In this section a back-test of the expected shortfall at a 99% confidence level for New York Stock Exchange (NYSE) daily returns is conducted. TheRcode is shown in Listing 8.1. This data set is a supplement to the monograph of Stock and Wat- son (2007) and is contained in the AERpackage (see Kleiber and Zeileis 2008).
Rcode 8.1Expected shortfall derived from GARCH(1, 1) models l i b r a r y (AER) 1
l i b r a r y ( f G a r c h ) 2
d a t a (NYSESW) 3
NYSELOSS <− t i m e S e r i e s (−1 . 0 ∗ d i f f ( l o g (NYSESW) ) ∗ 1 0 0 , 4
c h a r . v e c = t i m e (NYSESW) ) 5
# # F u n c t i o n f o r ES o f t−GARCH 6
E S g a r c h <− f u n c t i o n ( y , p = 0 . 9 9 ) { 7
g f i t <− g a r c h F i t ( f o r m u l a = ∼g a r c h ( 1 , 1 ) , d a t a = y , 8
cond . d i s t = " s t d " , t r a c e = FALSE ) 9
s i g m a <− p r e d i c t ( g f i t , n . a h e a d = 1 ) [ 3 ] 10
d f <− c o e f ( g f i t ) [ " s h a p e " ] 11
ES <− s i g m a ∗ ( d t ( q t ( p , d f ) , d f ) / ( 1 − p ) ) ∗ 12
( ( d f + ( q t ( p , d f ) ) ^ 2 ) / ( d f − 1 ) ) 13
r e t u r n ( ES ) 14
} 15
# # D a t e v e c t o r s f o r b a c k t e s t 16
f r o m <− t i m e ( NYSELOSS ) [−c ( ( nrow ( NYSELOSS ) − 9 9 9 ) : 17
nrow ( NYSELOSS ) ) ] 18
t o <− t i m e ( NYSELOSS ) [−c ( 1 : 1 0 0 0 ) ] 19
NYSEES <− f a p p l y ( NYSELOSS , f r om = from , t o = t o , FUN = E S g a r c h ) 20
NYSEESL1 <− l a g ( NYSEES , k = 1 ) 21
r e s <− na . o m i t ( c b i n d ( NYSELOSS , NYSEESL1 ) ) 22
c o l n a m e s ( r e s ) <− c ( "NYSELOSS" , " ES99 " ) 23
p l o t ( r e s [ , 2 ] , c o l = " r e d " , y l i m = r a n g e ( r e s ) , 24
main = "NYSE : t−GARCH( 1 , 1 ) ES 99%" , 25
y l a b = " p e r c e n t a g e s " , x l a b = " " ) 26
p o i n t s ( r e s [ , 1 ] , t y p e = " p " , c e x = 0 . 2 , pch = 1 9 , c o l = " b l u e " ) 27
l e g e n d ( " t o p l e f t " , l e g e n d = c ( " L o s s " , " ES " ) , 28
c o l = c ( " b l u e " , " r e d " ) , l t y = c (NA, 1 ) , pch = c ( 1 9 , NA) ) 29
k k
percentages
1992−01−01 1996−01−01 2000−01−01 2004−01−01
−4−20246
Figure 8.2 Time series plot for daily losses of NYSE.
The sample runs from 2 January 1990 to 11 November 2005 and contains 4003 observations. The necessary packages and the data set are loaded into the workspace with the first three commands in theRcode listing. Next, the daily compound losses are computed as percentages and expressed as positive numbers. The stylized facts of this series are pretty evident from Figure 8.2.
In order to conduct the back-test, the function ESgarch()is defined in lines 6–14. Within this function a GARCH(1, 1) model with a Student’st-distributed in- novation process is estimated first. The one-step-ahead forecast of the conditional standard deviation is computed next, and the fitted value of the degrees-of-freedom
NYSE: t−GARCH(1,1) ES 99%
percentages
0 500 1000 1500 2000 2500 3000
−50510 Loss
ES
Figure 8.3 Comparison of daily losses of NYSE and ES.
k k parameter is assigned to the objectdf. The expected shortfall is then computed for
the default confidence level ofp = 0.99. Incidentally, the mean equation of this GARCH model consists of a constant only and is omitted in the calculation of the ES.
Given that its estimate represents the mean of the series which is empirically close to zero, it can safely be discarded from the computation of the ES. The back-test itself is then conducted by utilizing a sliding window with 1000 observations. The function fapply()comes in handy for conducting back-tests of this kind. Two date vectors are created in which the start and end dates of this moving window through time are stored. In the next line the functionfapply()will callESgarch()with the subsamples of the losses according to the date values contained in the objectsfrom andto. It should be noted that the ES numbers are now associated with the date values offrom. But these conditional risk measures pertain to the next trading day and hence the series must be lagged by one period (objectNYSEESL1) for comparison with the actual losses. The size of the back-test therefore consists of 3001 risk measure–loss pairs. A graphical comparison of the actual losses and the conditional ES for a 99%
confidence level is produced in the final lines. The outcome is shown in Figure 8.3.
Not only are the spikes of the daily losses captured pretty well by this conditional risk measure, but also the level of the risk measure decreases rapidly during the more tranquil periods. During this back-test simulation only five violations occurred. This is a little too conservative, given that roughly 15 exceedances can be expected.
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k k
9
Modelling dependence