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. $$u=\left[\begin{array}{cccc}1&1&\dots&1\end{array}\right]$$.

Theorem

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

Theorem

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 $$wu^T=1$$.

Theorem

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

Example

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 $$1$$.