CHAPTER 2 Fundamentals of Asset Allocation

      THE FOUNDATION: PORTFOLIO THEORY

      E-V Maxim

Asset allocation is one of the most important and difficult challenges investors face, but thanks to Harry Markowitz, we have an elegant and widely accepted theory to guide us. In his classic article, “Portfolio Selection,” Markowitz reasoned that investors should not choose portfolios that maximize expected return, because this criterion by itself ignores the principle of diversification.¹ He proposed that investors should instead consider variances of return, along with expected returns, and choose portfolios that offer the highest expected return for a given level of variance. Markowitz called this rule the E-V maxim.

      Expected Return

      Markowitz showed that a portfolio's expected return is simply the weighted average of the expected returns of its component asset classes. A portfolio's variance is a more complicated concept, however. It depends on more than just the variances of the component asset classes.

      Risk

      The variance of an individual asset class is a measure of the dispersion of its returns. It is calculated by squaring the difference between each return in a series and the mean return for the series, and then averaging these squared differences. The square root of the variance (the standard deviation) is usually used in practice because it measures dispersion in the same units in which the underlying return is measured.

Variance provides a reasonable gauge of the risk of an asset class, but the average of the variances of two asset classes will not necessarily give a good indication of the risk of a portfolio comprising these two asset classes. The portfolio's risk also depends on the extent to which the two asset classes move together – that is, the extent to which their prices react in like fashion to new information.

      To quantify co-movement among security returns, Markowitz applied the statistical concept of covariance. The covariance between two asset classes equals the standard deviation of the first times the standard deviation of the second times the correlation between the two.

      The correlation, in this context, measures the association between the returns of two asset classes. It ranges in value from 1 to –1. If the returns of one asset class are higher than its average return when the returns of another asset class are higher than its average return, for example, the correlation coefficient will be positive, somewhere between 0 and 1. Alternatively, if the returns of one asset class are lower than its average return when the returns of another asset class are higher than its average return, then the correlation will be negative.

      The correlation, by itself, is an inadequate measure of covariance because it measures only the direction and degree of association between the returns of the asset classes. It does not account for the magnitude of variability in the returns of each asset class. Covariance captures magnitude by multiplying the correlation by the standard deviations of the returns of the asset classes.

      Consider, for example, the covariance of an asset class with itself. Obviously, the correlation in this case equals 1. The covariance of an asset class with itself thus equals the standard deviation of its returns squared, which is its variance.

      Finally, portfolio variance also depends on the weightings of its constituent asset classes – the proportion of a portfolio's wealth invested in each asset class. The variance of a portfolio consisting of two asset classes equals the variance of the first asset class times its weighting squared plus the variance of the second asset class times its weighting squared plus twice the covariance between the asset classes times the weighting of each asset class. The standard deviation of this portfolio equals the square root of the variance.

      From this formulation of portfolio risk, Markowitz was able to offer two key insights. First, unless the asset classes in a portfolio are perfectly inversely correlated (that is, have a correlation of –1), it is not possible to eliminate portfolio risk entirely through diversification. If a portfolio is divided equally among its component asset classes, for example, as the number of asset classes in the portfolio increases, the portfolio's variance will tend not toward zero but, rather, toward the average covariance of the component asset classes.

Second, unless all the asset classes in a portfolio are perfectly positively correlated with each other (a correlation of 1), a portfolio's standard deviation will always be less than the weighted average standard deviation of its component asset classes. Consider, for example, a portfolio consisting of two asset classes, both of which have expected returns of 10% and standard deviations of 20% and which are uncorrelated with each other. If we allocate the portfolio equally between these two asset classes, the portfolio's expected return will equal 10%, while its standard deviation will equal 14.1%. The portfolio offers a reduction in risk of nearly 30% relative to investment in either of the two asset classes separately. Moreover, this risk reduction is achieved without sacrificing any expected return.

      Efficient Frontier

      Markowitz also demonstrated that, for given levels of risk, we can identify particular combinations of asset classes that maximize expected return. He deemed these portfolios “efficient” and referred to a continuum of such portfolios in dimensions of expected return and standard deviation as the efficient frontier. According to Markowitz's E-V maxim, investors should choose portfolios located along the efficient frontier. It is almost always the case that there exists some portfolio on the efficient frontier that offers a higher expected return and less risk than the least risky of its component asset classes (assuming the least risky asset class is not completely risk free). However, the portfolio with the highest expected return will always be allocated entirely to the asset class with the highest expected return (assuming no leverage).

      The Optimal Portfolio

      Though all the portfolios along the efficient frontier are efficient, only one portfolio is most suitable for a particular investor. This portfolio is called the optimal portfolio. The theoretical approach for identifying the optimal portfolio is to specify how many units of expected return an investor is willing to give up to reduce the portfolio's risk by one unit. If, for example, the investor is willing to give up two units of expected return to lower portfolio variance (the squared value of the standard deviation) by one unit, his risk aversion would equal 2. The investor would then draw a line with a slope of 2 and find the point of tangency of this line with the efficient frontier (with risk defined as variance rather than standard deviation). The portfolio located at this point of tangency is theoretically optimal because its risk/return trade-off matches the investor's preference for balancing risk and return.

PRACTICAL IMPLEMENTATION

There are four steps to the practical implementation of portfolio theory. We must first identify eligible asset classes. Second, we need to estimate their expected returns, standard deviations, and correlations. Third, we must isolate the subset of efficient portfolios that offer the highest expected returns for different levels of risk. And fourth, we need to select the specific portfolio that balances our desire to increase wealth with our aversion to losses.

      Before