MPT: searching for the efficient frontier

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MPT: searching for the efficient frontier

Regularly monitoring how a portfolio is performing is how advisers and investors help ensure that the best long-term outcomes are achieved. The key principles that underpin how portfolio management is approached today are known as the “Modern Portfolio Theory” (MPT). This is an important contribution to investment analysis that was first developed by the famous academic Harry Markowitz seventy years ago.

Put simply, it aims to show that given all the different assets in an investment universe, there is an optimal portfolio for any investor depending on their level of risk and in fact for any investor this portfolio consists of some proportion of cash and the “market portfolio”.

Simplifying assumptions

Arriving at the best portfolio might seem a daunting task when a universe of thousands of potential investments need to be analysed but is made manageable by one main simplifying assumption. This is that all investments are “log-normally” distributed, meaning that their returns are normally distributed with the traditional bell shape curve. This assumption holds reasonably well for most stocks, bonds, currencies, indices, funds and ETFs. The reason that this theory has been so universally adopted is that it simplifies the analysis enormously and keeps the whole problem tractable in terms of running regular analysis. In the early days of computer power this was a critical requirement and is still a big advantage today.

The reason that the log normal assumption helps is that it removes the need to collect any more data on assets other than expected return, volatility and correlations. In addition, any combination of lognormally distributed assets are themselves approximately lognormally distributed, and so it is easy to measure thousands of candidate portfolios. The optimisation problem then can be formulated as the best portfolio that an investor should pick given their risk appetite. Since expected return is the only measure of the portfolio, an investor will seek to maximise return for their acceptable level of risk. The other way to formulate this is to minimise risk for a given target return.

Mean and volatility

The main data requirements are to calculate the mean (expected) return and volatility of any given asset and the correlations between any two assets. Estimating these parameters accurately has challenges. For volatility and correlation historical measures are used most commonly, leaving the hardest task to determine reliably the key input of expected asset return.

Creating a chart of all investment combinations gives a range of outcomes from which a curve can be drawn known as the “efficient frontier”. When the risk-free asset (high quality cash or very short dated bonds) is included the efficient frontier converts into a line connecting the risk-free asset to the market portfolio. This therefore arrives at the solution of choosing the desired level of risk between the risk-free asset and the market portfolio. This market portfolio is the one that maximises the risk adjusted measure known as the Sharpe ratio, a concept that was added to Modern Portfolio Theory a decade or so later.

All three properties of an asset will determine whether it will be included in the optimal market portfolio – high expected return and low volatility is clearly important, as is low correlation with other assets to help achieve risk reduction through diversification.

By way of example, suppose the market portfolio has a volatility of 15% and expected return of 7%, and that the risk-free rate in the market is 1%. For an investor whose risk limit is equivalent to 15%, their optimal portfolio is directly the market one, with the return of 7%. If the investor is lower risk and can only tolerate 9% volatility then the best choice will be 60% in the market portfolio and 40% in the risk-free asset (cash), giving an expected return of 4.6%.

Big leap forward

Modern Portfolio Theory was a ground-breaking concept that still has a lot of usage today. It is best thought of as a framework rather than a universal solution. It guides fund managers into broad asset allocation decisions, but its simplifying assumptions can have limitations. Quantitative strategies, hedge funds, smart beta and other disciplines all search for advantages outside what could be envisaged in the mean variance framework.

One major drawback of Modern Portfolio Theory is that it does not focus on avoiding losses, but on minimising volatility. By definition it is loss that hurts investors and so the alternative downside measure known as the Sortino ratio was created. The mathematics gets more complicated and so the Markowitz’s original formulation remains relevant and popular because of its simplicity.

Extensions for non-linear assets

Another problem that has been examined by many is how to extend the idea of Modern Portfolio Theory to non-linear (non-symmetric) assets where the normal distribution assumption breaks down. One important example of this is option and structured product payoffs. Access to these instruments will extend the universe and so it is interesting to see what this class of derived investments can contribute, and in what circumstances they should be included within an optimal portfolio.

The two extensions of downside risk control and non-linear assets naturally go together, since instruments such as structured products are designed for risk reduction and limiting losses. We will consider such extensions for a more sophisticated approach with different building blocks such as structured products in a follow-on article now published here.

Tags: Stress testing

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