We have observed that the objective function of multi-asset portfolio analysis is represented by the following indifference equation.

This provides investors and companies with a standard, against which they can compare their preferred risk-return profile for any efficient portfolio.

However, its interpretation, like other portfolio equations throughout the Chapter assumes that the efficiency frontier has been correctly defined. Unfortunately, this in itself is no easy task.

Based upon the pioneering work of Markowitz (op. cit.) we explained how a rational and risk-averse investor, or company, in an efficient capital market (characterised by a normal distribution of returns) who require an optimal portfolio of investments can maximise utility, having regard to the relationship between the expected returns and their dispersion (risk) associated with the covariance of returns within a portfolio.

Any combination of investments produces a trade-off between the two statistical parameters; expected return and standard deviation (risk) associated with the covariability of individual returns. And according to Markowitz, this statistical analysis can be simplified.

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.

The Markowitz portfolio selection model is theoretically sound. Unfortunately, even if we substitute the correlation coefficient into the covariance term of the portfolio variance, without the aid of computer software, the mathematical complexity of the variance-covariance matrix calculations associated with a multi-asset portfolio limits its applicability.

The constraints of Equation (25) are linear functions of the n variables x, whilst the objective function is an equation of the second degree in these variables. Consequently, methods of quadratic programming, rather than a simple linear programming calculation, must be employed by investors to minimise VAR(P) for various values of R (P) = K.

Once portfolio analysis extends beyond the two-asset case, the data requirements become increasingly formidable. If the covariance is used as a measure of the variability of returns, not only do we require estimates for the expected return and the variance for each asset in the portfolio but also estimates for the correlation matrix between the returns on all assets.

For example, if management invest equally in three projects, A, B and C, each deviation from the portfolio's expected return is given by:

If the deviations are now squared to calculate the variance, the proportion 1/3 becomes (1/3)2, so that:

VAR(P) = VAR[1/3 (A)+1/3 (B)+1/3 (C)]

= (1/3)2 (the sum of three variance terms, plus the sum of six covariances).

For a twenty asset portfolio:

VAR(P) = (1/20)2 (sum of twenty variance, plus the sum of 380 covariances). As a general rule, if there are E x. = n projects, we find that:

(26) VAR (P) = (1/n)2 (sum of n variance terms, plus the sum of n (n-1) covariances.

In the covariance matrix (xi ... xn), xi is paired in turn with each of the other projects (x2 ... xn) making (n-1) pairs in total. Similarly, (n -1) pairs can be formed involving x2 with each other xi and so forth, through to xn making n (n-1) permutations in total.

Of course, half of these pairs will be duplicates. The set x, x2 is identical with x2, x1. The n asset case therefore requires only 1/2 (n2 -n) distinct covariance figures altogether, which represents a substantial data saving in relation to Equation (26). Nevertheless, the decision-maker's task is still daunting, as the number of investments for inclusion in a portfolio increases.

Not surprising, therefore, that without today's computer technology, a search began throughout the late 1950s and early 1960s for simpler mathematical and statistical measures of Markowitz portfolio risk and optimum asset selection, as the rest of our text will reveal.

3.7. Selected References

1. Markowitz, H.M., "Portfolio Selection", The Journal of Finance, Vol. 13, No. 1, March 1952.