From our preceding discussion, rational investors in reasonably efficient markets can assess the likely profitability of individual corporate investments by a statistical weighting of their expected returns, based on a normal distribution (the familiar bell-shaped curve).

- Rational-risk averse investors expect either a maximum return for a given level of risk, or a given return for minimum risk.

- Risk is measured by the standard deviation of returns and the overall expected return is measured by its weighted probabilistic average.

Using mean-variance efficiency criteria, investors then have three options when managing a portfolio of investments depending on the performance of its individual components.

(i) To trade (buy or sell),

(ii) To hold (do nothing),

(iii) To substitute (for example, shares for loan stock).

However, it is important to note that what any individual chooses to do with their portfolio constituents cannot be resolved by statistical analyses alone. Ultimately, their behaviour depends on how they interpret an investment's risk-return trade off, which is measured by their utility curve. This calibrates the individual's current perception of risk concerning uncertain future gains and losses. Theoretically, these curves are simple to calibrate, but less so in practice. Risk attitudes not only differ from one investor to another and may be unique but can also vary markedly over time. For the moment, suffice it to say that there is no universally correct decision to trade, hold, or substitute one constituent relative to another within a financial investment portfolio.

Review Activity

1. Having read the fourth chapters of the following series from bookboon.com recommended in Activity 3:

- In SFM: pay particular attention to Section 4.5 onwards, which explains the relationship between mean-variance analyses, the concept of investor utility and the application of certainty equivalent analysis to investment appraisal.

- In SFME: work through Exercise 4.1.

2. Next download the free companion text to this e-book: Portfolio Theory and Financial Analyses; Exercises (PTFAE), 2010.

3. Finally, read Chapter One of PTFAE.

It will test your understanding so far. The exercises and solutions are presented logically as a guide to further study and are easy to follow. Throughout the remainder of the book, each chapter's exercises and equations also follow the same structure of this text. So throughout, you should be able to complement and reinforce your theoretical knowledge of modern portfolio theory (MPT) at your own pace.

1.5. Summary and Conclusions

Based on our Review Activity, there are two interrelated questions that we have not yet answered concerning any wealth maximising investor's risk-return trade off, irrespective of their behavioural attitude towards risk.

What if investors don't want "to put all their eggs in one basket" and wish to diversify beyond a single asset portfolio?

How do financial management, acting on their behalf, incorporate the relative risk-return trade-off between a prospective project and the firm's existing asset portfolio into a quantitative model that still maximises wealth?

To answer these questions, throughout the remainder of this text and its exercise book, we shall analyse the evolution of Modern Portfolio Theory (MPT).

Statistical calculations for the expected risk-return profile of a two-asset investment portfolio will be explained. Based upon the mean-variance efficiency criteria of Harry Markowitz (1952) we shall begin with:

- The risk-reducing effects of a diverse two-asset portfolio,

- The optimum two-asset portfolio that minimises risk, with individual returns that are perfectly (negatively) correlated.

We shall then extend our analysis to multi-asset portfolio optimisation, where John Tobin (1958) developed the capital market line (CML) to show how the introduction of risk-free investments define a "frontier" of efficient portfolios, which further reduces risk. We discover, however, that as the size of a portfolio's constituents increase, the mathematical calculation of the variance is soon dominated by covariance terms, which makes its computation unwieldy.

Fortunately, the problem is not insoluble. Ingenious, subsequent developments, such as the specific capital asset pricing model (CAPM) formulated by Sharpe (1963) Lintner (1965) and Mossin (1966), the option-pricing model of Black and Scholes (1973) and general arbitrage pricing theory (APT) developed by Ross (1976), all circumvent the statistical problems encountered by Markowitz.

By dividing total risk between diversifiable (unsystematic) risk and undiversifiable (systematic or market) risk, what is now termed Modern Portfolio Theory (MPT) explains how rational, risk averse investors and companies can price securities, or projects, as a basis for profitable portfolio trading and investment decisions. For example, a profitable trade is accomplished by buying (selling) an undervalued (overvalued) security relative to an appropriate stock market index of systematic risk (say the FT-SE All Share).This is measured by the beta factor of the individual security relative to the market portfolio. As we shall also discover it is possible for companies to define project betas for project appraisal that measure the systematic risk of specific projects.

So, there is much ground to cover. Meanwhile, you should find the diagram in the Appendix provides a useful road-map for your future studies.

1.6. Selected References

1. Jensen, M.C. and Meckling, W.H., "Theory of the Firm: Managerial Behaviour, Agency Costs and Ownership Structure", Journal of Financial Economics, 3, October 1976.

2. Fisher, I., The Theory of Interest, Macmillan (London), 1930.

3. Fama, E.F., "The Behaviour of Stock Market Prices", Journal of Business, Vol. 38, 1965.

5. Tobin, J., "Liquidity Preferences as Behaviour Towards Risk", Review of Economic Studies, February 1958.

6. Sharpe, W., "A Simplified Model for Portfolio Analysis", Management Science, Vol. 9, No. 2, January 1963.

7. Lintner, J., "The valuation of risk assets and the selection of risk investments in stock portfolios and capital budgets", Review of Economic Statistics, Vol. 47, No. 1, December, 1965.

8. Mossin, J., "Equilibrium in a capital asset market", Econometrica, Vol. 34, 1966.