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


Copyright 2011 - 2019 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

price = csvread('index.csv', 1, 0);
y=diff(log(price));                 % get returns
		
Listing 8.1/8.2: Load data in Python
Last updated June 2018

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

Listing 8.3/8.4: Set backtest up in MATLAB
Last updated August 2016

T = length(y);                          % number of obs for return y
WE = 1000;                              % estimation window length
p = 0.01;                               % probability
l1 = WE*p ;                             % HS observation
value = 1;                              % portfolio value
VaR = NaN(T,4);                         % matrix for forecasts
%% EWMA setup
lambda = 0.94;
s11 = var(y(1:30));
for t = 2:WE
    s11=lambda*s11+(1-lambda)*y(t-1)^2;
end
		
Listing 8.3/8.4: Set backtest up in Python
Last updated June 2018

import numpy as np
T = len(y)                            # number of obs for y
WE = 1000                             # estimation window length
p = 0.01                              # probability
l1 = int(WE * p)                      # HS observation
value = 1                             # portfolio value
VaR = np.full([T,4], np.nan)          # matrix for forecasts
## EWMA setup
lmbda = 0.94
s11 = np.var(y[1:30])
for t in range(1,WE):
    s11=lmbda*s11+(1-lmbda)*y[t-1]**2
		

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 Python
Last updated June 2018

import numpy as np
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')
    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)
## GARCH optimization in Python has some convergence issues
		

Listing 8.7/8.8: Backtesting analysis in MATLAB
Last updated 2011

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([i VR s])
end
plot([y(WE+1:T) VaR(WE+1:T,:)])
		
Listing 8.7/8.8: Backtesting analysis in Python
Last updated June 2018

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 ([i, m[i], VR, s])
plt.plot(y[W1:T])
plt.plot(VaR[W1:T])
plt.show()
plt.close()
		

Listing 8.9/8.10: Bernoulli coverage test in MATLAB
Last updated 2011

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 Python
Last updated June 2018

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))
		

Listing 8.11/8.12: Independence test in MATLAB
Last updated 2011

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 Python
Last updated June 2018

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))
		

Listing 8.13/8.14: Backtesting S&P 500 in MATLAB
Last updated June 2018

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([i,ber,1-chi2cdf(ber,1),in,1-chi2cdf(in,1)])
end
		
Listing 8.13/8.14: Backtesting S&P 500 in Python
Last updated June 2018

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 ([i, m[i], ber, 1 - stats.chi2.cdf(ber, 1), ind, 1 - stats.chi2.cdf(ind, 1)])
		

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 Python
Last updated June 2018

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
		

Listing 8.17/8.18: ES in MATLAB
Last updated 2011

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([i,nES])
end
		
Listing 8.17/8.18: ES in Python
Last updated June 2018

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 ([i, m[i], 'nES', nES])