# Style Analysis

StyleADVISOR implements returns based style analysis as set forth by Stanford professor and Nobel Prize winner William F. Sharpe. Returns based style analysis calculates a Style Benchmark for a manager from the manager’s return series and the return series of the indices that are to be used in the Style Benchmark. Thus, we are given:

• a sequence M = m1m2, ... , mt of manager returns over t periods, and
• N sequences of index returns over t periods:

C1 = c11, c12, ..., c1t
C2 = c21, c22, ..., c2t
...
CN = cN1, cN2, ..., cNt

The goal of style analysis is to determine weights x1, x2, ..., xN such that the composite series

S = x1C1 + x2C2 + ... + xNCN

becomes a best fit, or Style Benchmark, for the manager series M. The crucial question is which of the many possible mathematical definitions of �best fit� is to be chosen. William F. Sharpe proposed as the criterion for best fit to minimize the variance of excess return, i.e., one should determine the weights x1, x2, ... , xN so as to minimize the expression

Var( M - x1C1 + x2C2 + ... + xNCN ))

From the definition of variance, it is clear that the above expression is quadratic in the unknown weights x1, x2, ... , xN. Hence, determining the weights to minimize the expression requires a quadratic optimization. This is a very complex algorithm whose details are beyond this documentation. If you wish to verify StyleADVISOR’s calculation independently, you may want to use the “Solver” add-in that is provided with Microsoft Excel.

As a plausibility condition on the weights, the StyleADVISOR program always enforces the constraint that the weights add up to 1, i.e., the quadratic optimization is performed subject to the constraint

x1 + x2 + ... + xN = 1

By default, StyleADVISOR also constrains the weights to be between 0 and 1. However, this default can be changed in the Constraints group on the Parameters tab in the Analysis Parameters dialog to allow the optimizer to go short on some or all indices.

References:
Hardy, Steve. Returns Based Style Analysis. The Handbook of Equity Style Management, 1995, Chapter 3.
Sharpe, William F., Determining a fund’s effective asset mix, Investment Management Review, December 1988, pp. 59-69.
Sharpe, William F., Asset allocation:Management style and performance measurement, The Journal of Portfolio Management, Volume 18, Winter 1992, pp. 7-19.

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