## Style Drift

Since the inception of returns-based style analysis and the ensuing development of sophisticated software that brought returns-based style analysis to the masses, investors have used rolling asset allocation graphs and style maps to gain a visual feeling for the style consistency of managers and mutual funds. The opposite of style consistency is style inconsistency or style drift. Rolling asset allocation graphs and style maps are good tools for developing an intuitive understanding of a fund’s style consistency; however, they do not replace the need for a quantitative measure of style drift.

The Zephyr Style Drift Score measures the variability of style through time. A Zephyr Style Drift Score of 0 indicates perfect style consistency and is equivalent to buying and rebalancing the indices that constitute the style basis each period.

The Zephyr Style Drift Score is calculated by measuring the aggregate variability of the individual asset class coefficients.

where

is the kth asset class coefficient at time t.

Unfortunately this formula masks the underlying intuitiveness of the Zephyr Style Drift Score. An alternative calculation of the Zephyr Style Drift Score yields the same answer but reveals more of the intuition.

Start by calculating the average value of the asset class coefficients for time 1 through T. Let
be the average values of the K asset class coefficients. can be thought of as the center of gravity in the in K - dimensional space.

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Next, subtract the average values of the K asset class coefficients from the respective style coefficients at time 1 through T.

This results in a K x T matrix of calculations in which each row represents the component wise distance from the center of gravity for that time period. For each time period, the individual K distances are squared and the sum is calculated resulting in a T x 1 vector. Each element of the T x 1  vector represents the squared Euclidian distance between the asset class coefficients at time t and the center of gravity.

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Next, calculate the sample average of the T x 1 vector of squared Euclidian distances from the center of gravity.

Finally, calculate the square root of the average squared Euclidian distances.

To summarize, the Zephyr Style Drift Score is the square root of the average squared Euclidian distance of the T rolling window asset class coefficients from the center of gravity in K-dimensional space.

Additional mathematical details of the Zephyr Style Drift Score are available in a Zephyr research white paper, “The Style Drift Score: A Quantitative Measure,” by Zephyr’s Tom Idzorek, CFA, and Fred Bertsch, Ph.D., which is available upon request (support@styleadvisor.com).

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