Chapter 8. Backtesting and Stress Testing (in R/Julia)


Copyright 2011 - 2020 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/.


Listing 8.1/8.2: Load data in R
Last updated July 2020

p = read.csv('index.csv')
y = diff(log(p$Index)) # get returns
		
Listing 8.1/8.2: Load data in Julia
Last updated July 2020

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

Listing 8.3/8.4: Set backtest up in R
Last updated July 2020

WE = 1000  # estimation window length
p = 0.01   # probability
l1 = ceiling(WE*p)   # HS quantile
portfolio_value = 1  # portfolio value
VaR = matrix(nrow=length(y),ncol=4) # matrix for forecasts
## EWMA setup
lambda = 0.94
s11 = var(y)
for(t in 2:WE) {
    s11=lambda*s11+(1-lambda)*y[t-1]^2
}
		
Listing 8.3/8.4: Set backtest up in Julia
Last updated July 2020

using Statistics;
T = length(y)        # number of obs for return y
WE = 1000  # estimation window length
p = 0.01   # probability
l1 = ceil(Int, WE*p)   # HS observation
value = 1  # portfolio value
VaR = fill(NaN, (T,4)) # matrix for forecasts
## EWMA setup
lambda = 0.94
s11 = var(y)
for t in 2:WE
    s11=lambda*s11+(1-lambda)*y[t-1]^2
end
		

Listing 8.5/8.6: Running backtest in R
Last updated July 2020

## this will take several minutes to run
library(rugarch)
spec = ugarchspec(variance.model = list( garchOrder = c(1, 1)),
                  mean.model = list( armaOrder = c(0,0),include.mean = FALSE))
start_time <- Sys.time()
for (t in (WE+1):length(y)){
  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) * portfolio_value    # EWMA
  VaR[t,2] = - sd(window) * qnorm(p)*portfolio_value    # MA
  ys = sort(window)
  VaR[t,3] = -ys[l1]*portfolio_value     # HS
  res = ugarchfit(spec = spec, data = window,solver="hybrid")
  omega = res@fit$coef['omega']
  alpha = res@fit$coef['alpha1']
  beta = res@fit$coef['beta1']
  sigma2 = omega + alpha * tail(window,1)^2 + beta * tail(res@fit$var,1)
  VaR[t,4] = -sqrt(sigma2) * qnorm(p) * portfolio_value # GARCH(1,1)
}
end_time <- Sys.time()
print(end_time - start_time)
		
Listing 8.5/8.6: Running backtest in Julia
Last updated July 2020

using Distributions, ARCHModels;
for t in WE+1:T
    t1 = t - WE      # start of data window
    t2 = t - 1       # end of data window
    window = y[t1:t2]          # data for estimation
    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
    garch1_1 = fit(GARCH{1,1}, window; meanspec = NoIntercept);
    cond_vol = predict(garch1_1, :volatility)
    VaR[t,4]= -cond_vol * quantile(Normal(0,1),p) * value    # GARCH(1,1)
end
		

Listing 8.7/8.8: Backtesting analysis in R
Last updated August 2019

for (i in 1:4){
  VR = sum(y[(WE+1):length(y)]< -VaR[(WE+1):length(y),i])/(p*(length(y)-WE))
  s = sd(VaR[(WE+1):length(y),i])
  cat(i,"VR",VR,"VaR vol",s,"\n")
}
matplot(cbind(y,VaR),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")
		
Listing 8.7/8.8: Backtesting analysis in Julia
Last updated July 2020

using Plots;
W1 = WE + 1
names = ["Returns" "EWMA" "MA" "HS" "GARCH"]
for i in 1:4
    VR = sum(y[W1:T] .< -VaR[W1:T, i]) / (p * (T - WE))
    s = std(VaR[W1:T,i])
    println(names[i+1], "\n", "VR: ", round(VR, digits = 3), " VaR volatility: ", round(s, digits = 4))
end
plot([y, VaR[:,1], VaR[:,2], VaR[:,3], VaR[:,4]], label = names, title = "Backtesting")
		

Listing 8.9/8.10: Bernoulli coverage test in R
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))
}
		
Listing 8.9/8.10: Bernoulli coverage test in Julia
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
		

Listing 8.11/8.12: Independence test in R
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))
}
		
Listing 8.11/8.12: Independence test in Julia
Last updated July 2020

function ind_test(V)
    J = fill(0, (T,4))
    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)
    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
		

Listing 8.13/8.14: Backtesting S&P 500 in R
Last updated July 2020

W1=WE+1
ya=y[W1:length(y)]
VaRa=VaR[W1:length(y),]
m=c("EWMA","MA","HS","GARCH")
for (i in 1:4){
  q= y[W1:length(y)]< -VaR[W1:length(y),i]
  v=VaRa*0
  v[q,i]=1
  ber=bern_test(p,v[,i])
  ind=ind_test(v[,i])
  cat(i,m[i], "\n",
      "Bernoulli - ","Test statistic:",ber,"  p-value:",1-pchisq(ber,1),"\n",
      "Independence - ", "Test statistic:",ind,"  p-value:",1-pchisq(ind,1),"\n \n")
}
		
Listing 8.13/8.14: Backtesting S&P 500 in Julia
Last updated July 2020

W1 = WE+1
ya = y[W1:T]
VaRa = VaR[W1:T,:]
names = ["EWMA", "MA", "HS", "GARCH"]
for i in 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(names[i], "\n",
        "Bernoulli statistic: ", round(ber, digits = 2), " p-value: ", round(1-cdf(Chisq(1), ber), digits = 4), "\n",
        "Independence statistic: ", round(ind, digits = 2), " p-value: ", round(1-cdf(Chisq(1), ind), digits = 4), "\n")
end
		

Listing 8.15/8.16: Backtest ES in R
Last updated July 2020

VaR2 = matrix(nrow=length(y), ncol=2)    # VaR forecasts for 2 models
ES = matrix(nrow=length(y), ncol=2)      # ES forecasts for 2 models
for (t in (WE+1):length(y)){
  t1 = t-WE;
  t2 = t-1;
  window = y[t1:t2]
  s11 = lambda * s11  + (1-lambda) * y[t-1]^2
  VaR2[t,1] = -qnorm(p) * sqrt(s11) * portfolio_value # EWMA
  ES[t,1] = sqrt(s11) * dnorm(qnorm(p)) / p
  ys = sort(window)
  VaR2[t,2] = -ys[l1] * portfolio_value  # HS
  ES[t,2] = -mean(ys[1:l1]) * portfolio_value
}
		
Listing 8.15/8.16: Backtest ES in Julia
Last updated July 2020

VaR = fill(NaN, (T,2))         # VaR forecasts
ES = fill(NaN, (T,2))          # ES forecasts
for t in WE+1:T
    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
		

Listing 8.17/8.18: Backtest ES in R
Last updated August 2019

ESa = ES[W1:length(y),]
VaRa = VaR2[W1:length(y),]
for (i in 1:2){
  q = ya <= -VaRa[,i]
  nES = mean(ya[q] / -ESa[q,i])
  cat(i,"nES",nES,"\n")
}
		
Listing 8.17/8.18: ES in Julia
Last updated July 2020

ESa = ES[W1:T,:]
VaRa = VaR[W1:T,:]
m = ["EWMA", "HS"]
for i in 1:2
    q = ya .<= -VaRa[:,i]
    nES = mean(ya[q] ./ -ESa[q,i])
    println(m[i], " nES: ", round(nES, digits = 3))
end