3.1 Stylized facts of financial market returns
3.1.1 Stylized facts for univariate series
Before we turn to the topic of modelling financial market risks, it is worthwhile to consider and review typical characteristics of financial market data. These are sum- marized in the literature as “stylized facts” (see Campbell et al. 1997; McNeil et al.
2005). These observed properties have important implications for assessing whether the risk model chosen is appropriate or not. Put differently, a risk model that does not capture the time series characteristics of financial market data adequately will also not be useful for deriving risk measures. For observed financial market data, the following stylized facts can be stated:
•Time series data of returns, in particular daily return series, are in general not independent and identically distributed (iid). This fact is not jeopardized by low absolute values of the first-order autocorrelation coefficient.
•The volatility of return processes is not constant with respect to time.
•The absolute or squared returns are highly autocorrelated.
•The distribution of financial market returns is leptokurtic. The occurrence of extreme events is more likely compared to the normal distribution.
•Extreme returns are observed closely in time (volatility clustering).
•The empirical distribution of returns is skewed to the left; negative returns are more likely to occur than positive ones.
Financial Risk Modelling and Portfolio Optimization withR, Second Edition. Bernhard Pfaff.
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k k Rcode 3.1Stylized facts on the returns for Siemens.
l i b r a r y ( f B a s i c s ) 1
l i b r a r y ( e v i r ) 2
d a t a ( s i e m e n s ) 3
S i e D a t e s <− a s . c h a r a c t e r ( f o r m a t ( a s . POSIXct ( a t t r ( s i e m e n s , 4
" t i m e s " ) ) , "%Y−%m−%d " ) ) 5
S i e R e t <− t i m e S e r i e s ( s i e m e n s ∗ 1 0 0 , c h a r v e c = S i e D a t e s ) 6
c o l n a m e s ( S i e R e t ) <− " S i e R e t " 7
# # S t y l i s e d F a c t s I 8
p a r ( mfrow = c ( 2 , 2 ) ) 9
s e r i e s P l o t ( S i e R e t , t i t l e = FALSE , main = " D a i l y R e t u r n s o f 10
S i e m e n s " , c o l = " b l u e " ) 11
b o x P l o t ( S i e R e t , t i t l e = FALSE , main = " Box p l o t o f R e t u r n s " , 12
c o l = " b l u e " , c e x = 0 . 5 , pch = 1 9 ) 13
a c f ( S i e R e t , main = "ACF o f R e t u r n s " , l a g . max = 2 0 , y l a b = " " , 14
x l a b = " " , c o l = " b l u e " , c i . c o l = " r e d " ) 15
p a c f ( S i e R e t , main = "PACF o f R e t u r n s " , l a g . max = 2 0 , 16
y l a b = " " , x l a b = " " , c o l = " b l u e " , c i . c o l = " r e d " ) 17
# # S t y l i s e d F a c t s I I 18
S i e R e t A b s <− a b s ( S i e R e t ) 19
S i e R e t 1 0 0 <− t a i l ( s o r t ( a b s ( s e r i e s ( S i e R e t ) ) ) , 1 0 0 ) [ 1 ] 20
i d x <− w h i c h ( s e r i e s ( S i e R e t A b s ) > S i e R e t 1 0 0 , a r r . i n d = TRUE ) 21
S i e R e t A b s 1 0 0 <− t i m e S e r i e s ( r e p ( 0 , l e n g t h ( S i e R e t ) ) , 22
c h a r v e c = t i m e ( S i e R e t ) ) 23
S i e R e t A b s 1 0 0 [ i d x , 1 ] <− S i e R e t A b s [ i d x ] 24
a c f ( S i e R e t A b s , main = "ACF o f A b s o l u t e R e t u r n s " , l a g . max = 2 0 , 25
y l a b = " " , x l a b = " " , c o l = " b l u e " , c i . c o l = " r e d " ) 26
p a c f ( S i e R e t A b s , main = "PACF o f A b s o l u t e R e t u r n s " , 27
l a g . max = 2 0 , y l a b = " " , x l a b = " " , c o l = " b l u e " , 28
c i . c o l = " r e d " ) 29
q q n o r m P l o t ( S i e R e t , main = "QQ−P l o t o f R e t u r n s " , t i t l e = FALSE , 30
c o l = " b l u e " , c e x = 0 . 5 , pch = 1 9 ) 31
p l o t ( S i e R e t A b s 1 0 0 , t y p e = " h " , main = " V o l a t i l i t y C l u s t e r i n g " , 32
y l a b = " " , x l a b = " " , c o l = " b l u e " ) 33
As an example, we will now check whether these stylized facts are applicable to the returns of Siemens stock. The data set of daily returns is contained in the package evir(see Pfaff and McNeil 2012). This series starts on 2 January 1973 and ends on 23 July 1996, and comprises 6146 observations.
In Listing 3.1 the necessary packagesfBasics(see Würtz et al. 2014) andevirare loaded first. Functions contained in the former package will be utilized to produce some of the graphs. Next, the siemensdata set is loaded and converted into an object of classtimeSerieswith the function of the same name. The time series plot of the percentage returns, a box plot thereof, and the autocorrelation and partial autocorrelation are produced next and exhibited in Figure 3.1. As can been deduced from the time series plot, volatility clustering does exist. This is more pronounced
k k 1975−01−01 1990−01−01
−1005
Daily Returns of Siemens
−1005
Box plot of Returns
0 5 10 15 20
0.00.40.8
ACF of Returns
5 10 15 20
−0.040.02
PACF of Returns
Figure 3.1 Stylized facts for Siemens, part one.
in the second half of the sample period. Furthermore, by mere eyeball econometrics the returns are skewed to the left and heavy tails are evident, as can be seen from the box plot (upper right panel). The largest loss occurred on 16 October 1989, at
−12.01%. The highest return of 7.67% occurred on 17 January 1991. The skewness is−0.52 and the excess kurtosis 7.74, clearly indicating heavy tails. The autocor- relation function (ACF) and partial autocorrelation function (PACF) hint at a slight first-order autocorrelation. Incidentally, the series shows some systematic variation on the weekly frequency; though significant, it is much less pronounced than the daily autocorrelation.
Figure 3.2 further investigates whether the stylized facts about financial market returns hold in the case of Siemens. In the upper panels of this figure, the autocorre- lations and partial autocorrelations of the absolute returns are plotted. Clearly, these are significantly different from zero and taper off only slowly. In the lower left panel a quantile–quantile (QQ) plot compared to the normal distribution is produced. The negative skew and heavy tails are mirrored from their quantitative values in this graph.
Finally, in Listing 3.1 the 100 largest absolute returns have been retrieved from the objectSieRet. These values are shown in the lower right-hand panel. This time series plot vindicates more clearly what could already be deduced from the upper left-hand panel in Figure 3.1: first, the existence of volatility clustering; and second, that the returns become more volatile in the second half of the sample period.
Although these stylized facts have only been exemplified by the stock returns of Siemens, they not only hold for basically all stock returns, but also are applicable to other asset classes, such as bonds, currencies, and commodity futures.
k k
0 5 10 15 20
0.00.40.8
ACF of Absolute Returns
−4 −2 0 2 4
−1005
QQ−Plot of Returns
Normal Quantiles
SieRet Ordered Data Confidence Intervals: 95%
5 10 15 20
0.000.10
PACF of Absolute Returns
Volatility Clustering
1975−01−01 1990−01−01
04812
Figure 3.2 Stylized facts for Siemens, part two.