Backtesting VaR and CVaR from DCC-GARCH Forecasts
Article Quant Q&A · Author: meteoeliot
Summary
The document describes a practitioner’s attempt to extend univariate GARCH risk analysis to a three-series DCC-GARCH model. The workflow fits separate conditional volatility models, combines them through a dynamic conditional correlation specification, and asks how to plot and backtest portfolio VaR and CVaR, including Kupiec and Christoffersen tests. It also includes example code for calculating conditional expected shortfall from fitted distribution parameters in the univariate setting.
The response reports that, in the package version examined, DCC rolling output appeared not to retain the requested VaR forecasts; the VaR-related arguments seemed to be ignored, while one plot showed an EWMA VaR series. It suggests recovering the DCC covariance forecasts and calculating VaR manually, but the author is uncertain how to obtain the forecasts and provides no completed implementation or test results. The package behavior described may be version-specific, and CVaR backtesting is not resolved.
Key ideas
- The question concerns extending VaR and CVaR plotting and backtesting from univariate GARCH to DCC-GARCH forecasts.
- The example workflow fits univariate conditional volatility models and combines them with a dynamic correlation model.
- The response reports that VaR arguments appeared to be ignored by the rolling DCC function in the version examined.
- A proposed workaround is to recover conditional covariance forecasts and calculate risk measures manually.
- The document does not provide a complete DCC backtesting implementation or resolve CVaR testing.
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Full text
# Manually calculating and backtesting VaR and CVaR from DCC-GARCH R
# Manually calculating and backtesting VaR and CVaR from DCC-GARCH R
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 downloadable with `quantmod`:
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:
```
#loading libraries and data:
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))
cac <- as.data.frame(Ad(getSymbols("^FCHI", src = "yahoo", adjust = T, 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 VaR and 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.
## Answer by Tamir Einhorn Salem (score 1)
https://quant.stackexchange.com/a/50334
Apparently, the VaR arguments are simply ignored by the function. If you try running something absurd, such as specifying that calculate.VaR = 'banana', the function still runs. The object has the same size no matter the argumentos for calculate.VaR and VaR.alpha, and I was unable to find any trace of the values. The plot(DCCroll, which = 5) will give you a EWMA plot of 2.5% VaR, but I've been unable to find any way of recovering the actual values for backtesting. I'm currently facing the same problem, and, as far as I can tell, this paper (https://repositorio.iscte-iul.pt/bitstream/10071/17461/1/Master_Vladimir_Krecmer.pdf) recovered the covariance through DCC, used it to forecast the actual returns (I'm not sure how), then manually did the VaR. If, by any chance, you ever get an update, do tell. I'm considering working a function for doing it, but I don't think I have the skills.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.