Matlab and Julia Chapter 5. Implementing Risk Forecasts

# Chapter 5. Implementing Risk Forecasts

### Matlab and Julia

Copyright 2011 - 2023 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: www.gnu.org/licenses.

##### Listing 5.1/5.2
stocks = csvread('stocks.csv',1,0);
p1 = stocks(:,1);             % consider first two stocks
p2 = stocks(:,2);
y1=diff(log(p1));             % convert prices to returns
y2=diff(log(p2));
y=[y1 y2];
T=length(y1);
value = 1000;                 % portfolio value
p = 0.01;                     % probability

##### Listing 5.1/5.2
using CSV, DataFrames;
y1 = diff(log.(p[:,1]));
y2 = diff(log.(p[:,2]));
y = hcat(y1,y2);
T = size(y,1);
value = 1000; # portfolio value
p = 0.01;     # probability


##### % Univariate HS VaR in MATLAB
ys = sort(y1);   % sort returns
op = ceil(T*p);  % p percent smallest, rounded up to meet VaR probability requirement
VaR1 = -ys(op)*value

##### Univariate HS in Julia
ys = sort(y1)            # sort returns
op = ceil(Int, T*p)      # p percent smallest, rounding up
VaR1 = -ys[op] * value
println("Univariate HS VaR ", Int(p*100), "%: ", round(VaR1, digits = 3), " USD")


##### % Multivariate HS VaR in MATLAB
w = [0.3; 0.7];    % vector of portfolio weights
yp = y*w;          % portfolio returns
yps = sort(yp);
VaR2 = -yps(op)*value

##### Multivariate HS in Julia
w = [0.3; 0.7]          # vector of portfolio weights
yp = y * w              # portfolio returns
yps = sort(yp)
VaR2 = -yps[op] * value
println("Multivariate HS VaR ", Int(p*100), "%: ", round(VaR2, digits = 3), " USD")


##### % Univariate ES in MATLAB
ES1 = -mean(ys(1:op))*value

##### Univariate ES in Julia
using Statistics;
ES1 = -mean(ys[1:op]) * value
println("ES: ", round(ES1, digits = 3), " USD")


##### % Normal VaR in MATLAB
sigma = std(y1); % estimate volatility
VaR3 = -sigma * norminv(p) * value

##### Normal VaR in Julia
using Distributions;
sigma = std(y1);      # estimate volatility
VaR3 = -sigma * quantile(Normal(0,1),p) * value
println("Normal VaR", Int(p*100), "%: ", round(VaR3, digits = 3), " USD")


##### % Portfolio normal VaR in MATLAB
sigma = sqrt(w' * cov(y) * w); % portfolio volatility
VaR4 = - sigma * norminv(p) *  value

##### Portfolio normal VaR in Julia
sigma = sqrt(w'*cov(y)*w)   # portfolio volatility
VaR4 = -sigma * quantile(Normal(0,1), p) * value
println("Portfolio normal VaR", Int(p*100), "%: ", round(VaR4, digits = 3), " USD")


##### % Student-t VaR in MATLAB
scy1=y1*100;          % scale the returns
res=mle(scy1,'distribution','tlocationscale');
sigma1 = res(2)/100;  % rescale the volatility
nu = res(3);
VaR5 = - sigma1 * tinv(p,nu) * value

##### Student-t VaR in Julia



##### % Normal ES in MATLAB
sigma = std(y1);
ES2=sigma*normpdf(norminv(p))/p * value

##### Normal ES in Julia
sigma = std(y1)
ES2 = sigma * pdf(Normal(0,1), (quantile(Normal(0,1), p))) / p * value
println("Normal ES: ", round(ES2, digits = 3), " USD")


##### % Direct integration ES in MATLAB
VaR = -norminv(p);

##### Direct integration ES in Julia
using QuadGK;
VaR = -quantile(Normal(0,1), p)
integrand(x) = x * pdf(Normal(0,1), x)
ES3 = -sigma * quadgk(integrand, -Inf, -VaR) / p * value
println("Normal integrated ES: ", round(ES3, digits = 3), " USD")


##### % MA normal VaR in MATLAB
WE=20;
for t=T-5:T
t1=t-WE+1;
window=y1(t1:t);  % estimation window
sigma=std(window);
VaR6 = -sigma * norminv(p) * value
end

##### MA normal VaR in Julia
WE = 20
for t in T-5:T
t1 = t-WE
window = y1[t1+1:t] # estimation window
sigma = std(window)
VaR6 = -sigma*quantile(Normal(0,1),p)*value
println("MA Normal VaR", Int(p*100), "% using observations ", t1, " to ", t, ": ",
round(VaR6, digits = 3), " USD")
end


##### % EWMA VaR in MATLAB
lambda = 0.94;
s11 = var(y1(1:30)); % initial variance
for t = 2:T
s11 = lambda * s11  + (1-lambda) * y1(t-1)^2;
end
VaR7 = -norminv(p) * sqrt(s11) * value

##### EWMA VaR in Julia
lambda = 0.94
s11 = var(y1) # initial variance
for t in 2:T
s11 = lambda * s11 + (1-lambda) * y1[t-1]^2
end
VaR7 = -sqrt(s11) * quantile(Normal(0,1), p) * value
println("EWMA VaR ", Int(p*100), "%: ", round(VaR7, digits = 3), " USD")


##### % Two-asset EWMA VaR in MATLAB
s = cov(y);               % initial covariance
for t = 2:T
s = lambda * s +  (1-lambda) * y(t-1,:)' * y(t-1,:);
end
sigma = sqrt(w' * s * w); % portfolio vol
VaR8 = - sigma * norminv(p) * value

##### Two-asset EWMA VaR in Julia
s = cov(y) # initial covariance
for t in 2:T
s = lambda * s + (1-lambda) * y[t-1,:] * (y[t-1,:])'
end
sigma = sqrt(w'*s*w) # portfolio vol
VaR8 = -sigma * quantile(Normal(0,1), p) * value
println("Two-asset EWMA VaR ", Int(p*100), "%: ", round(VaR8, digits = 3), " USD")


##### % GARCH in MATLAB
[parameters,ll,ht]=tarch(y1,1,0,1);
omega = parameters(1)
alpha = parameters(2)
beta = parameters(3)
sigma2 = omega + alpha*y1(end)^2 + beta*ht(end) % calc sigma2 for t+1
VaR9 = -sqrt(sigma2) * norminv(p) * value

##### GARCH VaR in Julia
using ARCHModels;
garch1_1 = fit(GARCH{1,1}, y1; meanspec = NoIntercept);
garch_VaR_in = VaRs(garch1_1, :0.01)
cond_vol = predict(garch1_1, :volatility)     # 1-day-ahead conditional volatility
garch_VaR_out = -cond_vol * quantile(garch1_1.dist, p) * value
println("GARCH VaR ", Int(p*100), "%: ", round(garch_VaR_out, digits = 3), " USD")


##### Financial Risk Forecasting
Market risk forecasting with R, Julia, Python and Matlab. Code, lecture slides, implementation notes, seminar assignments and questions.