# Modern Portfolio Theory

## Arbitrage Pricing Theory and Beyond

### Introduction

Previous chapters have presented a series of mathematical models representing a body of work termed Modern Portfolio Theory (MPT) available to financial management when making strategic investment decisions. MPT was originally developed for use by investors in securities, primarily fund managers and professional analysts with the time, resources and expertise to implement the models and interpret their findings. Today, anybody with access to a computer, the appropriate software and a reasonable financial education can model quite complex tasks. Ultimately, however, it is people who should interpret the results and not the computer. One lesson to be learnt from the1987 stock market crash is the catastrophic effect of automated trading. Another from the 2007 meltdown and ongoing financial crises is that computer driven models can be so complex that hardly anybody understands what is going on anymore.

Like all financial theories, MPT should therefore be a guide to human action and not a substitute. And while the benefits of IT cannot be overstressed, you should always understand the financial model that underpins the computer program you are running. So, let us review the original purpose of MPT, notably the CAPM and then outline its subsequent development, notably Arbitrage Pricing Theory (APT).

### Portfolio Theory and the CAPM

You will recall that portfolio theory was initially developed by Harry Markowitz in the early 1950s to explain how rational investors in perfect markets can minimise the risk of investment without comprising return by diversifying and building up an efficient portfolio of investments. The risk of each portfolio is measured by the variability of possible returns about the mean measured by the standard deviation. Investor risk-return attitudes can be expressed by indifference curves.

In 1958, John Tobin explained how the introduction of risk-free investments into Markowitz' theory further reduces the risk of a portfolio. According to Tobin, the Capital Market Line (CML) defines a new "efficient frontier" of investments for all investors.

Applied to project appraisal, Markowitz theory reveals that an individual project's risk is not as important as its effect on the portfolio's overall risk. So, whenever management evaluate a risky project they must correlate the individual project risk with that for the existing portfolio it will join to assess its suitability.

Without the benefit of today's computer technology, the mathematical complexity of the Markowitz model arising from its covariance calculations prompted other theorists to develop alternative approaches to efficient portfolio diversification. In the early 1960s by common consensus, the CAPM emerged as a means whereby investors in financial securities were able to reduce their total risk by constructing portfolios that discriminate between systematic (market risk) and unsystematic (specific) risk.

The CAPM (usually associated with its prime advocate William Sharpe) states that the return on a security or portfolio depends on whether their prices follow prices in the market as a whole by reference to a suitable index, such as the FT-SE 100. The closer the correlation between the price of either an individual security or a portfolio and this market proxy (measured by the beta factor) the greater will be their expected returns. Thus, if an investor knows the beta factor (relative risk) of a security or portfolio, their returns can be predicted with accuracy. Profitable trading of portfolios is then accomplished by buying (selling) undervalued (overvalued) securities relative to their systematic or market risk.

The CAPM also states that rational investors would choose to hold a portfolio that comprises the stock market as a whole. By definition, the market portfolio has a beta of one and is the most "efficient" in the sense that no other combination of securities would provide a higher return for the same risk. You will recall that it is a benchmark by which the CAPM establishes the Security Market Line (SML) in order to compare other beta factors and returns. From this linear relationship, rational investors can ascertain whether individual shares are underpriced or overpriced and determine other efficient portfolios that balance their personal preference for risk and return.

According to the CAPM:

Any security with the same risk as the market will have a beta of 1.0; half as risky it will have a beta of 0.5; twice as risky it will have a beta of two.

The required rate of return given by the CAPM formula is composed of the return on risk-free investments, plus a risk premium measured by the difference between the market return and the risk free rate multiplied by an appropriate beta factor. For example, using Equation (45) for an investment with a beta of b:

If we use the CAPM for project appraisal, rather than stock market analysis, the procedure remains the same. Essentially, we are substituting an investment project for a security into a company's portfolio of investments, rather than a market portfolio. Risk relates to the cost of capital and management's objective is to obtain a discount rate to appraise individual projects.