Markowitz Portfolio Optimization

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Introduction
Markowitz (1952, 1956) pioneered the development of a quantitative method that takes the diversification benefits of portfolio allocation into account. Modern portfolio theory is the result of his work on portfolio optimization. Ideally, in a mean-variance optimization model, the complete investment opportunity set, i.e. all assets, should be considered simultaneously. However, in practice, most investors distinguish between different asset classes within their portfolio-allocation frameworks. In our analysis, we view the process of asset allocation as a four-step exercise like Bodie, Kane and Marcus (2005). It consists of choosing the asset classes under consideration, moving forward to establishing capital market expectations, followed by deriving the efficient frontier until finding the optimal asset mix. We take the perspective of an asset-only investor in search of the optimal portfolio. An asset-only investor does not take liabilities into account. The investment horizon is 5 - 10 years and the opportunity set consists of twelve asset classes. The investor pursues wealth maximization and no other particular investment goals are considered. We solve the asset-allocation problem using a mean-variance optimization based on excess returns. The goal is to maximize the Sharpe ratio (risk-adjusted return) of the portfolio, bounded by the restriction that the exposure to any risky asset class is greater than or equal to zero and that the sum of the weights adds up to one. The focus is on the relative allocation to risky assets in the optimal portfolio. In the mean-variance analysis, we use arithmetic excess returns. Geometric returns are not suitable in a mean-variance framework. The weighted average of geometric returns does not equal the geometric return of a simulated portfolio with the same composition. The observed difference can be explained by the diversification benefits of the portfolio allocation. We derive the arithmetic returns from the geometric returns and the volatility.

a) The CIO has sent some of the results you have done above to the IPC. After the members of the IPC perused the results, some of them asked the CIO to explain why the equal-weighted portfolio underperformed the mean-variance optimal portfolio for the periods studied. Explain to the CIO using only the whole period results.

First, let’s quickly look at some of the values of the fields that are used to draw the capital allocation line. As an example to my explanation let’s go through 2 possible capital allocation lines from the risk-free rate (rf = 3.5%).

The first possible CAL is drawn for naively diversified portfolio for the whole period with rf = 3.5%. The expected return for this portfolio is 0.006224053, and its standard deviation is 0.025002148, the reward-to-volatility ratio, which is the slope of the CAL is 0.132284095.

The second CAL is drawn for the Optimal portfolio for the whole period with rf = 3.5%. The expected return for this portfolio is 0.009508282, and its standard deviation is 0.00734826, the re- reward-to-volatility ratio is 0.897030832.

We can see from the numbers that the optimal portfolio does better than the naively diversified portfolio because the RTV is higher for the optimal portfolio. The reason for that is that we’ve identified the optimal portfolio of risky assets by finding the portfolio weights that result in steepest CAL. The CAL that is supported by the optimal portfolio is tangent to the efficient frontier.

The bottom line is that we have chosen the optimal portfolio that has the portfolio weights that lie on the capital allocation line that is tangent to the efficient frontier. Which means a portfolio of risky assets that provides the lowest risk for the expected return and thus this selected portfolio is bound to outperform the naively diversified.

b) The IPC has noticed that the optimal allocations of sub-period 1 and...
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