Copyright 2011, 2016, 2018 Jon Danielsson. This code is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This code is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. The GNU General Public License is available at: https://www.gnu.org/licenses/.

The original 2011 R code will not fully work on a recent R because there have been some changes to libraries. The latest version of the Matlab code only uses functions from Matlab toolboxes.

The GARCH functionality in the econometric toolbox in Matlab is trying to be too clever, but can't deliver and could well be buggy. If you want to try that, here are the docs (estimate). Besides, it can only do univariate GARCH and so can't be used in Chapter 3. Kevin Sheppard's MFE toolbox is much better, while not as user friendly, it is much better written and is certainly more comprehensive. It can be downloaded here and the documentation here is quite detailed.

Last updated August 2016

```
library(tseries)
library(zoo)
p = zoo(read.csv('index.csv',header=TRUE,sep=','))
y=diff(log(p)) # get returns
y=coredata(y) # strip date information
```

Last updated June 2018

```
using CSV;
price = CSV.read("index.csv", nullable = false);
y = diff(log.(price[:,1])); # get returns
```

Last updated August 2016

```
TT = length(y) # number of obs for y
WE = 1000 # estimation window length
p = 0.01 # probability
l1 = WE*p # HS observation
value = 1; # portfolio value
VaR = matrix(nrow=TT,ncol=4) # matrix for forecasts
## EWMA setup
lambda = 0.94;
s11 = var(y[1:30]);
for(t in 2:WE) s11=lambda*s11+(1-lambda)*y[t-1]^2
library(fGarch)
```

Last updated June 2018

```
T = length(y) # number of obs for return y
WE = 1000 # estimation window length
p = 0.01 # probability
l1 = convert(Int, WE*p) # HS observation
value = 1 # portfolio value
VaR = fill!(Array{Float64}(T,4), NaN) # matrix for forecasts
## EWMA setup
lambda = 0.94
s11 = var(y[1:30])
for t in range(2,WE-1)
s11=lambda*s11+(1-lambda)*y[t-1]^2
end
```

Last updated August 2016

```
for (t in (WE+1):TT){
t1 = t-WE; # start of the data window
t2 = t-1; # end of the data window
window = y[t1:t2] # data for estimation
s11=lambda*s11+(1-lambda)*y[t-1]^2
VaR[t,1] = -qnorm(p) * sqrt(s11) * value # EWMA
VaR[t,2] = - sd(window) * qnorm(p)*value # MA
ys = sort(window)
VaR[t,3] = -ys[l1]*value # HS
g=garchFit(formula=~garch(1,1),window,
trace=FALSE,include.mean=FALSE)
par=g@fit$matcoef
s4=par[1]+par[2]*window[WE]^2+par[3]*g@h.t[WE]
VaR[t,4] = -qnorm(p) * sqrt(s4) * value # GARCH(1,1)
}
```

Last updated June 2018

```
using Distributions;
using FRFGarch;
for t in range(WE+1, T-WE)
t1 = t - WE # start of data window
t2 = t - 1 # end of data window
window = y[t1:t2] # data for estimation
## EWMA
s11 = lambda * s11 + (1-lambda) * y[t-1]^2
VaR[t,1]=-quantile(Normal(0,1),p)*sqrt(s11)*value # EWMA
VaR[t,2]=-std(window)*quantile(Normal(0,1),p)*value # MA
ys = sort(window)
VaR[t,3] = -ys[l1] * value # HS
res = GARCHfit(window)
s4 = res.seForecast
VaR[t,4]=-s4*quantile(Normal(0,1),p)*value # GARCH(1,1)
end
## GARCH VaR estimation will be slightly different from other languages
## this is due to GARCHfit choosing initial conditional vol = sample vol
```

Last updated June 2018

```
W1=WE+1
for (i in 1:4){
VR = sum(y[W1:TT]< -VaR[W1:TT,i])/(p*(TT-WE))
s = sd(VaR[W1:TT,i])
cat(i,"VR",VR,"VaR vol",s,"\n")
}
matplot(cbind(y[W1:TT],VaR[W1:TT,]),type='l',col=1:5,las=1,ylab="",lty=1:5)
legend("topleft",legend=c("Returns","EWMA","MA","HS","GARCH"),lty=1:5,col=1:5,bty="n")
```

Last updated June 2018

```
W1 = WE + 1
for i in range(1,4)
VR = sum(y[W1:T] .< -VaR[W1:T, i]) / (p * (T - WE))
s = std(VaR[W1:T,i])
println([i, "VR", VR, "VaR vol", s])
end
using Plots;
return plot([y, VaR[:,1], VaR[:,2], VaR[:,3], VaR[:,4]])
```

Last updated August 2016

```
bern_test=function(p,v){
lv=length(v)
sv=sum(v)
al=log(p)*sv+log(1-p)*(lv-sv)
bl=log(sv/lv)*sv +log(1-sv/lv)*(lv-sv)
return(-2*(al-bl))
}
```

Last updated June 2018

```
function bern_test(p,v)
lv = length(v)
sv = sum(v)
al = log(p)*sv + log(1-p)*(lv-sv)
bl = log(sv/lv)*sv + log(1-sv/lv)*(lv-sv)
return (-2*(al-bl))
end
```

Last updated June 2018

```
ind_test=function(V){
J=matrix(ncol=4,nrow=length(V))
for (i in 2:length(V)){
J[i,1]=V[i-1]==0 & V[i]==0
J[i,2]=V[i-1]==0 & V[i]==1
J[i,3]=V[i-1]==1 & V[i]==0
J[i,4]=V[i-1]==1 & V[i]==1
}
V_00=sum(J[,1],na.rm=TRUE)
V_01=sum(J[,2],na.rm=TRUE)
V_10=sum(J[,3],na.rm=TRUE)
V_11=sum(J[,4],na.rm=TRUE)
p_00=V_00/(V_00+V_01)
p_01=V_01/(V_00+V_01)
p_10=V_10/(V_10+V_11)
p_11=V_11/(V_10+V_11)
hat_p=(V_01+V_11)/(V_00+V_01+V_10+V_11)
al = log(1-hat_p)*(V_00+V_10) + log(hat_p)*(V_01+V_11)
bl = log(p_00)*V_00 + log(p_01)*V_01 + log(p_10)*V_10 + log(p_11)*V_11
return(-2*(al-bl))
}
```

Last updated June 2018

```
function ind_test(V)
J = fill!(Array{Float64}(T,4), 0)
for i in range(2,length(V)-1)
J[i,1] = (V[i-1] == 0) & (V[i] == 0)
J[i,2] = (V[i-1] == 0) & (V[i] == 1)
J[i,3] = (V[i-1] == 1) & (V[i] == 0)
J[i,4] = (V[i-1] == 1) & (V[i] == 1)
end
V_00 = sum(J[:,1])
V_01 = sum(J[:,2])
V_10 = sum(J[:,3])
V_11 = sum(J[:,4])
p_00=V_00/(V_00+V_01)
p_01=V_01/(V_00+V_01)
p_10=V_10/(V_10+V_11)
p_11=V_11/(V_10+V_11)
hat_p = (V_01+V_11)/(V_00+V_01+V_10+V_11)
al = log(1-hat_p)*(V_00+V_10) + log(hat_p)*(V_01+V_11)
bl = log(p_00)*V_00 + log(p_01)*V_01 + log(p_10)*V_10 + log(p_11)*V_11
return (-2*(al-bl))
end
```

Last updated August 2016

```
W1=WE+1
ya=y[W1:TT]
VaRa=VaR[W1:TT,]
m=c("EWMA","MA","HS","GARCH")
for (i in 1:4){
q= y[W1:TT]< -VaR[W1:TT,i]
v=VaRa*0
v[q,i]=1
ber=bern_test(p,v[,i])
ind=ind_test(v[,i])
cat(i,m[i],'Bernoulli',ber,1-pchisq(ber,1),"independence",ind,1-pchisq(ind,1),"\n")
}
```

Last updated June 2018

```
using Distributions;
W1 = WE+1
ya = y[W1:T]
VaRa = VaR[W1:T,:]
m = ["EWMA", "MA", "HS", "GARCH"]
for i in range(1,4)
q = y[W1:T] .< -VaR[W1:T,i]
v = VaRa .* 0
v[q,i] = 1
ber = bern_test(p, v[:,i])
ind = ind_test(v[:,i])
println([i, m[i], ber, 1-cdf(Chisq(1), ber), ind, 1-cdf(Chisq(1), ind)])
end
```

Last updated August 2016

```
VaR = matrix(nrow=TT,ncol=2) # VaR forecasts for 2 models
ES = matrix(nrow=TT,ncol=2) # ES forecasts for 2 models
for (t in (WE+1):TT){
t1 = t-WE;
t2 = t-1;
window = y[t1:t2]
s11 = lambda * s11 + (1-lambda) * y[t-1]^2
VaR[t,1] = -qnorm(p) * sqrt(s11) * value # EWMA
ES[t,1] = sqrt(s11) * dnorm(qnorm(p)) / p
ys = sort(window)
VaR[t,2] = -ys[l1]*value # HS
ES[t,2] = -mean(ys[1:l1]) * value
}
```

Last updated June 2018

```
using Distributions;
VaR = fill!(Array{Float64}(T,2), NaN) # VaR forecasts
ES = fill!(Array{Float64}(T,2), NaN) # ES forecasts
for t in range(WE+1, T-WE)
t1 = t - WE
t2 = t - 1
window = y[t1:t2]
s11 = lambda*s11+(1-lambda)*y[t-1]^2
VaR[t,1]=-quantile(Normal(0,1),p)*sqrt(s11)*value # EWMA
ES[t,1]=sqrt(s11)*pdf(Normal(0,1),quantile(Normal(0,1),p))/p
ys = sort(window)
VaR[t,2] = -ys[l1] * value # HS
ES[t,2] = -mean(ys[1:l1]) * value
end
```

Last updated August 2016

```
ESa = ES[W1:TT,]
VaRa = VaR[W1:TT,]
for (i in 1:2){
q = ya <= -VaRa[,i]
nES = mean(ya[q] / -ESa[q,i])
cat(i,"nES",nES,"\n")
}
```

Last updated June 2018

```
ESa = ES[W1:T,:]
VaRa = VaR[W1:T,:]
m = ["EWMA", "HS"]
for i in range(1,2)
q = ya .<= -VaRa[:,i]
nES = mean(ya[q] ./ -ESa[q,i])
println([i, m[i], "nES", nES])
end
```