# [R-SIG-Finance] Manually calculating and backtesting VaR and CVaR from DCC-GARCH

Eliot Tabet e||ot@t@bet @end|ng |rom meteoprotect@com
Tue Sep 17 10:21:09 CEST 2019

```I estimated a GARCH fit to the log returns of three series (CAC 40, a french
real estate index and french T10 bond yield series) using `rugarch`. I then
manually calculated and backtested the VaR and CVaR measures. I also fitted
a DCC-GARCH(1,1) to the log returns of the 3 series using `rmgarch` and now
I would like to backtest the VaR and CVaR measures in a similar way as I did
for the univariate GARCH cases.

We'll need to specify the following functions for the CVaR before we
proceed:

#This function calculates the CVaR at a certain position gdist list
cvar <- function(p=0.05, s = "CAC", dist_params = gdist_var, pos = l, v
= df, dist = "jsu"){

ES <- abs((integrate(qdist, lower = 0, upper = p, distribution = dist,
mu = gdist_var[[s]][, 'Mu'][pos], sigma = gdist_var[[s]][, 'Sigma'][pos],
shape = gdist_var[[s]][, 'Shape'][pos], skew =
gdist_var[[s]][, 'Skew'][pos])\$value)/p * v[nrow(v),s])

return(ES)
}

#This function calculates the CVaR given the arguments
cvar_df <- function(p=0.01, dist = "jsu", mu = Mu, sigma = Sigma, shape
= Shape, skew = Skew){

ES <- (integrate(qdist, lower = 0, upper = p, distribution = dist, mu
= mu, sigma = sigma, shape = shape, skew = skew)\$value)/p

return(ES)
}

#This function is a vectorized form of the above
vcvar_df <- Vectorize(cvar_df)

The data can be found on dropbox under the following links (one for french
real estate index data and the other for french bonds) the CAC 40 data is

https://www.dropbox.com/s/vy8sl88fs5opmi3/IEIF%20SIIC%20FRANCE_quote_chart.csv?dl=0

https://www.dropbox.com/s/xljxk5izy6pt1ds/entre_obligations.csv?dl=0

The commented code is the following:

require(tidyquant)
require(reshape2)
require(astsa)
require(GGally)
require(forecast)
source("functions.R", local = T)

#
#
https://www.banque-france.fr/statistiques/taux-et-cours/les-indices-obligataires

obli_10 <- read.csv("entre_obligations.csv", sep = ";", na.strings =
"-", stringsAsFactors = F) %>%
rename(Date = 1) %>%
mutate(Date = dmy(Date)) %>%
mutate_at(vars(-Date), funs(gsub("\\,", ".", .))) %>%
mutate_at(vars(-Date), funs(as.numeric)) %>%
dplyr::select(c(1,2))

# #https://live.euronext.com/en/product/indices/QS0010980447-XPAR/quotes
indices nu de:
#
#
https://www.ieif.fr/wp-content/plugins/aa-indices/datas/histo/index.php?IndiceNu=SIICNu&IndiceNet=SIICNet&IndiceBrut=SIICBrut&Indice=Euronext%20IEIF%20SIIC%20France
reit <- read.csv("IEIF SIIC FRANCE_quote_chart.csv", stringsAsFactors =
F) %>%
dplyr::select(1,2) %>%
rename(Date = 1) %>%
mutate(Date = substr(Date, 1, 10)) %>%
mutate(Date = ymd(Date))

auto.assign = FALSE)))

cac <- cac %>%
mutate(Date = rownames(.)) %>%
mutate(Date = ymd(Date)) %>%
dplyr::select(Date, everything())

#Calculate the log returns

lr_df <- as.data.frame(sapply(df[2:ncol(df)], function(x) diff(log(x))))

lr_df <-cbind(df\$Date[2:nrow(df)], lr_df) %>%
dplyr::rename(Date = !!names(.)[1])

#Specification of GARCH models

cac_egarch_spec <- ugarchspec(mean.model = list(armaOrder = c(3, 3),
include.mean = T, archm = F, archpow = 1),
variance.model = list(model = "eGARCH",
garchOrder = c(2, 1)),
distribution.model="jsu")

reit_egarch_spec <- ugarchspec(mean.model = list(armaOrder = c(3, 1),
include.mean = T, archm = F, archpow = 1),
variance.model = list(model = "eGARCH",
garchOrder = c(2, 1)),
distribution.model="nig")

obli_apgarch_spec <- ugarchspec(mean.model = list(armaOrder = c(2, 1),
include.mean = T, archm = F, archpow = 1),
variance.model = list(model = "apARCH",
garchOrder = c(1, 1)),
distribution.model="jsu")

#Get VaR and CVaR

cac_roll <- ugarchroll(cac_egarch_spec, lr_df[,2],n.start = 750,
refit.every = 50, refit.window = "moving",
solver = "hybrid", calculate.VaR =
TRUE, VaR.alpha = c(0.01, 0.025, 0.05), keep.coef = T,
fit.control = list(scale = 1))

reit_roll <- ugarchroll(reit_egarch_spec, lr_df[,3],n.start = 750,
refit.every = 50, refit.window = "moving",
solver = "hybrid", calculate.VaR = TRUE,
VaR.alpha = c(0.01, 0.025, 0.05), keep.coef = T,
fit.control = list(scale = 1))

obli_roll <- ugarchroll(obli_apgarch_spec, lr_df[,4],n.start = 750,
refit.every = 50, refit.window = "moving",
solver = "hybrid", calculate.VaR = TRUE,
VaR.alpha = c(0.01, 0.025, 0.05), keep.coef = T,
fit.control = list(scale = 1))

gdist_var <- list()

gdist_var[["CAC"]] <- as.data.frame(cac_roll, which = 'density')
gdist_var[["REIT"]] <- as.data.frame(reit_roll, which = 'density')
gdist_var[["OBLI_10"]] <- as.data.frame(obli_roll, which = 'density')

#VaR and CVaR calculations
p <- c(0.05, 0.025, 0.01)
l <- nrow(gdist_var[["CAC"]])

for(j in p){
for(i in 1:3){
print(paste("VaR", names(gdist_var)[i], 1-j))
print(abs(qdist(dg[[i]], p=j, mu=gdist_var[[i]]\$Mu[l],
sigma=gdist_var[[i]]\$Sigma[l], skew=gdist_var[[i]]\$Skew[l],
shape=gdist_var[[i]]\$Shape[l]))*df[nrow(df),i+1])
}
}

for(j in p){
for(i in 1:3){
print(paste("CVaR", names(gdist_var)[i], 1-j))
print(cvar(p = j, s = names(gdist_var)[i], dist_params = gdist_var,
pos = l, v = df, dist = dg[[i]]))
}
}

#VaR plots for cac only but will be done for others the same way
var_cac <- gdist_var\$CAC
var_cac <- cbind.data.frame(tail(lr_df[,c("Date","CAC")],2438), var_cac)
%>%
dplyr::select(-`Shape(GIG)`, -Realized) %>%
dplyr::mutate(VaR_99 = qdist("jsu", p = 0.01, mu = Mu, sigma = Sigma,
skew = Skew, shape = Shape)) %>%
dplyr::select(-Mu, -Sigma, -Skew, -Shape)
var_cac <- melt(var_cac, id.vars = "Date")
ggplot(data = var_cac, aes(x = Date, value)) + geom_line(aes(colour =
variable)) +
ggtitle("Series with 1% 1D VaR Limit") +
theme(plot.title = element_text(hjust = 0.5))

#VaR backtesting reports using report function
report(cac_roll, type = "VaR", VaR.alpha = 0.05, conf.level = 0.95)
report(reit_roll, type = "VaR", VaR.alpha = 0.05, conf.level = 0.95)
report(obli_roll, type = "VaR", VaR.alpha = 0.05, conf.level = 0.95)

#CVaR plots for CAC only but will be done for others
cvar_cac <- gdist_var\$CAC
cvar_cac <- cbind.data.frame(tail(lr_df[,c("Date","CAC")],2438),
cvar_cac) %>%
dplyr::select(-`Shape(GIG)`, -Realized) %>%
dplyr::mutate(CVaR_99 = vcvar_df(p = 0.01, dist = "jsu", mu = Mu,
sigma = Sigma, shape = Shape, skew = Skew)) %>%
dplyr::select(-Mu, -Sigma, -Skew, -Shape)

mcvar_cac <- melt(cvar_cac, id.vars = "Date")
ggplot(data = mcvar_cac, aes(x = Date, value)) + geom_line(aes(colour =
variable)) +
ggtitle("Series with 1% 1D CVaR Limit") +
theme(plot.title = element_text(hjust = 0.5))

#Bactesting CVaRby calculating nuber of times CVaR crossed
cvar_cac <- gdist_var\$CAC
cvar_cac <- cbind.data.frame(tail(lr_df[,c("Date","CAC")],2438),
cvar_cac) %>%
dplyr::select(-`Shape(GIG)`, -Realized) %>%
dplyr::mutate(CVaR_99 = vcvar_df(p = 0.01, dist = "jsu", mu = Mu,
sigma = Sigma, shape = Shape, skew = Skew)) %>%
dplyr::mutate(CVaR_975 = vcvar_df(p = 0.025, dist = "jsu", mu = Mu,
sigma = Sigma, shape = Shape, skew = Skew)) %>%
dplyr::mutate(CVaR_95 = vcvar_df(p = 0.05, dist = "jsu", mu = Mu,
sigma = Sigma, shape = Shape, skew = Skew)) %>%
mutate(depasse_99 = case_when(CVaR_99 >= .[[2]] ~ 1, TRUE ~ 0)) %>%
mutate(depasse_975 = case_when(CVaR_975 >= .[[2]] ~ 1, TRUE ~ 0)) %>%
mutate(depasse_95 = case_when(CVaR_95 >= .[[2]] ~ 1, TRUE ~ 0)) %>%
mutate(sum_99 = sum(depasse_99)) %>%
mutate(sum_975 = sum(depasse_975)) %>%
mutate(sum_95 = sum(depasse_95))

#DCC GARCH of GARCH models above:
require(rmgarch)

dcc_garch <- multispec(c(cac_egarch_spec, reit_egarch_spec,
obli_apgarch_spec))
dcc_multfit <- multifit(dcc_garch, lr_df[,2:ncol(lr_df)]) #fitting many
univariate models
dcc_spec <- dccspec(uspec = dcc_garch, dccOrder = c(1,1), distribution =
"mvnorm")
dcc_fit <- dccfit(dcc_spec, lr_df[,2:ncol(lr_df)], fit.control =
list(eval.se = TRUE), fit = dcc_multfit) #fit = dcc_multfit not really
necessary but more robust
dcc_roll <- dccroll(dcc_spec, lr_df[,2:4],n.start = 750, refit.every =
50, refit.window = "moving",
solver = "solnp", calculate.VaR = TRUE, VaR.alpha
= c(0.01, 0.025, 0.05), keep.coef = T,
fit.control = list(scale = 1))

Now I want to do the backtesting and plotting steps for both 1Day VaR and
1Day CVaR measures. Ideally I would also conduct the Kupiec and
Christoffersen test just like in the function `report` of the package
`rugarch`. I am realy stumped as I tried to find an answer online but
couldn't.

--
Eliot TABET | Structurer/Quantitative Analyst

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