Mean Variance Optimization

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Mean-Variance Analysis
Mean-variance portfolio theory is based on the idea that the value of investment opportunities can be meaningfully measured in terms of mean return and variance of return. Markowitz called this approach to portfolio formation mean-variance analysis. Mean-variance analysis is based on the following assumptions: 1. All investors are risk averse; they prefer less risk to more for the same level of expected return. 2. Expected returns for all assets are known.

3. The variances and covariances of all asset returns are known. 4. Investors need only know the expected returns, variances, and covariances of returns to determine optimal portfolios. They can ignore skewness, kurtosis, and other attributes. 5. There are no transaction costs or taxes.

The Mean-Variance Approach
The mean-variance theory postulated that in determining a strategic asset allocation an investor should choose from among the efficient portfolios consistent with that investor’s risk tolerance amongst other constraints and objectives. Efficient portfolios make efficient use of risk by offering the maximum expected return for specific level of variance or standard deviation of return. Therefore, the asset returns are considered to be normally distributed. Efficient portfolios plot graphically on the efficient frontier, which is part of the minimum-variance frontier (MVF). Each portfolio on the minimum-variance frontier represents the portfolio with the smallest variance of return for given level of expected return. The graph of a minimum-variance frontier has a turning point that represents the Global Minimum Variance (GMV) portfolio that has the smallest variance of all the minimum-variance portfolios. Economists often say that portfolios located below the GMV portfolio are dominated by others that have the same variances but higher expected returns. Because these dominated portfolios use risk inefficiently, they are inefficient portfolios. The portion of the minimum-variance frontier beginning with and continuing above the GMV portfolio is the efficient frontier. Portfolios lying on the efficient frontier offer the maximum expected return for their level of variance of return. Efficient portfolios use risk efficiently: Investors making portfolio choices in terms of mean return and variance of return can restrict their selections to portfolios lying on the efficient frontier. This reduction in the number of portfolios to be considered simplifies the selection process. If an investor can quantify his risk tolerance in terms of variance or standard deviation of return, the efficient portfolio for that level of variance or standard deviation will represent the optimal mean-variance choice. Because standard deviation is easier to interpret than variance, investors often plot the expected return against the standard deviation rather than variance. The standard deviation is often plotted on the x-axis and the expected return on the y-axis. The trade-off between risk and return for a portfolio depends not only on the expected asset returns and variances, but also on the correlation of asset returns. The mean-variance theory can be extended to included nominally risk-free asset, where the theory points to choosing the asset allocation represented by the Tangency Portfolio given the investors can borrow or lend money at the risk free rate. The portfolio with the highest Sharpe Ratio amongst the efficient portfolio is called the tangency portfolio. The investor can then borrow money to increase the amount of leverage in tangency portfolio to achieve a higher expected return than the tangency portfolio, or split the money between the risk free asset and tangency portfolio to achieve a lower risk level than the tangency portfolio. The investor’s portfolio will fall on the so called Capital Allocation Line (CAL) that describes the combination of expected return and standard deviation available to an investor from combining risk free...
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