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investors who are unlucky because it exposes them to losses that might arise as a consequence of bad luck.⁴ As beneficial as it is for skillful investors to focus on activities that cause dispersion, it is equally important for unlucky investors to avoid activities that cause dispersion.

      Determining Relative Importance Analytically

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
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