times n\) matrix. Similarly, \(uC^{-1}\) is \(1\times n\) matrix. Therefore \(M^{-1}\left[\begin{array}{c}mC^{-1}\newline uC^{-1}\end{array}\right]\) is a \(2\times n\) matrix. Let us denote its entries by \(a_1\), \(a_2\), \(\dots\), \(a_n\), \(b_1\), \(b_2\), \(\dots\), \(b_n\), i.e. \[\left[\begin{array}{cccc}a_1&a_2&\dots&a_n\newline b_1&b_2&\dots&b_n\end{array}\right]=M^{-1}\left[\begin{array}{c}mC^{-1}\newline uC^{-1}\end{array}\right].\] Let us denote \(a=\left[\begin{array}{cccc}a_1&a_2&\dots&a_n\end{array}\right]\) and \(b=\left[\begin{array}{cccc}b_1&b_2&\dots&b_n\end{array}\right]\). Then the minimal variance portfolio satisfies \[w=\left[\begin{array}{cc} \mu & 1\end{array}\right]\left[\begin{array}{c}a\newline b\end{array}\right]=\mu a+b.\]

We have established a linear relationship between \(\mu\) and the portfolio of minimal variance whose expected return is \(\mu\). This linear relationship is called minimal variance line.

Since every line is uniquely determine by any two of its points, we immediately derive the following theorem:

Theorem (Two Fund Theorem)

If \(w_1\) and \(w_2\) are any two portfolios on the minimal variance line, then any other portfolio \(w\) on the minimal variance line can be expressed as \(w=\alpha w_1+(1-\alpha)w_2\).

Market portfolio

We now assume that in addition to the \(n\) risky securities, there is a risk-less security whose return is \(R > 0\).

Consider any risky portfolio \(P\) with expected return \(\mu\) and risk \(\sigma\) and construct a new portfolio that consists of this risky portfolio \(P\) and the risk-less security. If the weights of this new portfolio are \(\alpha\) and \(1-\alpha\) then the return is \(\alpha \mu+(1-\alpha)R\) and the risk is \(\alpha \sigma\).

Therefore, if we keep a portfolio with expected return \(\mu\) and risk \(\sigma\) fixed, we can vary \(\alpha\) and build a portfolio with expected return \(\mu^{\prime}=\alpha \mu+(1-\alpha)R\) and risk \(\sigma^{\prime}=\alpha \sigma\). The quantity \((\sigma^{\prime},\mu^{\prime})=\left(\alpha\sigma, \alpha\mu+(1-\alpha)R\right)\) represents the parametrization of a line (with parameter \(\alpha)\) in the \((\sigma,\mu)\) plane that connects the point \((0,R)\) with the point \((\sigma,\mu)\).

We want to determine the line with the highest slope, i.e. we want to determine \(w\) for which the slope of the line that connects \((\sigma,\mu)\) with \((0,R)\) is the maximal possible. In other words we want to maximize the function \(S(w)=\frac{\mu(w)-R}{\sigma(w)}=\frac{wm^T-R}{\sqrt{wCw^T}}\) under the condition \(wu^T=1\).


The function \(S(w)=\frac{wm^T-R}{\sqrt{wCw^T}}\) attains its maximum on \(wu^T=1\) for the following choice of \(w\): \[w=\frac{mC^{-1}-RuC^{-1}}{mC^{-1}u^T-RuC^{-1}u^T}.\]


Assume that \(\overrightarrow a\) and \(\overrightarrow b\) are the vectors such that for each \(\mu\) the minimal variance portfolio with expected return \(\mu\) is given by \(\overrightarrow{w}_{\mu}=\mu\overrightarrow a+\overrightarrow b\). Prove that the sum of all components of \(\overrightarrow a\) is equal to \(0\) and the sum of all components of the vector \(\overrightarrow b\) is equal to \(1\).