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

Copyright 2011 - 2022 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 MATLAB Last updated August 2016

y=diff(log(price)); % get returns

##### Listing 8.1/8.2: Load data in Julia Last updated June 2022

using CSV, DataFrames;
y = diff(log.(price[:,1])); # get returns


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

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

##### 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 MATLAB Last updated June 2018

for t = WE+1:T
t1 = t-WE;       % start of the 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) = -norminv(p) * sqrt(s11)  *value; % EWMA
VaR(t,2) = -std(window)*norminv(p)*value;   % MA
ys = sort(window);
VaR(t,3) = -ys(l1)*value;  % HS
[par,ll,ht]=tarch(window,1,0,1);
h=par(1)+par(2)*window(end)^2+par(3)*ht(end);
VaR(t,4) = -norminv(p)*sqrt(h)*value;          % GARCH(1,1)
end

##### 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 MATLAB Last updated July 2020

names = ["EWMA", "MA", "HS", "GARCH"];
for i=1:4
VR = length(find(y(WE+1:T)<-VaR(WE+1:T,i)))/(p*(T-WE));
s = std(VaR(WE+1:T,i));
disp([names(i), "Violation Ratio:", VR, "Volatility:", s])
end
plot([y(WE+1:T) VaR(WE+1:T,:)])

##### 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 MATLAB Last updated July 2020

%% To run this code block in Jupyter notebook:
%% delete all lines above the line with bern_test.m, then run
%%file bern_test.m
function res=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)
res=-2*(al-bl);
end

##### 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 MATLAB Last updated July 2020

%% To run this code block in Jupyter notebook:
%% delete all lines above the line with ind_test.m, then run
%%file ind_test.m
function res=ind_test(V)
T=length(V);
J=zeros(T,4);
for i = 2:T
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;
res= -2*(al-bl);
end

##### 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 MATLAB Last updated July 2020

names = ["EWMA", "MA", "HS", "GARCH"];
ya=y(WE+1:T);
VaRa=VaR(WE+1:T,:);
for i=1:4
q=find(y(WE+1:T)<-VaR(WE+1:T,i));
v=VaRa*0;
v(q,i)=1;
ber=bern_test(p,v(:,i));
in=ind_test(v(:,i));
disp([names(i), "Bernoulli Statistic:", ber, "P-value:", 1-chi2cdf(ber,1),...
"Independence Statistic:", in, "P-value:", 1-chi2cdf(in,1)])
end

##### 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 MATLAB Last updated 2011

VaR = NaN(T,2);      % VaR forecasts for 2 models
ES = NaN(T,2);       % ES forecasts for 2 models
for t = 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) = -norminv(p) * sqrt(s11)  *value; % EWMA
ES(t,1) = sqrt(s11) * normpdf(norminv(p)) / p;
ys = sort(window);
VaR(t,2) = -ys(l1) * value;          % HS
ES(t,2) = -mean(ys(1:l1)) * value;
end

##### 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: ES in MATLAB Last updated July 2020

names = ["EWMA", "HS"];
VaRa = VaR(WE+1:T,:);
ESa = ES(WE+1:T,:);
for i = 1:2
q = find(ya <= -VaRa(:,i));
nES = mean(ya(q) ./ -ESa(q,i));
disp([names(i), nES])
end

##### 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