**EFFICIENT FRONTIER**

Higher return for higher risk but not just any risk!

Learning objectives:

Introduce the Markowitz Theory and the Capital Asset Pricing Model.

Graph 1

Here we will introduce the concept of Efficient Frontier that has been developed by Harry Markowitz a recipient of the Nobel Memorial Prize in Economic Sciences. His work on the efficient frontier is the corner stone of the modern portfolio theory that is still in use in the investment world. We will also introduce the Capital Asset Pricing Model (CAPM) that was built on the Harry Markowitz work. Note that those concepts are not easy to understand and we will not go into the details of the mathematical model but we will try to present the main ideas as well as the main assumptions and limitations.

Instead of having two assets only like in the section on Diversification and Correlation let’s consider a set of several assets in which you can invest. Those can be shown on a graph based on their expected risk and return characteristics. Basically the efficient frontier tells you that with the available assets you can build a set of mean-variant efficient portfolios that for a given level of risk will have the highest return or for a given return will have the lowest risk.

Graph 2

The curve represents all the possible assets combinations that are mean variant efficient. The upper part of the graph shown in blue is called the Efficient Frontier. When the Efficient Frontier is determined we can select a unique portfolio that is optimal with respect to an investor risk and return profile.

The Markowitz theory implies several assumptions:

- All investors aim to maximize the expected return of total wealth.

- All investors are rational and risk-adverse, that is to say is they will only accept greater risk if they are compensated with a higher expected return.

- All investors have the same expected single period investment horizon.

- All investors have access to the same information at the same time.

- All Investors base their investment decisions on the expected return and risk (i.e. the standard deviation of an assets historical returns).

- All investors are price takers.

- All markets are perfectly efficient (e.g. no taxes and no transaction costs).

- Asset returns are normally distributed variables

To implement this model we need to know the expected return and expected return variance of every investable asset as well as every asset pair covariance. This information is of course not available and here is where we need to make some estimates. Historical information can be used as a good starting point to perform those estimates.

Note also that very often, applications that implement this model allow the user to constrain the weights of assets that composed the portfolio. For instance the application user can decide that one asset shall not weight more than 15% or less than 5%.

Graph 3

The Markowitz model can be extended to the Capital Asset Pricing Model (CAPM) by adding the following assumptions:

- The investor is able to borrow cash at the market rate. In other words he is able to borrow money from the market in order to invest in risky assets.

- The investor is able to lend cash at the market rate in order to un-leverage his position in risky assets.

- The lending and the borrowing market rates are the same.

In this model, cash is by definition risk free (variance equal to zero) and has a return equal to Rf. Therefore cash asset is located on the Y axis of the Risk-Return graph. Also by nature a risk free asset is not correlated with risky assets, therefore any combination of a risky assets and the risk free asset will be graphed by a straight line.

Basically, by introducing the risk free asset, the CAPM model shows that the optimal portfolios are located on the straight line known as Capital Market Line (CML). The optimal portfolios are not located on an Efficient Frontier curve defined by Markowitz anymore but rather on a straight line. The straight line is defined by the risk free asset point and the tangency point with the Efficient Frontier. The tangency point is also called the Market Portfolio (PM). The Market Portfolio is the optimal portfolio at its level of risk σm and its level of return Rm. If an investor wants to get a portfolio with a higher expected return, instead of selection an optimal portfolio located on the Efficient Frontier curve, the CPAM model tells you to borrow money with zero risk in order to increase your position in the Market Portfolio. The Market Portfolio is the best way in terms of risk to get a higher expected return.

Graph 4

The graph illustrates the scenario where the investor borrows some cash to increase his position in the Market Portfolio and build the portfolio PH. CAPM tells you that this is the best way to get a higher expected return and at the same time get the lowest risk. Note that at the same level of risk than PH, the optimal portfolio as defined by the Markowitz theory lies on the Efficient Frontier which is below the straight line.

Graph 5

The graph illustrates the scenario where the investor does not require the level of expected return provided by the Market Portfolios and thus does not need to take the corresponding risk. The CAPM tells you that the best way to build your portfolio is to sell part of your holdings in the Market Portfolio and lend the money at the market rate.

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