Capitulo 3 Propuesta de Procedimiento
3.11 Fase Depuración y Cambios; Ejecución de las pruebas de Integración
In this section, we estimate VaR with our two new methods: the time-varying GLDs introduced in Section 5.1 and the moving window GLDs introduced in the previous section. As a benchmark, we estimate VaR with two classical methods, historical simulation (Section 2.4) and the GARCH model (Section 2.6). Both methods are well known and widely used because they are easy and fast to apply but they still provide satisfactory results. In the case of GARCH(1,1), we consider two dierent situations. Firstly, we use the whole sample to estimate the parameters and we test the coverage on the same data (GARCHis, where is means in-sample). Secondly, to have more realistic results (following a suggestion of Poon and Granger, 2003), we divide the sample in two parts: the rst to estimate the parameters and the second to check the coverage sample (GARCHos, where os means out-of-sample). As regards the historical simulation method, we apply it using four dierent windows of length 30, 50, 100, 200 (HS30, HS50, HS100, HS200, respectively), so that we can consistently compare the results with the moving window GLDs.
Last but not least, we test each technique on the whole sample (1 January 2003 to 31 December 2008) even though the process is not stationary and a large structural change occurred around July 2008 because of the world
nancial crisis. However, a method is more appealing if it can react and adapt quickly to new situations. We also compare the methodologies during more stable periods. Each stationary period was selected in the previous chapter using simple empirical graphical techniques.
The results are shown in Tables 5.3-5.5. The best forecast (highlighted in boldface) is contended between historical simulation and moving window GLD. In selecting the best option we did not include the GARCH-is method
85 Estimating Value-at-Risk
because it is not a realistic situation (usually we do not forecast something that already happened).
For the ASX 200 index, the two GLDs are always the best choice. His-torical simulation provides results only slightly worse. The S&P 500 index oers a similar scenario with historical simulation being the best choice just in one case. It is worth mentioning the excellent results in the stationary period with window length 200, where all the three observed coverages are slightly conservative. The third index, FT 30, has a dierent situation. The moving window GLD-rs turns out to be the best choice in two cases, even though the other performances are slightly inferior than the best one. It would have probably been better to analyse the FTSE 100 index which is consistent with the other indices and it is more commonly used, whereas the FT 30 is a bit obsolete. This may also explain the dierences in the results mentioned above. Overall, RS parameterisation seems to work better but the dierence with FMKL parameterisation is always very small.
Dividend eects on indices were not taken into account, however a more realistic evaluation of risk should include these in the future.
The time-varying GLDrs does not give any good result. The VaR is al-ways very underestimated even if we are considering in-sample forecasting.
We suspect that the time-varying GLDrs could be improved by improving the estimation procedure. As regards the time-varying with FMKL parame-terisation, it is necessary to nd a good estimation method, because the one we used does not converge to suitable values.
It is still an open problem for moving window GLDs, as for historical simulation, the choice of the best window. The choice of the window length depends on the characteristics of the series. If the series of returns is station-ary, it is better to base VaR estimation on a large window, which gives a more
accurate description of the distribution. On the other hand, if the series is very unstable with frequent structural changes, a short window adapts faster to new scenarios. It appears that long windows are, in general, preferable even during unstable periods. We must not forget that the GLDs require the estimate of four parameters and we gain much in precision with long win-dows. It is also known from the literature that estimating the parameters of the GLDs is not an easy task because the likelihood function has many local maxima.
In the Tables 5.3-5.5, an asterisk highlight results that were rejected by the independence test, whereas the value in boldface indicates the best method not rejected by the test for independence. Since the coverage of the intervals to estimate VaR are often far from the real coverage, we decided not to test for coverage because this would reject most of the methods.
Overall, we can say that the moving window GLDs bring improvements in the estimation of VaR if compared to the two traditional techniques consid-ered here. Moving window GLDs have the advantage of being implemented in the GLDEX package (Su, 2007) of free software R (RDevelopmentCoreTeam, 2009).
87 Estimating Value-at-Risk
−5050
0.51
1.52x 10−3λ 1, FMKL
−5050.5
11.5
22.5x 10−3λ 1, RS −505150200250300350400λ 2, FMKL
−505−140
−120
−100−80
−60
−40λ 2, RS −505−0.5−0.4−0.3−0.2−0.1
0λ 3, FMKL
−505−0.45
−0.4−0.35
−0.3−0.25
−0.2λ 3, RS −505−0.4−0.3−0.2−0.1
0
0.1λ 4, FMKL
−505−0.4
−0.35
−0.3−0.25
−0.2−0.15λ 4, RSASX 200 Figure5.1:Q-Qplotofthebootstrapdistributionsoftheparametersλ1,λ2,λ3,λ4fortheindexASX200,toprowRS parameterisation,bottomrowFMKLparameterisation.
−505 0.5 1 1.5 2 x 10 −3 λ1 , FMKL −505 0 1 2 3 4 x 10 −3λ1 , RS
−505 250 300 350 400 λ2 , FMKL −505 −60 −40 −20 0 20 40 λ2 , RS
−505 −0.3 −0.2 −0.1 0 0.1 λ3 , FMKL −505 −0.2 −0.1 0 0.1 0.2 λ3 , RS
−505 −0.1 −0.05 0 0.05 0.1 0.15 λ4 , FMKL −505 −0.2 −0.1 0 0.1 0.2 λ4 , RS ASX200
Figure5.2:Q-Qplotofthebootstrapdistributionsoftheparametersλ1,λ2,λ3,λ4fortheindexASX200,toprowRSparameterisation,bottomrowFMKLparameterisation.
89 Estimating Value-at-Risk
−5050
0.51
1.5x 10−3λ 1, FMKL
−5050
0.51
1.52x 10−3λ 1, RS −505200250300350λ 2, FMKL
−505−140
−120
−100−80
−60
−40λ 2, RS −505−0.5−0.4−0.3−0.2−0.1λ 3, FMKL
−505−0.5
−0.4
−0.3
−0.2
−0.1λ 3, RS −505−0.4−0.3−0.2−0.1
0λ 4, FMKL
−505−0.5
−0.4
−0.3
−0.2
−0.1λ 4, RSS&P 500 Figure5.3:Q-Qplotofthebootstrapdistributionsoftheparametersλ1,λ2,λ3,λ4fortheindexS&P500,toprowRS parameterisation,bottomrowFMKLparameterisation.
−505 −5 0 5 10 15 x 10 −4 λ1 , FMKL −505 −1 0 1 2 3 x 10 −3λ1 , RS
−505 200 220 240 260 280 300 λ2 , FMKL −505 −40 −20 0 20 40 λ2 , RS
−505 −0.2 −0.1 0 0.1 0.2 λ3 , FMKL −505 −0.2 −0.1 0 0.1 0.2 λ3 , RS
−505 −0.1 0 0.1 0.2 0.3 λ4 , FMKL −505 −0.1 −0.05 0 0.05 0.1 0.15 λ4 , RS S&P 500
Figure5.4:Q-Qplotofthebootstrapdistributionsoftheparametersλ1,λ2,λ3,λ4fortheindexS&P500,toprowRSparameterisation,bottomrowFMKLparameterisation.
91 Estimating Value-at-Risk
−5050
0.51
1.5x 10−3λ 1, FMKL
−5050
0.51
1.52x 10−3λ 1, RS −505200220240260280300λ 2, FMKL
−505−120
−100−80
−60
−40λ 2, RS −505−0.5−0.4−0.3−0.2−0.1λ 3, FMKL
−505−0.45
−0.4−0.35
−0.3−0.25
−0.2λ 3, RS −505−0.4
−0.35
−0.3−0.25
−0.2−0.15λ 4, FMKL
−505−0.4
−0.35
−0.3−0.25
−0.2−0.15λ 4, RSFT 30 Figure5.5:Q-Qplotofthebootstrapdistributionsoftheparametersλ1,λ2,λ3,λ4fortheindexFT30,toprowRS parameterisation,bottomrowFMKLparameterisation.
−505 0 0.5 1 1.5 2 x 10 −3 λ1 , FMKL −505 0 1 2 3 x 10 −3λ1 , RS
−505 200 250 300 350 λ2 , FMKL −505 −60 −40 −20 0 20 40 λ2 , RS
−505 −0.3 −0.2 −0.1 0 0.1 λ3 , FMKL −505 −0.3 −0.2 −0.1 0 0.1 0.2 λ3 , RS
−505 −0.2 −0.1 0 0.1 0.2 λ4 , FMKL −505 −0.2 −0.1 0 0.1 0.2 λ4 , RS FT 30
Figure5.6:Q-Qplotofthebootstrapdistributionsoftheparametersλ1,λ2,λ3,λ4fortheindexFT30,toprowRSparameterisation,bottomrowFMKLparameterisation.
93 Estimating Value-at-Risk
GLD−RS Quantiles −0.1−0.1 −0.05 0 0.05 0.1
−0.05 0 0.05 0.1
GLD−FMKL Quantiles
Figure 5.7: Q-Q plots comparing theoretical distribution of ASX 200 index and estimated distributions: standard Normal distribution, GLDrs distribu-tion and GLDfmkl distribudistribu-tion, respectively.
−5 0 5
GLD−RS Quantiles −0.2−0.2 −0.1 0 0.1 0.2
−0.15
Figure 5.8: Q-Q plots comparing theoretical distribution of S&P 500 index and estimated distributions: standard Normal distribution, GLDrs distribu-tion and GLDfmkl distribudistribu-tion, respectively.
−5 0 5
Figure 5.9: Q-Q plots comparing theoretical distribution of FT 30 index and estimated distributions: standard Normal distribution, GLDrs distribu-tion and GLDfmkl distribudistribu-tion, respectively.
−0.0500.05 0 50 100 24−Oct−2003
−0.0200.02 0 50 100 10−Jun−2004
−0.0200.02 0 50 100 25−Jan−2005
−0.0200.02 0 50 100 13−Sep−2005
−0.0500.05 0 20 40 60 04−May−2006
−0.0500.05 0 20 40 60 15−Dec−2006
−0.0500.05 0 20 40 60 08−Aug−2007
−0.100.1 0 10 20 30 27−Mar−2008
−0.100.1 0 10 20 30 10−Nov−2008 ASX 200
Figure5.10:Comparisonbetweentheobserveddistribution(solidline)andthedistributionsobtainedbytheGLDesti-mateswithRSparameterisationfortwowindows,w=100,200fortheindexASX200(dottedanddashedlines,respec-tively).
95 Estimating Value-at-Risk
−0.0200.020
5010024−Oct−2003 −0.0200.020
50100
10−Jun−2004 −0.0200.020
5010025−Jan−2005 −0.0200.020
5010013−Sep−2005 −0.0500.050
20
40
6004−May−2006 −0.0500.050
20
40
6015−Dec−2006 −0.0500.050
20
40
6008−Aug−2007 −0.100.10
20
4027−Mar−2008 −0.100.10
10
20
10−Nov−2008
ASX 200 Figure5.11:Comparisonbetweentheobserveddistribution(solidline)andthedistributionsobtainedbytheGLDes- timateswithFMKLparameterisationfortwowindows,w=100,200fortheindexASX200(dottedanddashedlines, respectively).
−0.0500.05 0 20 40 30−Oct−2003
−0.0500.05 0 20 40 60 22−Jun−2004
−0.0500.05 0 20 40 60 80 08−Feb−2005
−0.0200.02 0 20 40 60 27−Sep−2005
−0.0200.02 0 50 100 17−May−2006
−0.0200.02 0 20 40 60 80 05−Jan−2007
−0.0500.05 0 50 100 24−Aug−2007
−0.100.1 0 20 40 15−Apr−2008
−0.200.2 0 10 20 30 01−Dec−2008 S&P 500
Figure5.12:Comparisonbetweentheobserveddistribution(solidline)andthedistributionsobtainedbytheGLDesti-mateswithRSparameterisationfortwowindows,w=100,200fortheindexS&P500(dottedanddashedlines,respectively).
97 Estimating Value-at-Risk
−0.0500.050
20
4030−Oct−2003 −0.0200.020
20
40
6022−Jun−2004 −0.0200.020
20
40
60
8008−Feb−2005 −0.0200.020
20
40
60
27−Sep−2005 −0.0200.020
5010017−May−2006 −0.0200.020
50100
05−Jan−2007 −0.0500.050
20
40
60
8024−Aug−2007 −0.0500.050
20
4015−Apr−2008 −0.200.20
10
20
3001−Dec−2008
S&P 500 Figure5.13:Comparisonbetweentheobserveddistribution(solidline)andthedistributionsobtainedbytheGLDes- timateswithFMKLparameterisationfortwowindows,w=100,200fortheindexS&P500(dottedanddashedlines, respectively).
−0.100.1 0 20 40 21−Oct−2003
−0.100.1 0 20 40 60 01−Jun−2004
−0.0500.05 0 20 40 60 11−Jan−2005
−0.0200.02 0 50 100 23−Aug−2005
−0.0200.02 0 20 40 60 80 04−Apr−2006
−0.04−0.0200.020.04 0 20 40 60 14−Nov−2006
−0.0500.05 0 20 40 60 26−Jun−2007
−0.100.1 0 20 40 05−Feb−2008
−0.100.1 0 10 20 30 16−Sep−2008 FT 30
Figure5.14:Comparisonbetweentheobserveddistribution(solidline)andthedistributionsobtainedbytheGLDesti-mateswithRSparameterisationfortwowindows,w=100,200fortheindexFT30(dottedanddashedlines,respectively).
99 Estimating Value-at-Risk
−0.100.10
20
4021−Oct−2003 −0.0500.050
20
40
6001−Jun−2004 −0.0500.050
20
40
60
11−Jan−2005 −0.0200.020
5010023−Aug−2005 −0.0500.050
20
40
60
8004−Apr−2006 −0.0500.050
20
40
60
14−Nov−2006 −0.0500.050
20
40
60
8026−Jun−2007 −0.100.10
20
4005−Feb−2008 −0.100.10
10
20
3016−Sep−2008
FT 30 Figure5.15:Comparisonbetweentheobserveddistribution(solidline)andthedistributionsobtainedbytheGLDesti- mateswithFMKLparameterisationfortwowindows,w=100,200fortheindexFT30(dottedanddashedlines,respec- tively).
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.1 −0.05 0 0.05 ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.05 0 0.05
λ1
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −1000 −500 0
λ2
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −1 0 1
λ3
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −1 0 1 2
λ4
Figure5.16:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=30forASX200index.
101 Estimating Value-at-Risk
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009−0.1
−0.05
0
0.05
ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
0
0.050.1
µ −10 10 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
100
2 σ −500 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
0
500 γ 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
100
κ
Figure5.17:Seriesofpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength,w=30for ASX200index.
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.1 −0.05 0 0.05 ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.01 0 0.01
λ1
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 0 500λ2 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −1 0 1
λ3
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −1 0 1
λ4
Figure5.18:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=30forASX200index.
103 Estimating Value-at-Risk
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009−0.1
−0.05
0
0.05
ASX200 −3 x 10 5 0 −5 −10 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
µ −6 10 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
10−4
10−2
2 σ −20 −40 −60 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
0
γ 0 10 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
102
κ
Figure5.19:Seriesofpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength,w=30 forASX200index.
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.1 −0.05 0 0.05 ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.04 −0.02 0 0.02
λ1
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −300 −200 −100 0 100
λ2
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −1 0 1
λ3
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.5 0 0.5 1
λ4
Figure5.20:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=50forASX200index.
105 Estimating Value-at-Risk
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009−0.1
−0.05
0
0.05
ASX200 −3 x 10 4 2 0 −2 −4 −6 −8 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
µ
12−Oct−200309−Jul−200506−Apr−200701−Jan−200910−5
100
2 σ −500 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
0
500 γ 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
100
κ
Figure5.21:Seriesofpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength,w=50for ASX200index.
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.1 −0.05 0 0.05 ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −5 0 5 x 10 −3
λ1
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 0 500λ2 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.5 0 0.5 1
λ3
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.5 0 0.5 1
λ4
Figure5.22:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=50forASX200index.
107 Estimating Value-at-Risk
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009−0.1
−0.05
0
0.05
ASX200 −3 x 10 5 0 −5 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
µ
12−Oct−200309−Jul−200506−Apr−200701−Jan−200910−6
10−4
10−2
2 σ −20 −40 −60 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
0
γ 0 10 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
102
κ
Figure5.23:Seriesofpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength,w=50for ASX200index.
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.1 −0.05 0 0.05 ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.01 0 0.01
λ1
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −50 0 50
λ2
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.2 0 0.2 0.4
λ3
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.2 0 0.2
λ4
Figure5.24:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=100forASX200index.
109 Estimating Value-at-Risk
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009−0.1
−0.05
0
0.05
ASX200 −3 x 10 2 0 −2 −4 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
µ
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
10−4
2 σ −500 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
0
500 γ 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
100
κ
Figure5.25:Seriesofpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ4,forwindowslength,w=100for ASX200index.
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.1 −0.05 0 0.05 ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −4 −2 0 2 x 10 −3
λ1
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 0 500λ2 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.2 0 0.2
λ3
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.2 0 0.2 0.4
λ4
Figure5.26:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=100forASX200index.
111 Estimating Value-at-Risk
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009−0.1
−0.05
0
0.05
ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
−4
−20
2x 10−3
µ
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
10−4
10−3
2 σ 1 0 −1 −2 −3 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
γ
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009100
102
κ
Figure5.27:Seriesofpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength,w=100 forASX200index.
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.1 −0.05 0 0.05 ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −5 0 5 x 10 −3
λ1
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −50 0 50
λ2
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.2 0 0.2
λ3
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.2 0 0.2
λ4
Figure5.28:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=200forASX200index.
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12−Oct−200309−Jul−200506−Apr−200701−Jan−2009−0.1
−0.05
0
0.05
ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
−20
2x 10−3
µ
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
10−4
2 σ −500 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
0
500 γ 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
100
κ
Figure5.29:Seriesofpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength,w=200for ASX200index.
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.1 −0.05 0 0.05 ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −2 0 2 x 10 −3
λ1
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 0 500λ2 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.2 0 0.2
λ3
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009 −0.2 0 0.2
λ4
Figure5.30:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=200forASX200index.
115 Estimating Value-at-Risk
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009−0.1
−0.05
0
0.05
ASX200 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
−20
2x 10−3
µ
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
10−4
10−3
2 σ 1 0 −1 −2 12−Oct−200309−Jul−200506−Apr−200701−Jan−2009
γ
12−Oct−200309−Jul−200506−Apr−200701−Jan−2009100
102
κ
Figure5.31:Seriesofpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength,w=200 forASX200index.
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.1 0 0.1 S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.05 0 0.05
λ1
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −800 −600 −400 −200 0
λ2
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −1 0 1
λ3
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.5 0 0.5 1 1.5
λ4
Figure5.32:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=100forS&P500index.
117 Estimating Value-at-Risk
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009−0.1
00.1
S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
−0.01
0
0.01 µ 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
10−4
10−2
2 σ −200 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
0
200 γ 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
100
κ
Figure5.33:Seriesofgeneralisedpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=100forS&P500index.
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.1 0 0.1 S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.02 0 0.02
λ1
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 0 500 1000λ2 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −1 0 1
λ3
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.5 0 0.5 1 1.5
λ4
Figure5.34:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=100forS&P500index.
119 Estimating Value-at-Risk
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S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
−0.01
0
0.01 µ 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
10−4
10−2
2 σ 5 0 −5 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
γ
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009100
102
κ
Figure5.35:Seriesofgeneralisedpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=100forS&P500index.
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.1 0 0.1 S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.04 −0.02 0 0.02
λ1
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −400 −200 0
λ2
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −1 0 1
λ3
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.5 0 0.5 1
λ4
Figure5.36:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=200forS&P500index.
121 Estimating Value-at-Risk
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S&P500 −3 x 10 0 −5 −10 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
µ 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
10−4
10−2
2 σ −200 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
0
200 γ 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
100
κ
Figure5.37:Seriesofgeneralisedpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=200forS&P500index.
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.1 0 0.1 S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.02 −0.01 0 0.01
λ1
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 0 500λ2 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.5 0 0.5 1
λ3
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 0 0.5 1 1.5λ4
Figure5.38:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=200forS&P500index.
123 Estimating Value-at-Risk
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S&P500 −3 x 10 0 −5 −10 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
µ 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
10−4
10−2
2 σ 5 0 −5 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
γ
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009100
102
κ
Figure5.39:Seriesofgeneralisedpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=200forS&P500index.
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.1 0 0.1 S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.01 0 0.01
λ1
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −800 −600 −400 −200 0
λ2
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.5 0 0.5
λ3
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.5 0 0.5
λ4
Figure5.40:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=100forS&P500index.
125 Estimating Value-at-Risk
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S&P500 −3 x 10 2 0 −2 −4 −6 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
µ
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
10−4
10−2
2 σ −200 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
0
200 γ 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
100
κ
Figure5.41:Seriesofgeneralisedpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=100forS&P500index.
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.1 0 0.1 S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −4 −2 0 2 x 10 −3
λ1
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 0 500λ2 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.5 0 0.5
λ3
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.4 −0.2 0 0.2 0.4
λ4
Figure5.42:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=100forS&P500index.
127 Estimating Value-at-Risk
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S&P500 −3 x 10 2 0 −2 −4 −6 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
µ
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
10−4
2 σ 0 −5 −10 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
γ 0 10 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
102
κ
Figure5.43:Seriesofgeneralisedpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=100forS&P500index.
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.1 0 0.1 S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −2 0 2 4 x 10 −3
λ1
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −400 −200 0
λ2
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.9 −0.25 0.4
λ3
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.6 −0.4 −0.2 0 0.2
λ4
Figure5.44:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=200forS&P500index.
129 Estimating Value-at-Risk
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S&P500 −3 x 10 2 0 −2 −4 −6 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
µ
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
10−4
10−2
2 σ −200 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
0
200 γ 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
100
κ
Figure5.45:Seriesofgeneralisedpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=200forS&P500index.
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.1 0 0.1 S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −1 0 1 2 x 10 −3
λ1
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 0 500λ2 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.4 −0.2 0 0.2
λ3
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009 −0.4 −0.2 0 0.2
λ4
Figure5.46:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=200forS&P500index.
131 Estimating Value-at-Risk
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S&P500 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
−20
2x 10−3
µ
15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
10−4
2 σ 0 −5 −10 −15 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
γ 0 10 15−Oct−200311−Jul−200507−Apr−200701−Jan−2009
102
κ
Figure5.47:Seriesofgeneralisedpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=200forS&P500index.
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.1 0 0.1 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.05 0 0.05 FT30
λ1
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −400 −200 0
λ2
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −1 0 1
λ3
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.5 0 0.5 1
λ4
Figure5.48:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=100forFT30index.
133 Estimating Value-at-Risk
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FT30 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
−0.01
0
0.01 µ 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
10−4
10−2
2 σ −500 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
0
500 γ 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
100
κ
Figure5.49:Seriesofgeneralisedpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=100forFT30index.
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.1 0 0.1 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.02 −0.01 0 0.01 FT30
λ1
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 0 500λ2 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −1 0 1
λ3
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.5 0 0.5 1 1.5
λ4
Figure5.50:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=100forFT30index.
135 Estimating Value-at-Risk
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FT30 −0.02 −0.04 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
0
µ 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
10−4
10−2
2 σ −10 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
010
γ 0 10 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
102
κ
Figure5.51:Seriesofgeneralisedpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=100forFT30index.
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.1 0 0.1 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.04 −0.02 0 0.02 FT30
λ1
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −200 −100 0 100
λ2
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.5 0 0.5 1
λ3
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.5 0 0.5 1
λ4
Figure5.52:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=200forFT30index.
137 Estimating Value-at-Risk
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FT30 −3 x 10 5 0 −5 −10 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
µ 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
10−4
10−2
2 σ −500 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
0
500 γ 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
100
κ
Figure5.53:Seriesofgeneralisedpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=200forFT30index.
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.1 0 0.1 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −10 −5 0 5 x 10 −3 FT30
λ1
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 0 500λ2 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.5 0 0.5 1
λ3
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.5 0 0.5 1
λ4
Figure5.54:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=200forFT30index.
139 Estimating Value-at-Risk
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FT30 −3 x 10 5 0 −5 −10 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
µ 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
10−4
2 σ −10 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
010
γ 0 10 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
102
κ
Figure5.55:Seriesofgeneralisedpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=200forFT30index.
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.1 0 0.1 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.01 0 0.01 FT30
λ1
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −100 −50 0 50
λ2
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.5 0 0.5
λ3
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.4 −0.2 0 0.2 0.4
λ4
Figure5.56:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=100forFT30index.
141 Estimating Value-at-Risk
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FT30 −3 x 10 5 0 −5 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
µ
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
10−4
10−2
2 σ −500 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
0
500 γ 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
100
κ
Figure5.57:Seriesofgeneralisedpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=100forFT30index.
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.1 0 0.1 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −6 −4 −2 0 2 x 10 −3 FT30
λ1
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 0 200 400λ2 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.5 0 0.5
λ3
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.4 −0.2 0 0.2 0.4
λ4
Figure5.58:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=100forFT30index.
143 Estimating Value-at-Risk
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FT30 −3 x 10 2 0 −2 −4 −6 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
µ
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
10−4
2 σ 0 −5 −10 −15 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
γ 0 10 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
102
κ
Figure5.59:Seriesofgeneralisedpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=100forFT30index.
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.1 0 0.1 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −6 −4 −2 0 2 4 x 10 −3 FT30
λ1
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −100 −50 0
λ2
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.4 −0.2 0 0.2
λ3
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.2 −0.1 0 0.1
λ4
Figure5.60:SeriesofgeneralisedlambdaestimatesusingRSparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=200forFT30index.
145 Estimating Value-at-Risk
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FT30 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
−20
2x 10−3
µ
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
10−4
10−2
2 σ −500 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
0
500 γ 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
100
κ
Figure5.61:Seriesofgeneralisedpara-momentsestimatesusingRSparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=200forFT30index.
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.1 0 0.1 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −2 0 2 x 10 −3 FT30
λ1
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 0 200 400λ2 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.4 −0.2 0 0.2
λ3
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009 −0.2 0 0.2
λ4
Figure5.62:SeriesofgeneralisedlambdaestimatesusingFMKLparameterisation,λ1,λ2,λ3,λ4,forwindowslength,w=200forFT30index.
147 Estimating Value-at-Risk
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FT30 −3 x 10 2 0 −2 −4 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
µ
06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
10−4
2 σ 0 −5 −10 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
γ 0 10 06−Oct−200305−Jul−200504−Apr−200701−Jan−2009
102
κ
Figure5.63:Seriesofgeneralisedpara-momentsestimatesusingFMKLparameterisation,µ,σ2 ,γ,κ,forwindowslength, w=200forFT30index.