Procedimiento

99% 95% 90% 99% 95% 90%

GARCHis 98.26 94.79 91.31 98.44 95.22 92.44 GARCHos 96.95 92.75 88.17 94.00 90.00 86.40 HS30 96.45 93.35

87.99 96.44 93.33

87.44 HS50 97.51 93.13 88.29 97.44 93.11 88.22 HS100 98.11 93.73 87.54 98.00

93.78 88.33 HS200 98.19

93.13 87.54 98.00

93.44 88.22 RS30 96.75 92.82 88.90 97.22 92.89 88.89 RS50 97.13 93.43 89.43 97.56

93.56 89.56 RS100 98.11 93.81 88.44 98.56

94.33 89.67 RS200 97.89

93.28 88.14 98.22

94.11 89.78 FMKL30 96.37 92.67 89.27 96.78 92.89 89.00 FMKL50 97.13 93.50 89.12 97.44 93.56 89.11 FMKL100 97.81

93.73 88.52 98.11

94.11 89.67 FMKL200 97.96

93.13 88.14 98.33

94.00 90.00

GLD-TV 94.50 87.60 80.30

Table 5.3: Actual coverage level of Value-at-Risk for ASX 200 at one day, estimated with GARCH in-sample, GARCH out-of-sample, historical sim-ulation (HS30, HS50, HS100, HS200), moving window GLD with RS pa-rameterisation (RS30, RS50, RS100, RS200), moving window GLD with FMKL parametrisation (FMKL30, FMKL50, FMKL100, FMKL200) and time-varying GLD (GLD-TV) for ASX 200 index. The percentages in bold-face highlight the techniques that gave the best coverage levels. GARCH

in-sample is not considered. ^∗Values rejected by the conditional coverage test

(Christoersen, 1998).

149 Estimating Value-at-Risk

all sampling period stationary period

99% 95% 90% 99% 95% 90%

GARCHis 98.24 94.50 90.53 99.11 95.67 91.22 GARCHos 95.49

90.98

85.88 99.00 97.20 93.80

HS30 96.26 93.21 88.63 96.44 93.33 89.11 HS50 97.48 93.66 88.63 97.89 93.78 89.33 HS100 97.86 93.21 88.32 98.33 93.78 88.67 HS200 97.94 93.59 88.47 98.33 94.33 89.67 RS30 97.86 94.05 87.56 98.11 94.78

88.44 RS50 98.24 93.51 87.40 98.56 94.78

89.00 RS100 98.40 93.13 87.18 98.56 94.67

88.89 RS200 98.47 92.98 87.10 99.00 95.67 90.22 FMKL30 96.87 93.66 88.40 96.78 94.11 89.44 FMKL50 97.71 93.21 87.86 97.78 94.44 89.56 FMKL100 98.24 93.05 87.25 98.33 94.56 89.11

FMKL200 98.40 92.82 87.18 98.89 95.56 90.44

GLD-TV 95.40 88.40 82.80

Table 5.4: Actual coverage level of Value-at-Risk for S&P 500 at one day, estimated with GARCH in-sample, GARCH out-of-sample, historical sim-ulation (HS30, HS50, HS100, HS200), moving window GLD with RS pa-rameterisation (RS30, RS50, RS100, RS200), moving window GLD with FMKL parametrisation (FMKL30, FMKL50, FMKL100, FMKL200) and time-varying GLD (GLD-TV) for S&P 500 index. The percentages in bold-face highlight the techniques that gave the best coverage levels. GARCH

in-sample is not considered. ^∗Values rejected by the conditional coverage test

(Christoersen, 1998).

all sampling period stationary period

99% 95% 90% 99% 95% 90%

GARCHis 97.94 94.93 91.34 98.13 94.88 91.00 GARCHos 96.09 93.24 89.86 98.75 95.25 91.25 HS30 96.55 93.25 88.77 96.50 93.50 89.50 HS50 97.94 93.39 88.11 98.00 93.63 89.25 HS100 98.24 93.61 88.77 98.63 94.25 90.50 HS200 98.46 93.25 88.11

98.88 95.00 90.63 RS30 97.14 93.17 89.06 97.25 92.63 88.50 RS50 97.72 93.47 88.84 98.25 93.25 89.50 RS100 98.02 93.39 88.55 98.38 94.00 89.88 RS200 98.24 93.02 87.89 98.75 94.88 90.00 FMKL30 96.70 93.17 89.28 96.63 92.75 88.75 FMKL50 97.58 93.47 88.77 97.75 93.38 89.38 FMKL100 98.09 93.02

88.47 98.50 93.75

89.75 FMKL200 98.24 92.95 87.81

98.75 94.75 89.88

GLD-TV 91.38 85.13 79.88

Table 5.5: Actual coverage level of Value-at-Risk for FT 30 at one day, estimated with GARCH in-sample, GARCH out-of-sample, historical sim-ulation (HS30, HS50, HS100, HS200), moving window GLD with RS pa-rameterisation (RS30, RS50, RS100, RS200), moving window GLD with FMKL parametrisation (FMKL30, FMKL50, FMKL100, FMKL200) and time-varying GLD (GLD-TV) for FT 30 index. The percentages in boldface highlight the techniques that gave the best coverage levels. GARCH in-sample

is not considered. ^∗Values rejected by the conditional coverage test

(Christof-fersen, 1998).

Chapter 6 Conclusions

In this thesis we addressed the following research questions.

• When dealing with nancial returns, it is important to have an idea of the underlying DGP in order to choose a good model. Which is a good strategy to conduct exploratory analyses?

• Is time-varying skewness relevant in the estimation of VaR?

• Until now GLDs have not been used, to our knowledge, to estimate VaR. Are these distributions bringing improvement in the estimation of VaR?

We applied our techniques to the returns of three famous nancial indexes, ASX 200, S&P 500 and FT 30, introduced in Chapter 4.

We have seen in Chapter 2 that the literature is very rich with methods to model and forecast nancial returns and VaR. There is not a universal best method, however, it is usually known that a particular method is more suitable to a particular situation. For this reason, we deem the importance of exploratory analyses to help highlight the characteristics of the underlying DGP and to choose a suitable model. In this thesis we used local linear

151

regression to model the rst three moments conditionally. The choice of a semi-parametric method has advantages and drawbacks. On one hand, it is not necessary to specify a model; thus there is no risk of misspecication.

On the other hand, we estimated the third moment of the residuals, i.e., the series of returns standardised with estimated mean and standard deviation:

the accumulated noise of the rst two estimation steps does not allow one to understand properly the eects of skewness on the returns. The local linear regression's estimates of mean and variance were quite encouraging:

they highlighted a growing conditional dependence when the market became more unstable due to the nancial crisis and the standardised series seemed to be closer to white noise, suggesting that some dependence was detected by the local linear regression analysis.

The issue of time-varying skewness is an open problem. Local linear regression analysis was not very helpful as we discussed in the previous para-graph. We handled the problem with a dierent technique in Chapter 5, using the GLDs. They are a family of distributions dened through their in-verse distribution function and depend on four parameters, λ1, λ₂, λ₃, λ₄. We considered the two parameterisations RS and FMKL. Estimating the four parameters over a moving window of variable length (w = 30, 50, 100, 200) allowed us to estimate skewness and kurtosis over time. The results were very interesting. The two para-moments seem to be highly correlated with each other and both varied over time. The skewness was mainly negative, whereas the kurtosis had large spikes corresponding to large negative val-ues of the skewness. For some valval-ues of the parameters the corresponding distribution may not have all moments, i.e., whenever min(λ3, λ₄) > −1/k then moments up to k order exist: it is often the case that third and fourth moments, and consequently skewness and kurtosis, do not exist. It would be

153

interesting in a future work to check if the GLDfmkl is able to recognise or predict the existence of moments. Generating independent data from Stu-dent t-distributions we could control the existence of the rst t moments and verify if the GLDfmkl recognises the existence of the same moments. We could observe some peaks in the series of skewness and kurtosis, followed by periods where the two moments do not exist. This nding contributes to the debate about existence of high order moments in nancial data: we deem that the third and fourth moments exist sometimes but not all the time.

This means also that the conditional distribution changes over time and one of the aspects that varies is the existence of the moments.

We conrmed the ndings of Karian and Dudewicz (1999), i.e., dierent sets of parameters for the GLDrs span very similar distributions and in par-ticular the maximum likelihood method of Su (2007) struggle in the conver-gence. The series of λ2, λ3, λ4 jump very often and simultaneously, however the series of the variance σ², which depends on these three parameters, is much smoother.

Overall the GLDs are exible and adapted very well in modeling the distribution of nancial returns, characterised but skewness and often heavy tails. Given this good results, we tried to estimate VaR with GLDs and the results were satisfying, very closed and often better than results obtained with historical simulation and with GARCH(1, 1) process.

In Chapter 5, we extended the GLDs to include a time-varying compo-nent. The time-varying GLDs did not provide interesting results in estimat-ing VaR. We deem that the time-varyestimat-ing GLDs could be useful tool, however it is necessary to improve the maximisation of the likelihood function, which is a challenging task since the function depends on seven parameters.

Appendix A

Likelihood equalities for the GLD