We have observed that mean-variance efficiency analyses, premised on investor rationality (maximum return) and risk aversion (minimum variability), are not always sufficient criteria for investment appraisal. Even if investments are considered in isolation, wealth maximising accept-reject decisions depend upon an individual's perception of the riskiness of its expected future returns, measured by their personal utility curve, which may be unique.

Your reading of the following material from the bookboon.com companion texts, recommended for Activity 3 and the Review Activity in the previous chapter, confirms this.

- SFM: Portfolio Theory and Analyses; Exercises (PTAE): Chapter One.

Any conflict between mean-variance efficiency and the concept of investor utility can only be resolved through the application of certainty equivalent analysis to investment appraisal. The ultimate test of statistical mean-variance analysis depends upon behavioural risk attitudes.

So far, so good, but there is now another complex question to answer in relation to the search for future wealth maximising investment opportunities:

Even if there is only one new investment on the horizon, including a choice that is either mutually exclusive, or if capital is rationed, (i.e. the acceptance of one precludes the acceptance of others).

How do individuals, or companies and financial institutions that make decisions on their behalf, incorporate the relative risk-return trade-off between a prospective investment and an existing asset portfolio into a quantitative model that still maximises wealth?

2.1. Mean-Variance Analyses: Markowitz Efficiency

Way back in 1952 without the aid of computer technology, H.M. Markowitz explained why rational investors who seek an efficient portfolio (one which minimises risk without impairing return, or maximises return for a given level of risk) by introducing new (or off-loading existing) investments, cannot rely on mean-variance criteria alone.

Even before behavioural attitudes are calibrated, Harry Markowitz identified a third statistical characteristic concerning the risk-return relationship between individual investments (or in management's case, capital projects) which justifies their inclusion within an existing asset portfolio to maximise wealth.

To understand Markowitz' train of thought; let us begin by illustrating his simple two asset case, namely the construction of an optimum portfolio that comprises two investments. Mathematically, we shall define their expected returns as Ri(A) and Ri(B) respectively, because their size depends upon which one of two future economic "states of the world" occur. These we shall define as S1 and S2 with an equal probability of occurrence. If S1 prevails, R1(A) > R1(B). Conversely, given S2, then R2(A) < R2(B). The numerical data is summarised as follows:

ReturnState S1 S2

Ri(A) 20% 10%

R(B) 10% 20%

Activity 1

The overall expected return R(A) for investment A (its mean value) is obviously 15 per cent (the weighted average of its expected returns, where the weights are the probability of each state of the world occurring. Its risk (range of possible outcomes) is between 10 to 20 per cent. The same values also apply to B.

Mean-variance analysis therefore informs us that because R(A) = R(B) and a (A) = a (B), we should all be indifferent to either investment. Depending on your behavioural attitude towards risk, one is perceived to be as good (or bad) as the other. So, either it doesn't matter which one you accept, or alternatively you would reject both.

- Perhaps you can confirm this from your reading for earlier Activities?

However, the question Markowitz posed is whether there is an alternative strategy to the exclusive selection of either investment or their wholesale rejection? And because their respective returns do not move in unison (when one is good, the other is bad, depending on the state of the world) his answer was yes.

By not "putting all your eggs in one basket", there is a third option that in our example produces an optimum portfolio i.e. one with the same overall return as its constituents but with zero risk.

If we diversify investment and combine A and B in a portfolio (P) with half our funds in each, then the overall portfolio return R(P) = 0.5R(A) + 0.5R(B) still equals the 15 per cent mean return for A and B, whichever state of the world materialises. Statistically, however, our new portfolio not only has the same return, R(P) = R(A) = R(B) but the risk of its constituents, a(A) = a(B), is also eliminated entirely. Portfolio risk; a(P) = 0. Perhaps you can confirm this?

Activity 2

As we shall discover, the previous example illustrates an ideal portfolio scenario, based upon your entire knowledge of investment appraisal under conditions of risk and uncertainty explained in the SFM texts referred to earlier. So, let us summarise their main points

- An uncertain investment is one with a plurality of cash flows whose probabilities are non-quantifiable.

- A risky investment is one with a plurality of cash flows to which we attach subjective probabilities.

- Expected returns are assumed to be characterised by a normal distribution (i.e. they are random variables).

- The probability density function of returns is defined by the mean-variance of their distribution.

- An efficient choice between individual investments maximises the discounted return of their anticipated cash flows and minimises the standard deviation of the return.

So, without recourse to further statistical analysis, (more of which later) but using your knowledge of investment appraisal:

Can you define the objective of portfolio theory and using our previous numerical example, briefly explain what Markowitz adds to our understanding of mean-variance analyses through the efficient diversification of investments?

For a given overall return, the objective of efficient portfolio diversification is to determine an overall standard deviation (level of risk) that is lower than any of its individual portfolio constituents.

According to Markowitz, three significant points arise from our simple illustration with one important conclusion that we shall develop throughout the text.

1) We can combine risky investments into a less risky, even risk-free, portfolio by "not putting all our eggs in one basket"; a policy that Markowitz termed efficient diversification, and subsequent theorists and analysts now term Markowitz efficiency (praise indeed).

2) A portfolio of investments may be preferred to all or some of its constituents, irrespective of investor risk attitudes. In our previous example, no rational investor would hold either investment exclusively, because diversification can maintain the same return for less risk.

3) Analysed in isolation, the risk-return profiles of individual investments are insufficient criteria by which to assess their true value. Returning to our example, A and B initially seem to be equally valued. Yet, an investor with a substantial holding in A would find that moving funds into B is an attractive proposition (and vice versa) because of the inverse relationship between the timing of their respective risk-return profiles, defined by likely states of the world. When one is good, the other is bad and vice versa.

According to Markowitz, risk may be minimised, if not eliminated entirely without compromising overall return through the diversification and selection of an optimum combination of investments, which defines an efficient asset portfolio.

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