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 K-dimensional space.
.
.
.
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.
.
.
.
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).
Related Statistics:
Style Analysis
Manager Style Graph Style Points
