The exponential utility function has two convenient properties. Because of the delta property it is not necessary to make explicit the initial state of wealth. It is sufficient to maximize the expected utility of any new investment. Furthermore, if the returns of the new investment are normally distributed with mean u and variance op then it is easy to show that

Qp = »p- ( 2) 2 -

In other words, maximizing the expected utility (or equivalently, maximizing the certainty equivalent) of an investment is the same as satisfying a mean-variance criterion with a constant trade-off ratio between mean and variance equal to X/2. That is, with an exponential utility the capital allocation problem becomes the simple optimization:

maxwr - (V2)wVw such that >, = 1. (7.12)

If an exponential utility function is not appropriate, perhaps because an investor has constant relative risk aversion,17 there is no such simple link between the expected utility of a portfolio and its mean-variance characteristics.

Mean-variance analysis requires that an investors preferences are defined purely in terms of the risk and return of the portfolio. It is unfortunate that so few utility functions allow the link to be made from preferences in a risky environment to mean-variance analysis. Some portfolio models will just assume from the outset that preferences are represented by indifference curves in risk-return space and, unfortunately, throw away the idea of expected utility of a portfolio defined in terms of end-of-period wealth. Indifference curves are isoquants of a utility function < ( , rj) that is defined over risk rj and return \x. That is, they are curves in risk-return space that join all points having the same utility, like contour lines on a map. A risk-averse investor has diminishing utility with respect to risk, and diminishing marginal utility of extra returns:

It is unfortunate that so few utility functions allow the link to be made from preferences in a risky environment to mean-variance analysis

dU/d\i > 0, dU/ < 0; d2U/d\i2 < 0, d2U/da2 < 0.

In this case the indifference curve will be convex downwards, as illustrated in Figure 7.5a. The investor is indifferent between certain high-risk, high-return and certain low-risk, low-return portfolios, that is, he is willing to accept more risk only if it is accompanied by more return, because he is risk-averse. Since the efficient frontier is a concave curve, and the indifference curves of a risk-averse investor are convex downwards, there should be a unique optimal portfolio. Three curves are drawn on the figure representing increasing levels of

17 The exponential utility function has constant absolute risk aversion (the coefficient of absolute risk aversion is X) so the exponential utility is an appropriate choice for an investor who holds the same dollar amounts in risky assets as wealth increases. An investor who does not change the percentage invested in risky assets as his wealth increases might be assumed to have the logarithmic utility function, because this has decreasing absolute risk aversion and constant relative risk aversion. Alternatively, if investment in risky assets decreases with wealth the quadratic utility function U{W) = W - aW1 would be appropriate, since this has increasing absolute and relative risk aversion.

Figure 7.5 (a) Risk-averse and (b) risk-loving indifference curves.

utility. The maximum achievable utility is on the indifference curve IC3 that is tangential to the efficient frontier, and the optimum portfolio X is at the point of tangency. Any other portfolio in the opportunity set will have a lower utilit} level.

Without the assumption of risk aversion, indifference curves will be straight lines (risk neutrality) or concave downwards (risk-loving). Neither of these assumptions can necessarily guarantee a unique optimal portfolio. In fact risk-loving indifference curves do not determine finite solutions, at least when short sales are allowed. The risk-loving investor would seek to go infinitely short the risk-free asset and long the risky asset to achieve the highest possible risk and return (Figure 7.5b).

7.2.5 Efficient Portfolios in Practice

It is not advisable to use mean-variance analysis as a black box technique. In particular, the recommended weights in optimal portfolios will require careful

scrutiny of high-risk, high-return investments. High-risk, high-return assets can have a huge effect on the shape of the efficient frontier and therefore the construction of optimal portfolios, particularly for an investor who is not very risk-averse and can go short on other assets. To see why, consider Figure 7.6 which shows an opportunity set for portfolios of up to 35 assets, whose risk and return are marked as points within the set. The efficient frontier, assuming short sales are allowed, is marked in bold and an optimal portfolio is indicated for a risk-averse investor with the indifference curves shown. Because of the high risk and return of asset B, and the high correlation between assets A and B, the efficient frontier is virtually a straight line through these two points and the optimal portfolio will just be a combination of these two assets. The less risk-averse the investor, the greater proportion will be invested in the high-risk, high-return asset B.

High-risk, high-return assets can have a huge effect on the shape of the efficient frontier and therefore the construction of optimal portfolios, particularly for an investor who is not very risk-averse and can go short on other assets

In the practical application of mean-variance analysis to stock selection and asset allocations it is quite common that optimal portfolio will be dominated by just a few assets that appear to have high-return, high-risk characteristics.18 For example, a risk-neutral investor in Nikkei 225 stocks would have chosen the portfolio shown in Table 7.2 that is dominated by the high-risk, high-return power stocks.

If measured over the short term the estimates of parameters that determine the shape of the efficient frontier (the mean, variance and correlation between the asset returns) can be extremely unstable and the efficient frontier will be changing all the time. For example, a risk-neutral investor who employs an efficient frontier based on 30-day averages of FTSE 100 stocks would have made the allocations shown in Tables 7.3a and 7.3b. The optimal portfolio on 6 April is quite well diversified, with no more than about 10% or 11% in any one

y When the covariance matrix and mean returns are calculated using equally weighted averages, an asset can appear to go through a prolonged period of high risk and high return following single jump in the share price. That single extreme return will affect the efficient frontier, and the optimal portfolio selection, for as many days as it remains in the equally weighted average (§3.1).