Figure 2.2 displays the time series of original prices, of excess returns as defined by (2.2) and of squared excess returns for the overall market and the two sectors Insurance and Food & Beverage. An informal first look at these graphs suggests
2The dendrogram has been generated using thehclust()function of the open-source statistical software package R 2.1.1 (R Development Core Team, 2005) which can be downloaded from www.r-project.org. For an introductory outline of cluster analysis and its implementation, see, for example, Struyf et al. (1996) and Kaufman and Rousseeuw (2005).
100
1990 1995 2000 2005 −20
−10
1990 1995 2000 2005 0
100
1990 1995 2000 2005 0
50
Figure 2.2: Summaries of the weekly returns on the (i) broad market, (ii) the In-surance sector and (iii) Food & Beverages. Summaries from top to bottom are time series of original prices, excess returns and squared excess returns.
that the original sector series in levels are trending. Formally, the existence of a unit root can be confirmed by an augmented Dickey-Fuller test or the Phillips-Perron test. As the existence of unit roots in financial price series is extensively discussed in the empirical literature, the corresponding results are not reported here; for a detailed outline of unit root tests, see, for example, Pagan (1996). In the following, only excess returns will be considered. Another obvious property illustrated by the returns and squared return series is that weeks of large absolute movements are followed by weeks with the same characteristics. This phenomenon, commonly referred to asvolatility clustering, can be particularly observed in the second half of the sample, which includes the Asian crisis (1997), the Russian debt crisis (1998) and the boom and bust of the dotcom bubble (1998–2003).
Univariate descriptive statistics for the data and some standard test statistics, which are referred to in the following subsections, are provided in Table 2.2. It can be seen that over the entire sample, the highest average weekly excess returns are offered by Healthcare (0.17%), Oil & Gas (0.15%) and Utilities (0.15%). The lowest are realized for Automobiles & Parts (0.02%), Insurance (0.04%) and Retail (0.06%). The risk, as measured by the annualized standard deviation, ranges from 14.64% for the defensive Utilities to 30.45% for the high risk sector Technology.
Table 2.2: Summary statistics of weekly excess returns.
Sector T µa σb skc kud JBe Qsqf levg ρ0ih
Broad 897 0.10 16.62 −0.30 6.83 560 332 −0.20 –
Automobiles 897 0.02 23.77 −0.56 6.30 453 220 −0.15 0.80 Banks 897 0.14 19.47 −0.28 7.49 766 451 −0.18 0.91 Basics 897 0.12 20.47 −0.24 5.13 177 153 −0.13 0.73 Chemicals 897 0.09 18.52 −0.19 7.87 890 147 −0.14 0.81 Construction 897 0.08 17.69 −0.32 4.97 160 215 −0.14 0.83 Financials 897 0.07 18.69 −0.63 8.73 1287 378 −0.18 0.89
Food 897 0.10 15.29 −0.27 5.86 318 122 −0.14 0.70
Healthcare 897 0.17 18.23 0.18 5.52 243 146 −0.16 0.71 Industrials 897 0.07 17.86 −0.47 5.69 303 272 −0.19 0.91 Insurance 897 0.04 24.09 −0.85 13.97 4607 431 −0.18 0.88 Media 897 0.07 24.69 −0.62 9.89 1832 291 −0.09 0.82 Oil & Gas 897 0.15 19.22 −0.02 5.56 246 296 −0.16 0.66 Personal 683 0.09 18.57 −0.22 4.95 114 108 −0.22 0.86 Retail 683 0.06 21.50 −0.78 10.32 1594 49 −0.12 0.78 Technology 897 0.07 30.45 −0.55 6.68 553 603 0.01 0.81 Telecom 897 0.13 24.81 −0.18 5.36 213 325 −0.07 0.80
Travel 683 0.07 16.90 0.10 6.36 322 118 −0.19 0.80
Utilities 897 0.15 14.64 −0.45 5.15 203 171 −0.13 0.79
aThe mean is expressed in percentage terms.
bThe standard deviation is expressed in annualized percentage terms.
cskdenotes the skewness of the return series.
dkudenotes the kurtosis of the return series.
eJB is the Jarque-Bera statistic for testing normality. The test statistic is asymptoticallyχ2 distributed with 2 degrees of freedom. The relevant critical value at the 99% level is 9.21.
fQsq is the Ljung-Box portmanteau test for the null of no autocorrelation in the squared excess returns up to order 12. The test statistic is asymptoticallyχ2 distributed with 12 degrees of freedom. The relevant critical value at the 99% level is 26.22.
glevis the unconditional correlation between the squared excess return at datetand the excess return at datet−1 for indexi. Negative values forlevindicate that large volatility tends to follow upon negative returns.
hρ0i is the unconditional correlation between sectori and the overall market.
2.2.1 Thick tails
The observed degree of kurtosis (ku) of market and sector returns reported in Ta-ble 2.2 generally exceeds the normal value of three. Compared to a normal distribu-tion, the peaks are higher and the tails are heavier, which reflects that large outlying observations occur more often than can be expected under the assumption of nor-mality. According to the stated values of skewness (sk), the overall market as well as all sectors, except for Healthcare and Travel & Leisure, are negatively skewed.
This might be an indication that large negative returns occur more often than large positive returns. The Jarque-Bera test statistics, as reported in columnJB, confirm the departure from normality for all return series at the 1% significance level. Since the seminal works by Mandelbrot (1963) and Fama (1965) it is well known that
many asset returns follow a leptokurtic distribution. The shortcomings related to the regularly made normality assumption are commonly addressed either by em-ploying a fat-tailed distribution, such as Student-t, or by relying on a mixture of normals; for an overview of references on the subject, see Bollerslev (1987) or, more recently, Gettinby et al. (2004).
2.2.2 Volatility clustering
It appeared from Figure 2.2 that quiet periods, characterized by relatively small re-turns, alternate with relatively volatile periods, where price changes are rather large.
This can be confirmed by looking at the first-order autocorrelation (AC) function of returns and squared returns in Figure 2.3. While the autocorrelations of the return series only show minor activity, the autocorrelation function of squared returns show significant correlations up to an extended lag length. The corresponding Box-Ljung statistics, reported in column Qsq of Table 2.2, confirm significant correlations for the squared returns at the 1% level for all series.
0 5 10 15 20 25 30
Figure 2.3: Autocorrelation functions (first 30 lags) of the broad market, the In-surance sector and Food & Beverages. The top row displays correlograms of excess returns, the bottom row displays correlograms of the corresponding squares.
The phenomenon of persistently changing volatility over time has been first rec-ognized by Mandelbrot (1963) and Fama (1965) who related volatility clustering to the observation of fat-tailed returns. The two most important concepts to deal with time-varying volatility are (i) the autoregressive conditional heteroskedasticity (ARCH) model by Engle (1982) together with the generalized ARCH (GARCH) model by Bollerslev (1986), and (ii) the stochastic volatility (SV) model by Taylor (1982, 1986). An alternative way to deal with conditional volatility is to employ a Markov regime switching model: the generation of excess returns under differ-ent volatility regimes is governed by differdiffer-ent states, each driven by a first-order
Markov chain. Leading papers on the subject include Hamilton and Susmel (1994) and Turner et al. (1989).
2.2.3 Leverage effects
Another common property of financial return series is the so-called leverage ef-fect, where future volatility depends negatively on the sign of past returns (Black, 1976). Column lev of Table 2.2 reports the estimated correlation coefficients be-tween squared excess returns at datetand excess returns at datet−1 for all sectors and the overall market. With the exception of Technology, all estimates are neg-ative. This gives a first indication that negative sector returns are followed by a pick-up in volatility.
2.2.4 Volatility co-movements
From the squared excess returns shown in the bottom row of Figure 2.2, it can be seen that volatility peaks tend to occur at the same time for all three series (August 1990, January 1999, period between July 2002 and March 2003). This is a com-mon observation as certain newsflow usually affects the volatility of different assets simultaneously. This phenomenon might lead to a failure of joint stationarity of two return series even though stationarity for the individual series holds. A possi-ble solution to this propossi-blem is to employ a multivariate GARCH or a multivariate SV framework to deal withvolatility co-movements by modeling conditional covari-ances. Important papers dealing with multivariate conditional heteroskedasticity include Diebold and Nerlove (1989), Harvey et al. (1992) and Harvey et al. (1994).