Chapter 8. Backtesting and Stress Testing

p = read.csv('index.csv')
y = diff(log(p$Index)) # get returns 

import numpy as np
price = np.loadtxt('index.csv',delimiter=',',skiprows=1)
y = np.diff(np.log(price), n=1, axis=0) # get returns

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
lambda = 0.94       
s11 = var(y)
for(t in 2:WE) {
    s11=lambda*s11+(1-lambda)*y[t-1]^2
}

from math import ceil
T = len(y)                   # number of obs for y
WE = 1000                    # estimation window length
p = 0.01                     # probability
l1 = ceil(WE * p)             # HS observation
value = 1                    # portfolio value
VaR = np.full([T,4], np.nan) # matrix for forecasts
lmbda = 0.94
s11 = np.var(y)
for t in range(1,WE):
    s11=lmbda*s11+(1-lmbda)*y[t-1]**2

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)

from scipy import stats
from arch import arch_model
for t in range(WE, T): 
    t1 = t - WE           # start of data window
    t2 = t - 1            # end of data window
    window = y[t1:t2+1]   # data for estimation
    s11 = lmbda * s11 + (1-lmbda) * y[t-1]**2
    VaR[t,0] = -stats.norm.ppf(p)*np.sqrt(s11)*value # EWMA
    VaR[t,1] = -np.std(window,ddof=1)*stats.norm.ppf(p)*value # MA
    ys = np.sort(window)
    VaR[t,2] = -ys[l1 - 1] * value # HS
    am = arch_model(window, mean = 'Zero',vol = 'Garch',
                    p = 1, o = 0, q = 1, dist = 'Normal', rescale = False)
    res = am.fit(update_freq=0, disp = 'off', show_warning=False)
    par = [res.params[0], res.params[1], res.params[2]]
    s4 = par[0] + par[1] * window[WE - 1]**2 + par[
        2] * res.conditional_volatility[-1]**2
    VaR[t,3] = -np.sqrt(s4) * stats.norm.ppf(p) * value # GARCH(1,1)

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")

W1 = WE # Python index starts at 0
m = ["EWMA", "MA", "HS", "GARCH"]
for i in range(4):
    VR = sum(y[W1:T] < -VaR[W1:T,i])/(p*(T-WE))
    s = np.std(VaR[W1:T, i], ddof=1)
    print (m[i], "\n", 
           "Violation ratio:", round(VR, 3), "\n", 
           "Volatility:", round(s,3), "\n")
plt.plot(y[W1:T])
plt.plot(VaR[W1:T])
plt.title("Backtesting")
plt.show()
plt.close()

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))
}

def bern_test(p,v):
    lv = len(v)
    sv = sum(v)
    al = np.log(p)*sv + np.log(1-p)*(lv-sv)
    bl = np.log(sv/lv)*sv + np.log(1-sv/lv)*(lv-sv)
    return (-2*(al-bl))

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))
}

def ind_test(V):
    J = np.full([T,4], 0)
    for i in range(1,len(V)-1):
        J[i,0] = (V[i-1] == 0) & (V[i] == 0)
        J[i,1] = (V[i-1] == 0) & (V[i] == 1)
        J[i,2] = (V[i-1] == 1) & (V[i] == 0)
        J[i,3] = (V[i-1] == 1) & (V[i] == 1)
    V_00 = sum(J[:,0])
    V_01 = sum(J[:,1])
    V_10 = sum(J[:,2])
    V_11 = sum(J[:,3])
    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 = np.log(1-hat_p)*(V_00+V_10) + np.log(hat_p)*(V_01+V_11)
    bl = np.log(p_00)*V_00 + np.log(p_01)*V_01 + np.log(p_10)*V_10 + np.log(p_11)*V_11
    return (-2*(al-bl))

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")
}

W1 = WE
ya = y[W1:T]
VaRa = VaR[W1:T,]
m = ['EWMA', 'MA', 'HS', 'GARCH']
for i in range(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])
    print (m[i], "\n",
           "Bernoulli:", "Test statistic =", round(ber,3), "p-value =", round(1 - stats.chi2.cdf(ber, 1),3), "\n",
           "Independence:", "Test statistic =", round(ind,3), "p-value =", round(1 - stats.chi2.cdf(ind, 1),3), "\n")

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
}

VaR = np.full([T,2], np.nan) # VaR forecasts
ES = np.full([T,2], np.nan)  # ES forecasts
for t in range(WE, T):
    t1 = t - WE
    t2 = t - 1
    window = y[t1:t2+1]
    s11 = lmbda * s11 + (1-lmbda) * y[t-1]**2
    VaR[t,0] = -stats.norm.ppf(p) * np.sqrt(s11)*value       # EWMA
    ES[t,0]=np.sqrt(s11)*stats.norm.pdf(stats.norm.ppf(p))/p
    ys = np.sort(window)
    VaR[t,1] = -ys[l1 - 1] * value                           # HS
    ES[t,1] = -np.mean(ys[0:l1]) * value

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")
}

ESa = ES[W1:T,:]
VaRa = VaR[W1:T,:]
m = ["EWMA", "HS"]
for i in range(2):
    q = ya <= -VaRa[:,i]
    nES = np.mean(ya[q] / -ESa[q,i])
    print (m[i], 'nES = ', round(nES,3))