How to implement modern portfolio theory?

The entire fund management industry is based upon the precept that superior returns can be obtained by investing in a group of assets (i.e. a portfolio, or fund) rather than individual assets.

By superior it is held that a greater return can be obtained for an equivalent level of risk, or the same return can be gained for a lower risk. There are few who dispute this basic idea in theory but the full practical implementation of Modern Portfolio Theory is far from straightforward.

 

This overview aims to illustrate the concepts behind the theory using numerical examples. The areas of theory covered are diversified portfolios and the capital asset pricing model (CAPM).

 

Diversified Portfolio Concept: The basic idea behind the Modern Theory Portfolio is that the ‘set’ or ‘universe’ of securities representing the investment choice faced by an individual behaves differently to each other. If, for example, the price of oil rises significantly then oil firms (for whom stocks will immediately be worth considerably more and whose margins may rise) will tend to offer better returns than firms for whom oil is a significant cost factor.

 

When the price of oil falls oil firms may well offer inferior returns when compared to others. The idea behind Modern Portfolio Theory is that, one should spread one’s capital across different investments in order that the portfolio is not completely exposed to an adverse piece of news affecting one security (or group of securities).

 

Let’s consider four possible scenarios in the next investment period, with each scenario having the same probability. Let’s take the same oil firms as an example.

Scenario	Effect on Stock Price
Oil price falls by 25%	-10.50%
Oil price falls by 10%	-0.64%
Oil price rises by 10%	5.67%
Oil price rises by 25%	13.50%
Expected Return	2.00%
Standard Deviation	8.78%

 

The expected return is simply the weighted average of the returns for each scenario (notice that +2.00% is not one of the possible outcomes). Since we do not, a priori, know what the outcome will be, the unexpected return is the best estimate that we can make of the return that we will receive from the oil stock. The actual returns, given that there are only four scenarios, vary enormously (ranging from a negative return of 10.5% to a positive return of 13.5%).

 

This variation is generally described as risk, the metric for which is nothing more complex than standard deviation. In this case the standard deviation is 8.78% which is actually quite high given a mean of 2%. It is this uncertainty that according to this theory, we should avoid. Intuitively, any rational investor should minimize the risk of losing his hard earned capital, a ‘given’ being that he is taking a risk in the first place in order to benefit from an expected return.

Efficient Frontier Analysis: The series of values on the curved line below is known as the efficient frontier, the idea being that there is neither a higher return available for a given level of risk, nor a lower risk for the same return.

 

Point A plots the risk against the return for the final portfolio. Point B is the portfolio before the addition of another stock. In this theory this is not an efficient portfolio since the same return is available for a lower risk. At point C, we have a different situation; here the expected return is higher, but at a higher risk. In this type of scenario it is left to the investor to decide whether the additional return is warranted by the additional risk of loss of capital.

Given a very large number of potential investments, how many stocks should a fund manager hold in his portfolio?

 

Experts analyzed the average portfolio variance for portfolios consisting of different number of stocks listed on the New York Stock Exchange. This shows a diminishing marginal diversification effect through the addition of new securities to the portfolio.

 

Fund managers tend to have a minimum of 15 stocks in their portfolio, but will hold more securities if the overall value of the fund is higher. Certainly by the time 15 stocks been included, a very great proportion of specific risk has been diversified away as a result of differing behavior of securities given different market conditions.

 

CAPM and the Market Price Risk:

The theory that investors are not rewarded (i.e. there is no additional expected returns available) for holding any diversifiable risk, is taken to its logical limit in the Capital Asset Pricing Model (CAPM). This model is based on the premise that all investors will hold portfolios which are invested in every single asset in existence.

 

The rationale behind this is that if an investible asset is not included, then an opportunity for diversification, and therefore risk reduction, has been missed. This theoretical ‘market portfolio’ is weighted according to the relative overall market values of each asset.

 

Deriving the CAPM: The final building block of CAPM is to introduce the concept of investor risk preference. According to the theory, investors will combine the market portfolio with a risk free asset (eg a short term government debt instrument). The proportion of the risk free asset held will increase the greater the investor’s risk aversion.

 

The CAPM, which is concerned with pricing market risk, ie determining what additional expected return is required for additional market risk. The only risk considered by a rational investor is market risk, we need to measure each security’s risk in these terms.

 

The key elements here are as follows :

 

- The higher the weighting a security has, the greater will be its influence on the market return.
- The risk if measured in terms of market risk, the greater must be the compensating expected return.
- The higher the risk free rate, the higher will be the required expected return.

 

Having discussed these, its important to emphasize that despite their many simplifying assumptions, Modern Portfolio Theory still justifiably provides the cornerstones of portfolio management. In particular, the adoption of the principle of portfolio diversification has enabled fund managers to offer investors returns that are consistently higher than the risk free rate.

With regard to CAPM, it does appear that the number of different security holdings tend to increase as the fund size increases, i.e. fund managers are diversifying as far as they possibly can. This supports the assertion that CAPM could be viewed as being implemented as far as possible given real life constraints.

Tweet