6 – L3 08 The Efficient Frontier V3

Okay. So far you’ve learned about the importance of diversification and how to calculate portfolio mean and variance. You may wonder, what are the best ways to assign to each stock in your portfolio. Before we dive into portfolio optimization, let’s take a look at the set of all possible portfolios. This will lead us to an important concept, the efficient frontier. Before we start, let’s quickly recap. The expected return of a portfolio, equals the weighted sum of each stock’s expected returns. The variance of a portfolio, equals the sum of the pairwise covariances weighted by the products of the weights. Okay. I think we’re ready to go. Let’s first take a look at a simple example with five stocks. Stocks A, B, C, D and E with given annual mean returns and covariances. One day your manager gives you $10,000 and tells you to invest in these five stocks whichever way you want. But taking only long not short positions. What would you do with this money? Of course, you would want to impress your manager and create a portfolio with the highest returns and the lowest risk. Is this possible? How would you do it? You’ve given the task some thought and decided to just create a few portfolios with randomly assigned weights. For the first portfolio, you give each stock an equal weight of 20% of the total capital. The portfolio return equals 4% and the standard deviation equals 3%. Is this good enough? You are not sure. So, you just keep testing different combinations by assigning a little more weight to one stock and a little less to another and then rechecking the portfolio’s return and risk. The more simulations you do, the more interesting the problem gets. For example, both scenario one and two have the same risk, but scenario one has higher expected return. In this case, you will definitely prefer scenario one. Also, scenario four has the highest expected return of 0.07 but it also has the highest risk of 0.04. Scenario four is less attractive because it is a riskier portfolio. This is interesting; isn’t it? Let’s try to run thousands more scenarios and plot the expected portfolio return against portfolio volatility. Okay. Here is a plot representing 25,000 simulated portfolios. Each dot represents a possible risk return combination that can be generated by the basket of stocks. The x axis is the volatility and the y-axis is the return. What have you noticed? Let’s take a look at a quick example. Let’s take a look at these two portfolios represented by the red and yellow dots. Notice that both portfolios have the same risk of 0.035 but the red portfolio has significantly higher expected return of 0.06. Furthemore, the red dot outperforms all the portfolios with risk of 0.035 but there’s more. Have you noticed that all the dots that lie on the upper boundary outperform the dots below it? This is the efficient frontier. The efficient frontier is the upper boundary of the set of possible portfolios. Portfolios on this boundary have the maximum achievable return for a given level of risk. Any portfolios above the frontier are unachievable. It’s not possible to find a combination of stocks that produces a higher level of return at that same level of risk. On the flip side, portfolios below the frontier are certainly achievable but one might wonder why a rational investor would prefer a lower rate of return when she could achieve a higher return while taking on the same level of risk. The portfolio that achieves the lowest level of risk is called the minimum variance portfolio, and the portfolios on the efficient frontier are known as market portfolios.

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