**Author: Robert Sloma, Sigma Advantage**

**Covestor model: Aggressive Stock**

In my last post I introduced the concept of the capital asset pricing model (CAPM) and, as part of CAPM, the security market line (SML). The SML can be utilized to determine whether a security is under, over or fairly valued based on whether the security falls above, on or under the SML on a graph plotting expected return versus beta (β).

In this post, I will introduce the concepts of the efficient frontier and the capital allocation line (also sometimes referred to as the capital market line). These concepts were developed by Harry Markowitz in the early 1950’s when he was a graduate student at the University of Chicago. As part of his thesis, he was looking for a way to build portfolios that could maximize an investor’s return for a given risk level. His work resulted in the efficient frontier, for which he as well as several others received the Nobel Memorial Prize in Economics for their contributions to the field of financial economics.

As shown in Figure 1, the efficient frontier is made up of portfolios of the entire universe of marketable securities. It goes from 100% stocks to 100% bonds and defines the boundary of all possible portfolios, where the return is maximized for a given level of risk. Risk in this case is defined as the annualized standard deviation of monthly returns over a given time period. The efficient frontier also has a minimum risk portfolio, which for this time period was made up of 80% bonds and 20% stocks. You will also notice that the efficient frontier shows that it is possible to build portfolios that outperform the S&P 500 on a risk and return basis.

Sometime later after Markowitz published his thesis outlining the efficient frontier, several others asked the question, “What happens if you consider the rate you can earn with basically no risk?”. Although there is no investment without risk, due to other possible risks like reduced purchasing power due to inflation risk, reinvestment risk, and other risks, the closest investment to risk-free are U.S. Treasury Bills. Using 30-day Treasury bills and reinvesting the proceeds over time, the effective risk-free rate over the period shown above in Figure 2 was around 2.5% per year.

Figure 3

Illustrated in Figure 3, if we draw a line from the risk-free rate to where it is just touching, or is tangent to, the efficient frontier, the point on the efficient frontier is called the optimal risky portfolio. So what does this mean? It means that for every unit of risk beyond this point on the efficient frontier, we are getting less and less return. This is known as the law of diminishing returns. For every point below, we can obtain a better return for the same risk level by investing a portion of the portfolio in the risk-free asset and the rest in the optimal risky portfolio.

Now let’s say that the optimal risky portfolio is still too much risk for me. What can I do to reduce my risk level and still obtain the maximum return. Figure 4 shows an example of how to do this.

Figure 4

We can reduce the risk of the portfolio by investing 20% of the portfolio in the risk-free asset and the other 80% in the optimal risky portfolio and maximize the return possible for that risk level. By utilizing the capital market line and the efficient frontier, we can maximize the return for any given risk level according to this theory.

Now my portfolios utilize only long positions in stocks, mutual funds and ETFs, which means I do not utilize hedging or short positions to further manage risk and return. In my opinion, the “typical” efficient frontier includes this as an option. However, the same concept and mathematics behind it can be used to design portfolios that maximize returns for a given range of risk, with the additional constraint that only long positions (positive allocation weights) are used.

In my next post, I will present the concept of risk-adjusted return, which builds on the efficient frontier and the capital allocation/market line. Finally, I will post an analysis of the six portfolios I am currently managing to see how their risk-adjusted return compares over time with the S&P 500, from the time period January 2006 through July 2012.