Formulating and Implementing Investment Strategies Using Financial Econometrics 311
comes with the benefit of the additional description is the increase of estima-
tion error of an additional parameter.
In Table 15.1, we demonstrate the explanatory power of a typical mul-
tifactor model for stock returns by including one additional variable at a
time.^4 The second and third columns clearly show that, although by design
the R^2 increases with the number of variables, the adjusted R^2 , which also
reflects the impact of the additional estimation error, levels off and starts
decreasing after some point. This illustration suggests that in the process of
estimation, the cost of estimation error is even compounded when a new
prediction is further extended into the forecast period.
In Table 15.2, we also perform an out-of-sample prediction based on the
estimated multifactor model in each stage. The fourth and fifth columns in
the table show a more striking pattern that the risk-adjusted excess return,
in the form of information ratio,^5 deteriorates even more quickly when the
model becomes large.
(^4) This is based on Christopher K. Ma, “How Many Factors Do You Need?” (Research
Paper #96-4, KCM Asset Management, Inc., 2005 and 2010).
(^5) The information ratio is a reward-risk ratio. The reward is the average of the active
return, which is the portfolio return reduced by the return on a benchmark. This is
popularly referred to in finance as “alpha.” The risk is the standard deviation of the
active return, which is known as the tracking error. The higher the information ratio,
the better the portfolio manager performed relative to the risk assumed.
taBle 15.1 Marginal Contribution of Additional Explanatory Variables
In Sample Out of Sample
Additional
Explanatory
Variable
Explanatory
Power
(R^2 )
Explanatory
Power
(Adj. R^2 )
Annualized
Excess
Return (%)
Annualized
Standard
Deviation (%)
Information
Ratio
1st 0.086 0.082 2.52 7.15 0.352
2nd 0.132 0.105 2.98 7.10 0.420
3rd 0.175 0.117 3.61 6.97 0.518
4th 0.188 0.165 3.82 6.82 0.560
5th 0.202 0.174 4.05 6.12 0.662
6th 0.251 0.239 3.99 6.08 0.656
7th 0.272 0.221 3.76 6.19 0.607
8th 0.282 0.217 3.71 6.22 0.596
9th 0.291 0.209 3.64 6.37 0.571
10th 0.292 0.177 3.53 6.58 0.536