Markowitz Algorithm

Markowitz Algorithm

Modern portfolio theory (also called Markowitz model or mean-variance model) was introduced in a 1952 essay by Harry Markowitz, the Nobel Prize winner in Economics in 1990. It assumes that an investor wants to maximize a portfolio’s expected return contingent on any given amount of risk. For portfolios that meet this criterion, known as efficient portfolios, achieving a higher expected return requires taking on more risk, so investors are faced with a trade-off between risk and expected return. The investment decision is not merely which security to own, but how weight to own.

Math Forms

Let \(\mu \)= Expected returns on portfolio, \(r_i \)= Returns on asset i, \(\omega_i \) =Weight of component asset i, and \(\sigma_i \)=Standard deviation of asset i.

To simplify the above expression, introducing \(\Sigma\)to represent the covariance matrix. In this way, the optimization problem can be concluded to the following format with matrix form:

\(min \frac{1}{2} \mathbf{\omega}^\mathrm{T} \Sigma \mathbf{\omega} , s.t. \mathbf{\omega}^\mathrm{T} \mathbf{r}= \mu, \mathbf{\omega}^\mathrm{T} \mathbf{e} = 1 \)

In order to solve the above quadratic linear programming problem, the Lagrange Multiplier Method can be used. And finally, we gain the relationship between the investment ratio of various securities and the expected rate of return as follows:

\begin{array}{**rcl**}
      w &=&g + h \mu\\
      g &=& \frac{1}{D} (B \Sigma^\mathrm{-1} e – A \Sigma^\mathrm{-1} r)\\
      h &=& \frac{1}{D} (C \Sigma^\mathrm{-1} r – A  \Sigma^\mathrm{-1} e)
\end{array}

A, B, C, D are all constants, and the values are all affected by the returns and variances of each asset in the portfolio:

\(A=\mathbf{e}^\mathrm{T} \Sigma^\mathrm{-1} \mathbf{r} = \mathbf{r}^\mathrm{T} \Sigma^\mathrm{-1} \mathbf{e} , B = \mathbf{r}^\mathrm{T} \Sigma^\mathrm{-1} \mathbf{r}, C = \mathbf{e}^\mathrm{T} \Sigma^\mathrm{-1} \mathbf{e}, D = BC-A^2\)

Under the given conditions for the type of securities, investors can get unlimited kinds of products through different combinations of risks and returns. The portfolio effective frontier curve reflects the investment at a given level of expected returns, the smallest variance a portfolio can reach. For rational investors, the frontier portfolio can reach its investment to maximize the utility, and the equation of the effective frontier curve of the portfolio is as follows:

\(\sigma^2 = \frac{C}{D} (\mu – \frac{A}{C})^2 +\frac{1}{C} \)

Configuration of parameters

For estimation of returns:

• Sample Mean
• Equal Mean
• Linear Shrinkage

For estimation of covariance:

• Sample Covariance
• Equal Correlation
• Linear Shrinkage
• Nonlinear Shrinkage

Optional Constraints

• Position Limits

References

[1] Markowitz, H.M. (March 1952). “Portfolio Selection”. The Journal of Finance. 7(1): 77–91.

[2] Markowitz, H.M. (1959). Portfolio Selection: Efficient Diversification of Investments. New York: John Wiley & Sons. (reprinted by Yale University Press, 1970

[3] Abhijit. Ravipati Rutgers The State University of New Jersey – New Brunswick. Graduate School – New Brunswick. 2012, Masters Abstracts International 50-05.

[4] 张贺清. 均值和方差变动的马科维茨投资组合模型研究[D].哈尔滨工业大学,2015.

[5] Mandelbrot B.Paretian  Distributions and Income Maximization.  Quarterly Journal of Economics, 1962, (76): 57-85.

[6] Cont R. Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues. Quantitative Finance, 2001, 1(2): 223-236.