Back to topic: The Math and Models behind the Strategies### Efficient Frontier

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The efficient frontier is a risk versus return plot that illustrates which portfolio is optimal to hold for your given level of risk. This was inspired by academic work from Harry Markowitz in the 1950s.

A portfolio is just a weighted combination of a set of stocks. By definition, each stock in your portfolio has an expected return and expected risk level associated with it. These expected return and expected risks are key to deciding which stocks you want to hold in your portfolio. The risk and return can be estimated via complex mathematical models and this is what the alpha-generating hedge fund industry focuses on, where alpha refers to future expected returns.

*For this article, what can we do if we do not have access to these sophisticated estimation methods for risk and return ? *We can use a very simple heuristic which is that we expect future risk and return to equal the past risk and return - so let's go with that model assumption for this article.

The concept of domination relates to the comparison of two portfolios where one portfolio is clearly a better choice than the other. This better choice, assuming A dominates B, will either be based on

- Portfolio A has a greater return than Portfolio B, with both Portfolio A having the same or less risk than portfolio B
- Portfolio A has the same return as Portfolio B but it has lower risk

Mathematically we can write this as a mean-variance condition

*Essentially, any portfolio that does not sit on the efficient frontier is dominated by a portfolio that does sit on the efficient frontier.*

There are essentially an infinite number of ways to combine a set of N stocks into a portfolio P if we are allowed to use any weight combinations for our N stocks, no matter how small each weight is. If we take a simple two stock portfolio of stock A and stock B, then the two limits in this case are 100% weight in stock A and 0% in stock B or 100% in stock B and 0% in stock A - but in between these limits we can also invest x% in stock A and y% in stock B.

It is easy to show a risk-return graph of all the possible portfolios of stock A and B, if we have an assumption for the correlation between stock A and B. To make things simple, we can again suggest that the future correlation between stock A and B is equal to the past correlation. For a specific example, let us take Amazon and Walmart as our two stocks and calculate the average annual return, risk and correlation.

The average annual return, risk and correlation between Amazon and Walmart, all figures in percentages, are given by:

We can quickly look at the potential portfolios assuming we shift our portfolio in steps from 100% invested in Walmart to 100% invested in Amazon.

Investigating further to test the impact of the correlation parameter, we can vary it from a very unrealistic -70% all the way to 70%. Each point on each line is a potential portfolio for an investor to invest in, under the fixed correlation assumption for that line plot. This hopefully illustrates why a low to negative correlation is so desired from a diversification perspective and allows the investor to obtain superior risk-adjusted returns.

** Clearly the left-most blue line is our preferred portfolio (even though it is based on a non-existent -70% correlation). **This extreme example was shown to really illustrate the power of negative correlation, or even small positive correlation, and why it is so important at both the stock or portfolio level but also at the strategy level. If an investor can combine one strategy (or portfolio) with another that has low correlation or even negative correlation then the final combined portfolio will offer much better risk-adjusted returns than either of the investments independently.

Let us see a toy example and assume we have a small set of 6 US stocks, covering the technology sector, food and clothing. We can calculate the average annual return and the volatility of each of these stocks. Let us now assume that out of these 6 stocks we want a portfolio of 5 stocks only, then the question is

*What weight should each stock in our portfolio have to ensure we are maximising our return for a given risk level i.e we are on the efficient frontier ?*

We show in the table below the returns and risk for each stock where our goal is to obtain the optimal weights i.e the set of weights that lie on the efficient frontier. These sets of weights will be the vector of weights in the last column *Weight*.

One simple way to do this is to create a large set of random portfolio weights via a Monte Carlo simulation and then calculate the portfolio risk and return. If we plot all of those simulations, which in our case is 10,000 we get a plot like below. For each risk point, we have highlighted the best portfolio return and it is clear to see, a la Markowitz, that there is a frontier where portfolios on that frontier *dominate *and are the ideal portfolios.

Additionally, we have also highlighted with a red start the optimal sharpe ratio portfolio, which is the best risk-adjusted return portfolio a user could obtain theoretically.

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