 Chapter 5. Implementing Risk Forecasts (in R/Julia)

Copyright 2011 - 2020 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/.

## Calculate returns. Note that first column is dates
y = apply(log(p),2,diff)
## Specify portfolio value and VaR probability
portfolio_value = 1000
p = 0.01

using CSV;
## 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 R Last updated July 2020

y1 = y[,1]           # select one asset
ys = sort(y1)        # sort returns
op = ceiling(length(y1)*p) # p percent smallest, rounded up
VaR1 = -ys[op]*portfolio_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 R Last updated July 2020

w = matrix(c(0.3,0.2,0.5)) # vector of portfolio weights
## The number of columns of the left matrix must be the same as the number of rows of the right matrix
yp = y %*% w         # obtain portfolio returns
yps = sort(yp)
VaR2 = -yps[op]*portfolio_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 R Last updated August 2019

ES1 = -mean(ys[1:op])*portfolio_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 R Last updated August 2019

sigma = sd(y1) # estimate volatility
VaR3 = -sigma * qnorm(p) * portfolio_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 R Last updated August 2019

sigma = sqrt(t(w) %*% cov(y) %*% w) # portfolio volatility
## Note: the trailing  is to convert a single element matrix to float
VaR4 = -sigma * qnorm(p)*portfolio_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 R Last updated July 2020

library(QRM)
scy1 = (y1)*100      # scale the returns
res = fit.st(scy1)
sigma1 = unname(res$par.ests['sigma']/100) # rescale the volatility nu = unname(res$par.ests['nu'])
## Note: We are removing the names of the fit.st output using unname()
VaR5 = - sigma1 * qt(df=nu,p=p) *  portfolio_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 R Last updated June August 2019

sigma = sd(y1)
ES2 = sigma*dnorm(qnorm(p))/p * portfolio_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 R Last updated July 2020

VaR = -qnorm(p)
integrand = function(q){q*dnorm(q)}
ES = -sigma*integrate(integrand,-Inf,-VaR)$value/p*portfolio_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) / p * value println("Normal integrated ES: ", round(ES3, digits = 3), " USD") Listing 5.19/5.20: MA normal VaR in R Last updated June August 2019 WE=20 for (t in seq(length(y1)-5,length(y1))){ t1=t-WE+1 window= y1[t1:t] # estimation window sigma=sd(window) VaR6 = -sigma * qnorm(p) * portfolio_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 R Last updated July 2020 lambda = 0.94 s11 = var(y1) # initial variance, using unconditional for (t in 2:length(y1)){ s11 = lambda * s11 + (1-lambda) * y1[t-1]^2 } VaR7 = -qnorm(p) * sqrt(s11) * portfolio_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: Three-asset EWMA VaR in R Last updated August 2019 s = cov(y) # initial covariance for (t in 2:dim(y)){ s = lambda*s + (1-lambda)*y[t-1,] %*% t(y[t-1,]) } sigma = sqrt(t(w) %*% s %*% w) # portfolio vol ## Note: trailing is to convert single element matrix to float VaR8 = -sigma * qnorm(p) * portfolio_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: Univariate GARCH in R Last updated July 2020 library(rugarch) spec = ugarchspec(variance.model = list( garchOrder = c(1, 1)), mean.model = list( armaOrder = c(0,0),include.mean = FALSE)) res = ugarchfit(spec = spec, data = y1) omega = res@fit$coef['omega']
alpha = res@fit$coef['alpha1'] beta = res@fit$coef['beta1']
sigma2 = omega + alpha * tail(y1,1)^2 + beta * tail(res@fit\$var,1)
VaR9 = -sqrt(sigma2) * qnorm(p) * portfolio_value
names(VaR9)="VaR"
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")