In efficient capital markets where the returns from two investments are normally distributed (symmetrical) we have explained how rational (risk averse) investors and companies who require an optimal portfolio can maximise their utility preferences by diversification. Any combination of investments produce a trade-off between the statistical parameters that define a normal distribution; the expected return and standard deviation (risk) associated with the covariance of individual returns.

Efficient diversified portfolios are those which maximise return for a given level of risk, or minimise risk for a given level of return for different correlation coefficients.

However, most investors, or companies and financial managers (whether they control capital projects or financial services (such as insurance premiums, pension funds or investment trusts) may be responsible for numerous investments. It is important, therefore, that we extend our analysis to portfolios with more than two constituents.

Theoretically, this is not a problem. According to Markowitz (op cit.) if individual returns, standard deviations and the covariance for each pair of returns are known, the portfolio return R(P), portfolio variance VAR(P) and a probabilistic estimate of portfolio risk measured by the standard deviation s(P), can be calculated.

For a multi-asset portfolio where the number of assets equals n and xi represents the proportion of funds invested in each, such that:

We can define the portfolio return and variance as follows

The covariance term, COVij determines the degree to which variations in the return to one investment, i, can serve to offset the variability of another, j. The standard deviation is then derived in the usual manner.

Assuming we now wish to minimise portfolio risk for any given portfolio return; our financial objective is equally straightforward:

(23) MIN: a(P), Given R(P) = K (constant)

This mathematical function combines Equation (22) which is to be minimised, with a constraint obtained by setting Equation (20) for the portfolio return equal to a constant (K):

Figure 3.2 illustrates all the different risk-return combinations that are available from a hypothetical multi investment scenario.

Figure 3.2: The Portfolio Efficiency Frontier: The Multi-Asset Case

The first point to note is that when an investment comprises a large number of assets instead of two, the possible portfolios now lie within on area, rather than along a line or curve. The area is constructed by plotting (infinitely) many lines or curves similar to those in Figure 3.1.

However, like a two-asset portfolio, rational, risk-averse investors or companies are not interested in all these possibilities, but only those that lie along the upper boundary between F and F1. The portfolios that lie along this frontier are efficient because each produces the highest expected return for its given level of risk. To the right and below, alternative portfolios yield inferior results. To the left, no possibilities exist. Thus, an optimum portfolio for any investor can still be determined at an appropriate point on the efficiency frontier providing the individual's attitude toward risk is known.

So how is this calibrated?

The Optimum Portfolio

We have already observed that the calculation of statistical means and standard deviations is separate from their behavioural interpretation, which can create anomalies. For example, a particular problem we encountered within the context of investment appraisal was the "risk-return paradox" where one project offers a lower return for less risk, whilst the other offers a higher return for greater risk. Here, investor rationality (maximum return) and risk aversion (minimum variability) may be insufficient behavioural criteria for project selection. Similarly, with portfolio analysis:

If two different portfolios lie on the efficiency frontier, it is impossible to choose between them without information on investor risk attitudes.

One solution is for the investor or company to consider a value for the portfolio's expected return R(P), say R(pi) depicted schematically in Figure 3.3.

Figure 3.3: The Multi-Asset Efficiency Frontier and Investor Choice

All R(pi), a(pi) combinations for different portfolio mixes are then represented by points along the horizontal line R(pi) - R(pi)1 for which R(P) = R(pi). The leftmost point on this line, F then yields the portfolio investment mix that satisfies Equation (23) for our objective function:

By repeating the exercise for all other possible values of R(P) and obtaining every efficient value of R(pi) we can then trace the entire opportunity locus, F-Fl. The investor or company then subjectively select the investment combination yielding a maximum return, subject to the constraint imposed by the degree of risk they are willing to accept, say P* corresponding to R(P*) and s(P*) in the diagram.

Review Activity

As an optimisation procedure, the preceding model is theoretically sound. However, without today's computer technology and programming expertise, its practical application was a lengthy, repetitive process based on trial and error, when first developed in the 1950s. What investors and companies needed was a portfolio selection technique that actually incorporated their risk preferences into their analyses. Fortunately, there was a lifeline.

As we explained in the Summary and Conclusions of Chapter Two's Exercise text, (PTFAE) rational risk-averse investors, or companies, with a two-asset portfolio will always be willing to accept higher risk for a larger return, but only up to a point. Their precise cut-off rate is defined by an indifference curve that calibrates their risk attitude, based on the concept of expected utility.

We can apply this analysis to a multi-asset portfolio of investments. However, before we develop the mathematics, perhaps you might care to look back at Chapter Two (PTFAE)and the simple two-asset scenario before we continue.

In Chapter Two (PTFAE) we discovered that if an investor's or company's objective is to minimise the standard deviation of expected returns this can be determined by reference to a their utility indifference curve, which calibrates attitudes toward risk and return. Applied to portfolio analysis, the mathematical equation for any curve of indifference between portfolio risk and portfolio return for a rational investor can be written:

Graphically, the value of l indicates the steepness of the curve and a indicates the horizontal intercept. Thus, the objective of the Markowitz portfolio model is to minimise a. If we rewrite Equation (24), for any indifference curve that relates to a portfolio containing n assets, this objective function is given by:

For all possible values of l > 0, where R(P) = K(constant), subject to the non-negativity constraints:

And the essential requirement that sources of funds equals uses and x. be proportions expressed mathematically as:

Any portfolio that satisfies Equation (25) is efficient because no other asset combination will have a lower degree of risk for the requisite expected return.

An optimum portfolio for an individual investor is plotted in Figure 3.4. The efficiency frontier F - F1 of risky portfolios still reveals that, to the right and below, alternative investments yield inferior results. To the left, no possibilities exist. However, we no longer determine an optimum portfolio for the investor by trial and error.

Figure 3.4: the Determination of an Optimum Portfolio: The Multi-Asset Case

The optimum portfolio is at the point where one of the curves for their equation of indifference (risk-return profile) is tangential to the frontier of efficient portfolios (point E on the curve F-F1). This portfolio is optimal because it provides the best combination of risk and return to suit their preferences.

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