Modeling, estimation, and optimization of equity portfolios with heavy-tailed distributions 123
Generally, we can apply the PCA either to the variance – covariance matrix or to
the correlation matrix. Since returns are heavy-tailed dimensionless quantities,
we apply PCA to the correlation matrix obtaining 30 principal components,
which are linear combinations of the original series, r ( r 1 , ... , r 30 ).
Table 5.1 shows the total variance explained by a growing number of com-
ponents. Thus, the first component explains 41.2% of the total variance and
the first 14 components explain 80.65% of the total variance. Because all the
other components contribute no more than 1.75% of the global variance, we
implement a dimensionality reduction by choosing only the first 14 factors. As
a consequence of this principal component analysis, each series r i ( i 1, ... ,30)
can be represented as a linear combination of 14 factors plus a small uncorre-
lated noise.
Once we have identified 14 factors that explain more than 80% of the
global variance, then we can generate the future returns r i using the factor
model:
rfit i ij jt eti
j,, , it,,..., ; ,...,
αβ
114
∑^11837130
(5.2)Table 5.2 reports the coefficients α (^) i ; β (^) i , (^) j of factor model (5.2). The genera-
tion of future scenarios should take into account (1) all the anomalies observed
Table 5.1 Percentage of the total variance explained by a growing number
of components based on the covariance matrix
Principal
component
Percentage
of variance
explained
Percentage of
total variance
explained
Principal
component
Percentage
of variance
explained
Percentage of
total variance
explained
1 41.20 41.20 16 1.71 84.11
2 5.33 46.53 17 1.64 85.75
3 4.52 51.05 18 1.53 87.28
4 4.08 55.14 19 1.45 88.73
5 3.82 58.95 20 1.40 90.13
6 3.24 62.19 21 1.37 91.50
7 2.84 65.03 22 1.31 92.81
8 2.72 67.75 23 1.24 94.05
9 2.63 70.38 24 1.15 95.20
10 2.37 72.75 25 1.05 96.26
11 2.14 74.89 26 0.97 97.23
12 2.03 76.92 27 0.88 98.11
13 1.90 78.82 28 0.72 98.83
14 1.83 80.65 29 0.65 99.48
15 1.75 82.40 30 0.52 100.00