Factor rotation
2019-09-26
Giles Heywood
Unrotated factor loadings
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Unrotated factors
Factor loadings after rotation
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Factors after rotation
Factor loadings after rotation, reduced set
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Rotation matrix
| factor1 | factor2 | factor3 |
|---|---|---|
| 0.954 | 0.095 | -0.285 |
| -0.077 | 0.994 | 0.073 |
| 0.290 | -0.048 | 0.956 |
Rotation matrix self-inverse property verification : R’R = I
| factor1 | factor2 | factor3 |
|---|---|---|
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
Properties under rotation
Properties preserved
- eigenvectors (‘factor portfolio weights’) are orthogonal
- factor covariance is the identity
Properties NOT preserved
- variance of factor 1 is maximised
- for i in 2:k factor i+1 has maximal variance subject to factors 1:i
Why rotate?
- the default rotation is arbitrary and rotate of their own accord as the window rolls - this is undesirable
- factors 2,3 are already very close to pure cyclical - a small rotation makes them cyclical
- ‘pure cyclical’ has a special meaning which is ‘zero average risk premium’ which has meaning in finance
Why not rotate?
- I am not sure why rotation seems to alarm people?
What factors are these?
- 15-year monthly rolling 12-month uncentred covariance
- uncentred variance is unity on a one-year time-horizon
- Pat Burns package BurStFin::factor.model.stat