The Markowitz Portfolio Theory: Mean Variance Optimization

The Markowitz Portfolio Theory

Theory and Applications

Shafin Shabir Naik

AAA1325

Contents Introduction Portfolio Expected Value and Variance Diversification Mean Variance Optimization Efficient Frontier Efficient Frontier in Excel Bibliography

Introduction

People invest with the aim of earning returns on their investments. But these returns are uncertain which creates an element of risk for the investors. Nevertheless, investor is also interested in the total return and rate of return that he gets from his investment. The formulas for calculating them for a single asset are given below Total Return, R = amount received amount invested

We can now prove the claims that diversification really leads to lower risk. However we have assumed expected rates of returns to be fixed. In general, as our variance decreases due to diversification so does our expected rates of return. Therefore, blind diversification may decrease our returns. This is where Markowitz Portfolio Theory comes handy. Harry Markowitz solved this problem in a systematic way and helped investors to make rational decisions regarding how much to invest.

Mean Variance Optimization

Harry Markowitz was the first economist who formalized the trade-off between higher returns and lower risk. He proposed the following approach: for a given level of expected returns, find the portfolio allocation with smallest risk. His theory is summarized in the diagram given below. Mathematically, the optimization problem can be solved as follows.

Assume that there are n assets. The expected rates of return are µ1,µ2,…,µn and the covariances are σ_(i,j) for i,j = 1,2,3,…,n. A portfolio is defined by a set of n weights wi = 1,…,n that sum to 1. To find minimum-variance, we fix the expected value at µ. Therefore the problem will

It allows him to take informed decision about his investment. This theory also explains the trade-off between maximizing returns and minimizing the associated risk with the return. According to Bartvold and Begg, “The basic premise of portfolio theory is that the variance of returns for a portfolio of risky assets is a function not only of the variance of each individual asset, but also of the covariance [or correlation] between each asset (variance of return, or standard deviation of return, can be considered a measure of economic risk). When multiple risky assets are held within a portfolio, it can be expected that some properties will increase in value while at the same time others will decrease in value. By holding risky assets in groups, some of the risk of each asset may be reduced or eliminated through the process of