Читать книгу Asset Allocation - William Kinlaw, Mark P. Kritzman - Страница 49

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.