Читать книгу Asset Allocation - William Kinlaw, Mark P. Kritzman - Страница 69

It is simple to measure the relative importance of asset allocation and security selection analytically if we limit our investment universe to two asset classes, each of which contains two securities. We simply measure the potential for dispersion as the tracking error between two investments that differ either by asset class composition or by security composition. We should expect security selection to cause greater dispersion than asset allocation because individual securities are more volatile than the asset classes that comprise them unless the securities move in perfect unison. Therefore, if we argue that asset allocation causes greater dispersion, we necessarily believe that high correlations among individual securities offset their relatively high individual volatilities.

Consider two asset classes that contain two securities each. Asset class A includes securities A1 and A2, while asset class B includes B1 and B2. We measure the relative volatility and hence the importance of security selection within asset class A as shown:

(3.2)

In Equation 3.2, equals the relative volatility between A1 and A2, equals the standard deviation of A1, equals the standard deviation of A2, and is the correlation between A1 and A2. The same equation is used to calculate the relative volatility between securities B1 and B2.

We measure the importance of choosing between asset class A and asset class B the same way, but first we must calculate the standard deviation of each asset class. If we assume the individual securities are weighted equally within each asset class, the standard deviation of asset class A equals

(3.3)

Here, equals the standard deviation of asset class A, equals the standard deviation of A1, equals the standard deviation of A2, and is the correlation between A1 and A2.

We repeat the same calculation to derive the standard deviation of asset class B.

The relative volatility between asset class A and asset class B equals

(3.4)

TABLE 3.1 Standard Deviation, Correlation, and Relative Volatility

	Standard Deviation (%)	Correlation (%)	Relative Volatility (%)			Standard Deviation (%)	Correlation (%)	Relative Volatility (%)
A1	10.0				A1	10.0
A2	10.0	0.0	14.1		A2	10.0	50.0	10.0
B1	10.0				B1	10.0
B2	10.0	0.0	14.1		B2	10.0	50.0	10.0
A	7.1				A	8.7
B	7.1	0.0	10.0		B	8.7	50.0	8.7
	Standard Deviation (%)	Correlation (%)	Relative Volatility (%)			Standard Deviation (%)	Correlation (%)	Relative Volatility (%)
A1	10.0				A1	10.0
A2	10.0	50.0	10.0		A2	10.0	50.0	10.0
B1	10.0				B1	10.0
B2	10.0	50.0	10.0		B2	10.0	50.0	10.0
A	8.7				A	8.7
B	8.7	33.3	10.0		B	8.7	25.0	10.6

In Equation 3.3, equals the relative volatility between A and B, equals the standard deviation of A, equals the standard deviation of B, and is the correlation between A and B.

Suppose the four securities are uncorrelated with each other. Then security selection would be more important than asset allocation because the asset classes would be less risky than the average risk of the securities they comprise, which results in less relative volatility between the asset classes than between the securities within each asset class. Moreover, as more securities are added, the asset class standard deviations decline further, which in turn further reduces the relative volatility between the asset classes. If, for example, security returns are uncorrelated and the securities are equally weighted, then the asset class standard deviation diminishes with the square root of the number of securities included. It is only when the correlation between asset classes A and B is substantially less than the correlation between the individual securities within the asset classes that the relative volatility between asset classes is greater than the relative volatility between securities. These relationships are illustrated in Table 3.1.

The upper left panel shows that relative volatility between asset classes is less than relative volatility between securities when they are all uncorrelated with one another. The upper right panel shows the same result when they all are equally correlated with one another. The lower left panel shows the asset class and security correlations that lead to convergence between relative volatilities. Finally, the lower right panel provides an example in which the relative volatility between asset classes is higher than it is between securities.