12.4 Minimum Variance Portfolio

• A minimum variance (risk) portfolio

\[\begin{equation} Minimise\,\sigma_{p}^{2}=w^{\prime}\Sigma w \tag{12.14} \end{equation}\]

subject to

\[\begin{equation} w^{\prime}\mu=\mu_{p} \tag{12.15} \end{equation}\]

and

\[\begin{equation} w^{\prime}1=1 \tag{12.16} \end{equation}\]
  • The above constraints apply to long only portfolios. Summation of weights can be different for long and short combination

  • Portfolio can be formed for a target return and minimum risk or just for minimum risk or with maximum return with minimum risk (mean variance)

  • With different stocks and asset types, individual weight limits can be imposed.

12.4.1 Efficient Weights for Two Assets

  • For a portfolio with Two Risk Assets, the mean variance portfolio weights are calculated as
\[\begin{equation} w_{1}=\frac{\sigma_{2}^{2}-Cov(r_{1},r_{2})}{\sigma_{1}^{2}+\sigma_{2}^{2}-2Cov(r_{1},r_{2})}\\\\ w_{2}=1-w_{1} \tag{12.17} \end{equation}\] \[\begin{equation} w_{2}=1-w_{1} \tag{12.18} \end{equation}\]

The above is derived after taking the first derivative of the portfolio variance w.r.t the weight of asset 1,\(w_{1}\). Set that derivative equal to zero and solve for \(w_{1}\).

\[\begin{equation} \frac{\partial\sigma_{p}^{2}}{\partial w_{1}}=\frac{\partial}{\partial w_{1}}\left[w_{1}^{2}\sigma_{1}^{2}+(1-w_{1})^{2}\sigma_{2}^{2}+2w_{1}(1-w_{1})\sigma_{12}\right] \tag{12.19} \end{equation}\]

12.4.2 Portfolio with N Risky Assets

  • Optimisation using Quadratic Programming-Brief overview

  • To recap: Given a target return \(\mu_{p}\), the efficient portfolio minimises

\[\begin{equation} Minimise\,\sigma_{p}^{2}=w^{\prime}\Sigma w \tag{12.20} \end{equation}\]

subject to

\[w^{\prime}\mu=\mu_{p}\]

and

\[w^{\prime}1=1\]

  • Quadratic Programming is used to minimise a quadratic objective function subject to linear constraints.We focus on the implementation on the technique for optimisation not the mathematical details. Students are expected to learn how to use the method in R

  • In applications to portfolio optimization, the objective function is the variance of the portfolio return.

  • The objective function is a function of N variables, such as the weights of N assets, that are denoted by an N × 1 vector x. Suppose that the quadratic objective function to be minimized is

\[\begin{equation} \frac{1}{2}x^{\prime}Dx-d^{\,\prime}x \tag{12.21} \end{equation}\]

where D is an \(N\mathtt{X}N\) matrix and d is an \(N\mathtt{x}1\) vector. The factor of 1/2 is used make the optimisation consistent with R

  • Two types of linear constraints on x ; inequality and equality

    • Inequality constraint

\[A_{neq}^{\prime}x\geq b_{neq}\]

  • Equality

\[A_{eq}^{\prime}x=b_{eq}\]

  • To apply quadratic programming

    • \(x=w\)
    • \(D=2\Sigma\)
    • d is an \(N\mathtt{x}1\) vector of zeros so that (12.21) is \(w^{\prime}\Sigma w\), the return variance of the portfolio.

– R package quadprog has a function called solve.QP to achieve this.

– R package tseries has portfolio.optim function using the solve.QP to make it easier

References

Markowitz, Harry. 1952. “Portfolio Selection.” The Journal of Finance 7 (1): 77–91.
Markowitz, Harry M. 1991. “Foundations of Portfolio Theory.” The Journal of Finance 46 (2): 469–77.