6.7 Applications of the GLD to risk modelling
6.7.2 Shape triangle for FTSE 100 constituents
In the second example, the characteristics of FTSE 100 stocks are analyzed by means of a shape triangle for the standardized GLD as in (6.23)–(6.27). This kind of shape triangle was proposed in Chalabi et al. (2010) and applied to the constituent stocks of the NASDAQ 100. The shape triangle is depicted in the𝛿=𝜆4−𝜆3and𝛽 =𝜆3+𝜆4 plane. TheRcode is shown in Listing 6.5.
The robust estimation of the GLD parameters is covered in the packagefBasicsand the weekly price data of the FTSE 100 constituents is part of the packageFRAPO,
k k Rcode 6.5FTSE 100 stocks: shape triangle of standardized GLD.
l i b r a r y ( FRAPO ) 1
l i b r a r y ( f B a s i c s ) 2
# # L o a d i n g o f d a t a 3
d a t a ( INDTRACK3 ) 4
P <− INDTRACK3 [ , −1] 5
R <− r e t u r n s e r i e s ( P , method = " d i s c r e t " , t r i m = TRUE) 6
# # F i t t i n g and c a l c u l a t i n g b e t a and lambda 7
F i t <− a p p l y ( R , 2 , g l d F i t , method = " r o b " , d o p l o t = FALSE , 8
t r a c e = FALSE ) 9
D e l t a B e t a P a r a m <− m a t r i x ( u n l i s t ( l a p p l y ( F i t , f u n c t i o n ( x ){ 10
l <− x @ f i t $ e s t i m a t e [ c ( 3 , 4 ) ] 11
r e s <− c ( l [ 2 ] − l [ 1 ] , l [ 1 ] + l [ 2 ] ) 12
r e s}) ) , n c o l = 2 , byrow = TRUE) 13
# # S h a p e t r i a n g l e 14
p l o t ( D e l t a B e t a P a r a m , x l i m = c (−2 , 2 ) , y l i m = c (−2 , 0 ) , 15
x l a b = e x p r e s s i o n ( d e l t a == l a m b d a [ 4 ] − l a m b d a [ 3 ] ) , 16
y l a b = e x p r e s s i o n ( b e t a == l a m b d a [ 3 ] + l a m b d a [ 4 ] ) , 17
pch = 1 9 , c e x = 0 . 5 ) 18
s e g m e n t s ( x0 = −2 , y0 = −2 , x1 = 0 , y1 = 0 , 19
c o l = " g r e y " , lwd = 0 . 8 , l t y = 2 ) 20
s e g m e n t s ( x0 = 2 , y0 = −2 , x1 = 0 , y1 = 0 , 21
c o l = " g r e y " , lwd = 0 . 8 , l t y = 2 ) 22
s e g m e n t s ( x0 = 0 , y0 = −2 , x1 = 0 , y1 = 0 , c o l = " b l u e " , 23
lwd = 0 . 8 , l t y = 2 ) 24
s e g m e n t s ( x0 = −0 . 5 , y0 = −0 . 5 , x1 = 0 . 5 , y1 = −0 . 5 , 25
c o l = " r e d " , lwd = 0 . 8 , l t y = 2 ) 26
s e g m e n t s ( x0 = −1 . 0 , y0 = −1 . 0 , x1 = 1 . 0 , y1 = −1 . 0 , 27
c o l = " r e d " , lwd = 0 . 8 , l t y = 2 ) 28
hence these two packages are loaded into the workspace first. Next, the data object INDTRACK3is loaded and its first column—representing the FTSE 100 index—is omitted from further analysis. The percentage returns of the stocks are assigned to the objectR, which is then used to fit each of its columns to the GLD with the function gldFit(). This task is swiftly accomplished by utilizing theapply()function.
The objectFitis a list with the returned objects ofgldFit. In lines 10–13 a small function is defined that returns the(𝛿, 𝛽)parameter pairs, which are then plotted in the shape triangle. The output is shown in Figure 6.10.
Thex-axis represents the difference between the right- and left-tail shape param- eters, and the y-axis their sum. There are a total of six regions discernible in this triangle. The light gray dashed line discriminates between stocks that are character- ized by either a left-skewed or a right-skewed distribution. Points on that line refer to a symmetric distribution. As can easily be seen, the majority of stock returns are characterized by being skewed to the left, thus confirming a stylized fact of financial returns. Points in the top part of the triangle represent return distributions with finite variance and kurtosis, and points in the middle region, between the−0.5 and−1.0
k k
−2 −1 0 1 2
−2.0−1.5−1.0−0.50.0
δ = λ4 − λ3
β = λ3 + λ4
Figure 6.10 Shape triangle for FTSE 100 stock returns.
dark gray dashed lines, refer to return distributions with infinite kurtosis but finite variance. The(𝛿, 𝛽)pairs below the−1 line represent return processes where these moments are infinite, which is the case for one of the FTSE 100 constituents stocks.
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Extreme value theory