The document contains output from analyzing stock return data for Amazon (AMZN) and Coca-Cola (KO) using various GARCH and EVT models. Key results include: 1) Estimating a GARCH(1,1) model for AMZN returns, finding significance for the coefficients and a 99% VaR of 0.0630. 2) Fitting an EVT GEV model to positive AMZN returns, estimating shape and scale parameters and calculating a 99.9% VaR of 0.1584. 3) Estimating GPD models to AMZN returns above thresholds of 0.03 and 0.045, and calculating 99% VaRs of

1. #HW6, R output
> getSymbols("AMZN",from="2004-01-05",to="2012-05-18")
[1] "AMZN"
> amzn=diff(log(as.numeric(AMZN$AMZN.Adjusted)))
> source("Igarch.R")
> dim(AMZN)
[1] 2110 6
> length(amzn)
[1] 2109
> plot(amzn,type='l')
#Problem 1
> m1=Igarch(amzn)
Estimates: 0.999999
Maximized log-likehood: -4504.73
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
beta 0.999999 NA NA NA
Warning message:
In sqrt(diag(solve(Hessian))) : NaNs produced
> v1=m1$volatility
> v=.999999*v1[2109]^2+(1-.999999)*amzn[2109]^2
> v
[1] 0.0008170621
> vo=sqrt(v)
> vo
[1] 0.0285843
> source("RMeasure.R")
> RMeasure(0,vo)
Risk Measures for selected probabilities:
prob VaR ES
[1,] 0.950 0.04701699 0.05896120
[2,] 0.990 0.06649702 0.07618328

2. [3,] 0.999 0.08833212 0.09624591
> require(fGarch)
> t.test(amzn)
One Sample t-test
data: amzn
t = 1.0588, df = 2108, p-value = 0.2898
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
-0.0005615692 0.0018796360
sample estimates:
mean of x
0.0006590334
> m2=garchFit(~garch(1,1),data=amzn,include.mean=F,trace=F)
> summary(m2)
Title:
GARCH Modelling
Call:
garchFit(formula = ~garch(1, 1), data = amzn, include.mean =
F,
trace = F)
Mean and Variance Equation:
data ~ garch(1, 1)
[data = amzn]
Conditional Distribution:
norm
Coefficient(s):
omega alpha1 beta1
8.1295e-05 4.4142e-02 8.5634e-01

3. Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
omega 8.130e-05 2.587e-05 3.142 0.00168 **
alpha1 4.414e-02 1.458e-02 3.028 0.00246 **
beta1 8.563e-01 4.334e-02 19.759 < 2e-16 ***
---
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 22801.82 0
Shapiro-Wilk Test R W 0.877114 0
Ljung-Box Test R Q(10) 6.174051 0.8004343
Ljung-Box Test R Q(15) 11.60764 0.7084517
Ljung-Box Test R Q(20) 18.48538 0.555467
Ljung-Box Test R^2 Q(10) 1.863959 0.9972804
Ljung-Box Test R^2 Q(15) 2.104383 0.9999585
Ljung-Box Test R^2 Q(20) 2.546816 0.999999
LM Arch Test R TR^2 1.858131 0.9995944
> predict(m2,1)
meanForecast meanError standardDeviation
1 0 0.02708331 0.02708331
> RMeasure(0,.027083)
Risk Measures for selected probabilities:
prob VaR ES
[1,] 0.950 0.04454757 0.05586445
[2,] 0.990 0.06300448 0.07218200
[3,] 0.999 0.08369276 0.09119090
>
m3=garchFit(~garch(1,1),data=amzn,include.mean=F,trace=F,co
nd.dist="std")
> summary(m3)

4. Title:
GARCH Modelling
Call:
garchFit(formula = ~garch(1, 1), data = amzn, cond.dist =
"std",
include.mean = F, trace = F)
Mean and Variance Equation:
data ~ garch(1, 1)
[data = amzn]
Conditional Distribution:
std
Coefficient(s):
omega alpha1 beta1 shape
5.4010e-06 2.0039e-02 9.7171e-01 3.6910e+00
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
omega 5.401e-06 2.351e-06 2.297 0.0216 *
alpha1 2.004e-02 4.550e-03 4.404 1.06e-05 ***
beta1 9.717e-01 6.252e-03 155.428 < 2e-16 ***
shape 3.691e+00 2.805e-01 13.158 < 2e-16 ***
---
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 44179.67 0
Shapiro-Wilk Test R W 0.8497441 0
Ljung-Box Test R Q(10) 6.935119 0.7315556
Ljung-Box Test R Q(15) 11.39435 0.7241714

5. Ljung-Box Test R Q(20) 17.60512 0.6134033
Ljung-Box Test R^2 Q(10) 2.171012 0.9948534
Ljung-Box Test R^2 Q(15) 2.921834 0.9996587
Ljung-Box Test R^2 Q(20) 3.835627 0.9999671
LM Arch Test R TR^2 2.402731 0.9984912
> predict(m3,1)
meanForecast meanError standardDeviation
1 0 0.02602888 0.02602888
> RMeasure(0,0.02603,cond.dist="std",df=3.691)
Risk Measures for selected probabilities:
prob VaR ES
[1,] 0.950 0.05685405 0.08725013
[2,] 0.990 0.10220280 0.14580552
[3,] 0.999 0.20346447 0.28199876
>
#Problem 2
> require(evir)
> namzn=-amzn #Long position
> m4=gev(namzn,21)
> m4
$n.all
[1] 2109
$n
[1] 101
$data
[1] 0.07148198 0.03824780 0.04663128 0.05770617
0.02916481 0.03440878
...
[97] 0.08017129 0.03635603 0.02880931 0.02581640
0.02576880

6. $block
[1] 21
$par.ests
xi sigma mu
0.37144211 0.01645415 0.03269744
$par.ses
xi sigma mu
0.096305320 0.001609465 0.001883217
$varcov
[,1] [,2] [,3]
[1,] 9.274715e-03 4.039022e-06 -5.420633e-05
[2,] 4.039022e-06 2.590378e-06 2.064383e-06
[3,] -5.420633e-05 2.064383e-06 3.546507e-06
$converged
[1] 0
$nllh.final
[1] -233.9451
attr(,"class")
[1] "gev"
>
#
> source("evtVaR.R")
> evtVaR(0.37144211,0.01645415, 0.03269744)
[1] 0.06734551
> 0.06734551*10^(.37144)
[1] 0.1583976
#Problem 3
> m5=gpd(namzn,thres=0.03)
> m5

7. $n
[1] 2109
$data
[1] 0.03250727 0.07022383 0.05417415 0.04318093
0.07148198 0.03824780
.....
[169] 0.03549203 0.03011677 0.04870838 0.04579498
0.08017129 0.03635603
$threshold
[1] 0.03
$p.less.thresh
[1] 0.9174964
$n.exceed
[1] 174
$method
[1] "ml"
$par.ests
xi beta
0.34313804 0.01443334
$par.ses
xi beta
0.105037392 0.001811058
$varcov
[,1] [,2]
[1,] 0.0110328538 -1.194869e-04
[2,] -0.0001194869 3.279929e-06
$information

8. [1] "observed"
$converged
[1] 0
$nllh.final
[1] -503.7101
attr(,"class")
[1] "gpd"
> riskmeasures(m5,c(0.99))
p quantile sfall
[1,] 0.99 0.07470822 0.1200365
> m6=gpd(namzn,thres=0.045)
> m6
$n
[1] 2109
$data
[1] 0.07022383 0.05417415 0.07148198 0.04663128
0.05403311 0.05770617
....
[73] 0.04579498 0.08017129
$threshold
[1] 0.045
$p.less.thresh
[1] 0.9649123
$n.exceed
[1] 74
$method
[1] "ml"

9. $par.ests
xi beta
0.45764955 0.01647918
$par.ses
xi beta
0.20579974 0.00376538
$varcov
[,1] [,2]
[1,] 0.042353533 -5.446820e-04
[2,] -0.000544682 1.417809e-05
$information
[1] "observed"
$converged
[1] 0
$nllh.final
[1] -195.9504
attr(,"class")
[1] "gpd"
> riskmeasures(m6,c(0.99))
p quantile sfall
[1,] 0.99 0.07294936 0.1269185
>
#Problem 4
> getSymbols("KO",from="2004-01-05",to="2012-05-18")
[1] "KO"
> ko=diff(log(as.numeric(KO$KO.Adjusted)))
> mm1=Igarch(ko)
Estimates: 0.9462869
Maximized log-likehood: -6669.622

10. Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
beta 0.94628692 0.00850725 111.233 < 2.22e-16 ***
---
> v2=mm1$volatility
> length(ko)
[1] 2109
> vo=0.9463*v2[2109]^2+(1-.9463)*ko[2109]^2
> vo=sqrt(vo)
> vo
[1] 0.007926053
> RMeasure(0,vo)
Risk Measures for selected probabilities:
prob VaR ES
[1,] 0.950 0.01303720 0.01634917
[2,] 0.990 0.01843876 0.02112463
[3,] 0.999 0.02449335 0.02668773
> nko=-ko
> cor(namzn,nko)
[1] 0.3433976
>
> nko=-ko
> cor(namzn,nko)
[1] 0.3433976
>
> r1=amzn+ko
> r2=amzn-ko
> n1=Igarch(r1)
Estimates: 0.999999
Maximized log-likehood: -4098.254
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
beta 0.999999 NA NA NA

11. Warning message:
In sqrt(diag(solve(Hessian))) : NaNs produced
> vn1=n1$volatility
> n2=Igarch(r2)
Estimates: 0.999999
Maximized log-likehood: -4629.792
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
beta 0.999999 NA NA NA
Warning message:
In sqrt(diag(solve(Hessian))) : NaNs produced
> vn2=n2$volatility
> m1=Igarch(amzn)
Estimates: 0.999999
Maximized log-likehood: -4504.73
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
beta 0.999999 NA NA NA
Warning message:
In sqrt(diag(solve(Hessian))) : NaNs produced
> v1=m1$volatility
> cov1=(vn1[2109]^2-vn2[2109]^2)/4
> cov1
[1] 0.0001189115
> corr1=cov1/(v1[2109]*v2[2109])
> corr1
[1] 0.5624339
>
#Problem 5
> cor(amzn,nko)
[1] -0.3433976
>
> t.test(ko)

12. One Sample t-test
data: ko
t = 1.1073, df = 2108, p-value = 0.2683
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
-0.0002249640 0.0008085214
sample estimates:
mean of x
0.0002917787
>
k1=garchFit(~garch(1,1),data=nko,trace=F,include.mean=F,con
d.dist="std")
> summary(k1)
Title:
GARCH Modelling
Call:
garchFit(formula = ~garch(1, 1), data = nko, cond.dist = "std",
include.mean = F, trace = F)
Mean and Variance Equation:
data ~ garch(1, 1)
[data = nko]
Conditional Distribution:
std
Coefficient(s):
omega alpha1 beta1 shape
1.7747e-06 6.8873e-02 9.1784e-01 5.2960e+00
Std. Errors:

13. based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
omega 1.775e-06 6.549e-07 2.710 0.00673 **
alpha1 6.887e-02 1.454e-02 4.738 2.16e-06 ***
beta1 9.178e-01 1.680e-02 54.640 < 2e-16 ***
shape 5.296e+00 5.937e-01 8.920 < 2e-16 ***
---
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 2505.346 0
Shapiro-Wilk Test R W 0.9658619 0
Ljung-Box Test R Q(10) 9.347308 0.4994903
Ljung-Box Test R Q(15) 17.2665 0.3031844
Ljung-Box Test R Q(20) 19.89797 0.4643285
Ljung-Box Test R^2 Q(10) 7.080028 0.7178688
Ljung-Box Test R^2 Q(15) 7.736915 0.9338964
Ljung-Box Test R^2 Q(20) 10.5478 0.9571176
LM Arch Test R TR^2 7.44726 0.8266984
> predict(k1,1)
meanForecast meanError standardDeviation
1 0 0.008945565 0.008945565
> RMeasure(0,.008945565,cond.dist="std",df=5.296)
Risk Measures for selected probabilities:
prob VaR ES
[1,] 0.950 0.01780556 0.02529450
[2,] 0.990 0.02941036 0.03850407
[3,] 0.999 0.05054351 0.06368452
>
k2=garchFit(~garch(1,1),data=amzn,cond.dist="std",include.me
an=F,trace=F)
> summary(k2)

14. Title:
GARCH Modelling
Call:
garchFit(formula = ~garch(1, 1), data = amzn, cond.dist =
"std",
include.mean = F, trace = F)
Mean and Variance Equation:
data ~ garch(1, 1)
[data = amzn]
Conditional Distribution:
std
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
omega 5.401e-06 2.351e-06 2.297 0.0216 *
alpha1 2.004e-02 4.550e-03 4.404 1.06e-05 ***
beta1 9.717e-01 6.252e-03 155.428 < 2e-16 ***
shape 3.691e+00 2.805e-01 13.158 < 2e-16 ***
---
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 44179.67 0
Shapiro-Wilk Test R W 0.8497441 0
Ljung-Box Test R Q(10) 6.935119 0.7315556
Ljung-Box Test R Q(15) 11.39435 0.7241714
Ljung-Box Test R Q(20) 17.60512 0.6134033
Ljung-Box Test R^2 Q(10) 2.171012 0.9948534
Ljung-Box Test R^2 Q(15) 2.921834 0.9996587
Ljung-Box Test R^2 Q(20) 3.835627 0.9999671
LM Arch Test R TR^2 2.402731 0.9984912
> predict(k2,1)
meanForecast meanError standardDeviation

15. 1 0 0.02602888 0.02602888
> RMeasure(0,0.02602888,cond.dist="std",df=3.691)
Risk Measures for selected probabilities:
prob VaR ES
[1,] 0.950 0.0568516 0.08724638
[2,] 0.990 0.1021984 0.14579925
[3,] 0.999 0.2034557 0.28198662
>