Chapter 5. Implementing Risk Forecasts (in Python/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 5.1/5.2: Download stock prices in Python
Last updated July 2020

import numpy as np
from scipy import stats
p = np.loadtxt('stocks.csv',delimiter=',',skiprows=1)
p = p[:,[0,1]] # consider two stocks
## convert prices to returns, and adjust length
y1 = np.diff(np.log(p[:,0]), n=1, axis=0)
y2 = np.diff(np.log(p[:,1]), n=1, axis=0)
y = np.stack([y1,y2], axis = 1)
T = len(y1)
value = 1000   # portfolio value
p = 0.01   # probability
		
Listing 5.1/5.2: Download stock prices in Julia
Last updated June 2022

using CSV, DataFrames;
p = CSV.read("stocks.csv", DataFrame);
## Convert prices of first two stocks to returns
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
		

Listing 5.3/5.4: Univariate HS in Python
Last updated July 2020

from math import ceil
ys = np.sort(y1) # sort returns
op = ceil(T*p)   # p percent smallest
VaR1 = -ys[op - 1] * value
print(VaR1)
		
Listing 5.3/5.4: Univariate HS in Julia
Last updated July 2020

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")
		

Listing 5.5/5.6: Multivariate HS in Python
Last updated July 2020

w = [0.3, 0.7]       # vector of portfolio weights
yp = np.squeeze(np.matmul(y, w)) # portfolio returns
yps = np.sort(yp)
VaR2= -yps[op - 1] * value
print(VaR2)
		
Listing 5.5/5.6: Multivariate HS in Julia
Last updated July 2020

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")
		

Listing 5.7/5.8: Univariate ES in Python
Last updated June 2018

ES1 = -np.mean(ys[:op]) * value
print(ES1)
		
Listing 5.7/5.8: Univariate ES in Julia
Last updated July 2020

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

Listing 5.9/5.10: Normal VaR in Python
Last updated June 2018

sigma = np.std(y1, ddof=1) # estimate volatility
VaR3 = -sigma * stats.norm.ppf(p) * value
print(VaR3)
		
Listing 5.9/5.10: Normal VaR in Julia
Last updated July 2020

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")
		

Listing 5.11/5.12: Portfolio normal VaR in Python
Last updated June 2022

## portfolio volatility
sigma = np.sqrt(np.mat(w)*np.mat(np.cov(y,rowvar=False))*np.transpose(np.mat(w)))[0,0]
## Note: [0,0] is to pull the first element of the matrix out as a float
VaR4 = -sigma * stats.norm.ppf(p) * value
print(VaR4)
		
Listing 5.11/5.12: Portfolio normal VaR in Julia
Last updated July 2020

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")
		

Listing 5.13/5.14: Student-t VaR in Python
Last updated June 2018

scy1 = y1 * 100    # scale the returns
res = stats.t.fit(scy1)
sigma = res[2]/100 # rescale volatility
nu = res[0]
VaR5 = -sigma*stats.t.ppf(p,nu)*value
print(VaR5)
		
Listing 5.13/5.14: Student-t VaR in Julia
Last updated July 2020

## Julia does not have a function for fitting Student-t data yet
## Currently: there exists Distributions.jl with fit_mle
## usage: Distributions.fit_mle(Dist_name, data[, weights])
##
## using Distributions;
## res = fit_mle(TDist, y1)
## nu = res.ν (this is the Greek letter nu, not Latin v)
## sigma = sqrt(nu/(nu-2))
## VaR5 = -sigma * quantile(TDist(nu), p) * value
		

Listing 5.15/5.16: Normal ES in Python
Last updated June 2018

sigma = np.std(y1, ddof=1)
ES2 = sigma * stats.norm.pdf(stats.norm.ppf(p)) / p * value
print(ES2)
		
Listing 5.15/5.16: Normal ES in Julia
Last updated July 2020

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

Listing 5.17/5.18: Direct integration ES in Python
Last updated June 2018

from scipy.integrate import quad
VaR = -stats.norm.ppf(p)
integrand = lambda q: q * stats.norm.pdf(q)
ES = -sigma * quad(integrand, -np.inf, -VaR)[0] / p * value
print(ES)
		
Listing 5.17/5.18: Direct integration ES in Julia
Last updated July 2020

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

Listing 5.19/5.20: MA normal VaR in Python
Last updated June 2018

WE = 20
for t in range(T-5,T+1):
    t1 = t-WE
    window = y1[t1:t] # estimation window
    sigma = np.std(window, ddof=1)
    VaR6 = -sigma*stats.norm.ppf(p)*value
    print (VaR6)
		
Listing 5.19/5.20: MA normal VaR in Julia
Last updated July 2020

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
		

Listing 5.21/5.22: EWMA VaR in Python
Last updated June 2018

lmbda = 0.94
s11 = np.var(y1[0:30], ddof = 1) # initial variance
for t in range(1, T):
    s11 = lmbda*s11 + (1-lmbda)*y1[t-1]**2
VaR7 = -np.sqrt(s11)*stats.norm.ppf(p)*value
print(VaR7)
		
Listing 5.21/5.22: EWMA VaR in Julia
Last updated July 2020

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")
		

Listing 5.23/5.24: Two-asset EWMA VaR in Python
Last updated June 2022

## s is the initial covariance
s = np.cov(y, rowvar = False)
for t in range(1,T):
    s = lmbda*s+(1-lmbda)*np.transpose(np.asmatrix(y[t-1,:]))*np.asmatrix(y[t-1,:])
sigma = np.sqrt(np.mat(w)*s*np.transpose(np.mat(w)))[0,0]
## Note: [0,0] is to pull the first element of the matrix out as a float
VaR8 = -sigma * stats.norm.ppf(p) * value
print(VaR8)
		
Listing 5.23/5.24: Two-asset EWMA VaR in Julia
Last updated July 2020

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")
		

Listing 5.25/5.26: GARCH VaR in Python
Last updated July 2020

from arch import arch_model
am = arch_model(y1, mean = 'Zero', vol='Garch', p=1, o=0, q=1, dist='Normal', rescale = False)
res = am.fit(update_freq=5, disp = "off")
omega = res.params.loc['omega']
alpha = res.params.loc['alpha[1]']
beta = res.params.loc['beta[1]']
## computing sigma2 for t+1
sigma2 = omega + alpha*y1[T-1]**2 + beta * res.conditional_volatility[-1]**2
VaR9 = -np.sqrt(sigma2) * stats.norm.ppf(p) * value
print(VaR9)
		
Listing 5.25/5.26: GARCH VaR in Julia
Last updated July 2020

## We use the ARCHModels package
using ARCHModels;
garch1_1 = fit(GARCH{1,1}, y1; meanspec = NoIntercept);
## In-sample GARCH VaR - Use the VaRs function
garch_VaR_in = VaRs(garch1_1, :0.01)
## Out-of-sample GARCH VaR - Use the predict function
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")