Multiple Linear Regression 67
Sharpe benchmarks are determined by regressing periodic returns (e.g.,
monthly returns) on various market indexes. The Sharpe benchmark was
reported for one portfolio management firm based on performance from the
period January 1981 through July 1988 using monthly returns.^16 The result-
ing Sharpe benchmark based on monthly observations was
Sharpe benchmark=0.43 × (FRC Price-Driven Index)
+ 0.13 × (FRC Earnings-Growth Index)
+ 0.44 × (FRC 2000 Index)where FRC is an index produced by the Frank Russell Company.
The three indexes were selected because they were the only indexes of
the 10 that were statistically significant. Notice that the sum of the three
coefficients is equal to one. This is done by estimating a constrained regres-
sion, a topic we do not cover in this book. The R^2 for this regression is
97.6%. The intercept term for this regression is 0.365%, which represents
the average excess monthly return.
By subtracting the style benchmark’s monthly return from the manag-
er’s monthly portfolio return, performance can be measured. This difference,
which we refer to as “added value residuals,” is what the manager added
over the return from three “index funds” in the appropriate proportions.
For example, suppose that in some month the return realized by this man-
ager is 1.75%. In the same month, the return for the three indexes were as
follows: 0.7% for the FRC Price-Driven Index, 1.4% for the FRC Earnings-
Growth Index, and 2.2% for the FRC 2000 Index. The added value residual
for this month would be calculated as follows. First, calculate the value of
the Sharpe benchmark:
Sharpe benchmark=0.43 × (0.7%) + 0.13 × (1.4%) + 0.44 × (2.2%)
=1.45%
The added value residual is then:Added value residual = Actual return − Sharpe benchmark return
Since the actual return for the month is 1.75%,Added value residual = 1.75% − 1.45% = 0.3%
Notice that if this manager had been benchmarked against a single
investment style index such as the FRC Price-Driven Index, the manager
(^16) See H. Russell Fogler, “Normal Style Indexes—An Alternative to Manager Uni-
verses?” in Performance Measurement: Setting the Standards, Interpreting the Num-
bers (ICFA, 1989), 102.