Package 'quarks' reference manual

Title:	Simple Methods for Calculating and Backtesting Value at Risk and Expected Shortfall
Description:	Enables the user to calculate Value at Risk (VaR) and Expected Shortfall (ES) by means of various types of historical simulation. Currently plain-, age-, volatility-weighted- and filtered historical simulation are implemented in this package. Volatility weighting can be carried out via an exponentially weighted moving average model (EWMA) or other GARCH-type models. The performance can be assessed via Traffic Light Test, Coverage Tests and Loss Functions. The methods of the package are described in Gurrola-Perez, P. and Murphy, D. (2015) <https://EconPapers.repec.org/RePEc:boe:boeewp:0525> as well as McNeil, J., Frey, R., and Embrechts, P. (2015) <https://ideas.repec.org/b/pup/pbooks/10496.html>.
Authors:	Sebastian Letmathe [aut, cre]
Maintainer:	Sebastian Letmathe <[email protected]>
License:	GPL-3
Version:	1.1.4
Built:	2024-12-06 06:52:30 UTC
Source:	CRAN

Unconditional and Conditional Coverage Tests, Independence Test

Description

The conditional (Kupiec, 1995), the unconditional coverage test (Christoffersen, 1998) and the independence test (Christoffersen, 1998) of the Value-at-Risk (VaR) are applied.

Usage

cvgtest(obj = list(loss = NULL, VaR = NULL, p = NULL), conflvl = 0.95)
cvgtest(obj = list(loss = NULL, VaR = NULL, p = NULL), conflvl = 0.95)

Arguments

obj

a list that contains the following elements:

loss: a numeric vector that contains the values of a loss series ordered from past to present; is set to NULL by default.
VaR: a numeric vector that contains the estimated values of the VaR for the same time points of the loss series loss; is set to NULL by default.
p: a numeric vector with one element; defines the probability p stated in the null hypotheses of the coverage tests (see the section Details for more information); is set to NULL by default.

conflvl

a numeric vector with one element; the significance level at which the null hypotheses are evaluated; is set to 0.95 by default. Please note that a list returned by the rollcast function can be directly passed to cvgtest.

Details

With this function, the conditional and the unconditional coverage tests introduced by Kupiec (1995) and Christoffersen (1998) can be applied. Given a return series $r_t$ with $n$ observations, divide the series into $n-K$ in-sample and $K$ out-of-sample observations, fit a model to the in-sample data and obtain rolling one-step forecasts of the VaR for the out-of-sample time points.

Define

$I_t = 1,$

if $-r_t > \widehat{VaR}_t (\alpha)$ or

$I_t = 0,$

otherwise,

for $t = n + 1, n + 2, ..., n + K$ as the hit sequence, where $\alpha$ is the confidence level for the VaR (often $\alpha = 0.95$ or $\alpha = 0.99$ ). Furthermore, denote $p = \alpha$ and let $w$ be the actual covered proportion of losses in the data.

1. Unconditional coverage test:

$H_{0, uc}: p = w$

Let $K_1$ be the number of ones in $I_t$ and analogously $K_0$ the number of zeros (all conditional on the first observation). Also calculate $\hat{w} = K_0 / (K - 1)$ . Obtain

$L(I_t, p) = p^{K_0}(1 - p)^{K_1}$

and

$L(I_t, \hat{w}) = \hat{w}^{K_0}(1 - \hat{w})^{K_1}$

and subsequently the test statistic

$LR_{uc} = -2 * \ln \{L(I_t, p) / L(I_t, \hat{w})\}.$

$LR_{uc}$ now asymptotically follows a chi-square-distribution with one degree of freedom.

2. Conditional coverage test:

The conditional coverage test combines the unconditional coverage test with a test on independence. Denote by $w_{ij}$ the probability of an $i$ on day $t-1$ being followed by a $j$ on day $t$ , where $i$ and $j$ correspond to the value of $I_t$ on the respective day.

$H_{0, cc}: w_{00} = w{10} = p$

with $i = 0, 1$ and $j = 0, 1$ .

Let $K_{ij}$ be the number of observations, where the values on two following days follow the pattern $ij$ . Calculate

$L(I_t, \hat{w}_{00}, \hat{w}_{10}) = \hat{w}_{00}^{K_{00}}(1 - \hat{w}_{00})^{K_{01}} * \hat{w}_{10})^{K_{10}}(1 - \hat{w}_{10})^{K_{11}},$

where $\hat{w}_{00} = K_{00} / K_0$ and $\hat{w}_{10} = K_{10} / K_1$ . The test statistic is then given by

$LR_{cc} = -2 * \ln \{ L(I_t, p) / L(I_t, \hat{w}_{00}, \hat{w}_{10}) \},$

which asymptotically follows a chi-square-distribution with two degrees of freedom.

3. Independence test:

$H_{0,ind}: w_{00} = w_{10}$

The asymptotically chi-square-distributed test statistic (one degree of freedom) is given by

$LR_{ind} = -2 * \ln \{L(I_t, \hat{w}_{00}, \hat{w}_{10}) / L(I_t, \hat{w})\}.$

—————————————————————————–

The function needs four inputs: the out-of-sample loss series obj$loss, the corresponding estimated VaR series obj$VaR, the coverage level obj$p, for which the VaR has been calculated and the significance level conflvl, at which the null hypotheses are evaluated. If an object returned by this function is entered into the R console, a detailed overview of the test results is printed.

Value

A list of class quarks with the following four elements:

p: probability p stated in the null hypotheses of the coverage tests
p.uc: the p-value of the unconditional coverage test
p.cc: the p-value of the conditional coverage test
p.ind: the p-value of the independence test
conflvl: the significance level at which the null hypotheses are evaluated
model: selected model for estimation; only available if a list returned by the rollcast function is passed to cvgtest
method: selected method for estimation; only available if a list returned by the rollcast) function is passed to cvgtest

References

Christoffersen, P. F. (1998). Evaluating interval forecasts. International economic review, pp. 841-862.

Kupiec, P. (1995). Techniques for verifying the accuracy of risk measurement models. The J. of Derivatives, 3(2).

Examples


prices <- DAX$price_close
returns <- diff(log(prices))
n <- length(returns)
nout <- 250 # number of obs. for out-of-sample forecasting
nwin <- 500 # window size for rolling forecasts
results <- rollcast(x = returns, p = 0.975, method = 'age', nout = nout,
                     nwin = nwin)
cvgtest(results)

prices <- DAX$price_close
returns <- diff(log(prices))
n <- length(returns)
nout <- 250 # number of obs. for out-of-sample forecasting
nwin <- 500 # window size for rolling forecasts
results <- rollcast(x = returns, p = 0.975, method = 'age', nout = nout,
                     nwin = nwin)
cvgtest(results)

German Stock Market Index (DAX) Financial Time Series Data

Description

A dataset that contains the daily financial data of the DAX from 2000 to December 2023 (currency in EUR).

Usage

DAX
DAX

Format

A data frame with 6094 rows and 11 variables:

ticker: id of the stock
ref_date: date in format YY-MM-DD
price_open: opening price (daily)
price_high: highest price (daily)
price_low: lowest price (daily)
price_close: closing price (daily)
volume: trading volume
price_adjusted: adjusted closing price (daily)
ret_adjusted_prices: returns obtained from the adj. closing prices
ret_closing_prices: returns obtained from the closing prices
cumret_adjusted_prices: accumulated arithmetic/log return for the period

Source

The data was obtained from Yahoo Finance.

Dow Jones Industrial Average (DJI) Financial Time Series Data

Description

A dataset that contains the daily financial data of the DJI from 2000 to December 2023 (currency in USD).

Usage

DJI
DJI

Format

A data frame with 6037 rows and 11 variables:

ticker: id of the stock
ref_date: date in format YY-MM-DD
price_open: opening price (daily)
price_high: highest price (daily)
price_low: lowest price (daily)
price_close: closing price (daily)
volume: trading volume
price_adjusted: adjusted closing price (daily)
ret_adjusted_prices: returns obtained from the adj. closing prices
ret_closing_prices: returns obtained from the closing prices
cumret_adjusted_prices: accumulated arithmetic/log return for the period

Source

The data was obtained from Yahoo Finance.

Exponentially weighted moving average

Description

Estimates volatility of a return series by means of an exponentially weighted moving average.

Usage

ewma(x, lambda = 0.94)
ewma(x, lambda = 0.94)

Arguments

`x`	a numeric vector of asset returns
`lambda`	decay factor for the calculation of weights; default is `0.94`

Value

Returns a numerical vector vol that contains the computed volatility.

Examples

prices <- DAX$price_close
returns <- diff(log(prices))
date <- DAX$ref_date[-1]
cvar <- ewma(x = returns, lambda = 0.94)
csig <- sqrt(cvar)
plot(date, csig, type = 'l',
     main = 'conditional standard deviations for the DAX30 return series')
prices <- DAX$price_close
returns <- diff(log(prices))
date <- DAX$ref_date[-1]
cvar <- ewma(x = returns, lambda = 0.94)
csig <- sqrt(cvar)
plot(date, csig, type = 'l',
     main = 'conditional standard deviations for the DAX30 return series')

Filtered historical simulation

Description

Calculates univariate Value at Risk and Expected Shortfall (Conditional Value at Risk) by means of filtered historical simulation. Volatility can be estimated with an exponentially weighted moving average or a GARCH-type model.

Usage

fhs(x, p = 0.975, model = c("EWMA", "GARCH"), lambda = 0.94, nboot = NULL, ...)
fhs(x, p = 0.975, model = c("EWMA", "GARCH"), lambda = 0.94, nboot = NULL, ...)

Arguments

`x`	a numeric vector of asset returns
`p`	confidence level for VaR calculation; default is `0.975`
`model`	model for estimating conditional volatility; options are `'EWMA'` and `'GARCH'`; if `model = 'GARCH'`, additional arguments can be adjusted via `...`; default is `'EWMA'`
`lambda`	decay factor for the calculation of weights; default is `0.94`
`nboot`	size of bootstrap sample; must be a single non-NA integer value with `nboot > 0`; default is `NULL`
`...`	additional arguments of the `ugarchspec` function from the `rugarch`-package; only applied if `model = 'GARCH'`; default settings for the arguments `variance.model` and `mean.model` are: `variance.model` = `list(model = 'sGARCH', garchOrder = c(1, 1))` `mean.model` = `list(armaOrder = c(0, 0))`

Value

Returns a list with the following elements:

VaR: Calculated Value at Risk
ES: Calculated Expected Shortfall (Conditional Value at Risk)
p: Confidence level for VaR calculation
garchmod: The model fit. Is the respective GARCH fit for model = "GARCH" (see rugarch documentation) and 'EWMA' for model = "EWMA"

Examples

prices <- DAX$price_close
returns <- diff(log(prices))
# volatility weighting via EWMA
ewma <- fhs(x = returns, p = 0.975, model = "EWMA", lambda = 0.94,
            nboot = 10000)
ewma
# volatility weighting via GARCH
garch <- fhs(x = returns, p = 0.975, model = "GARCH", variance.model =
list(model = "sGARCH"), nboot = 10000)
garch
prices <- DAX$price_close
returns <- diff(log(prices))
# volatility weighting via EWMA
ewma <- fhs(x = returns, p = 0.975, model = "EWMA", lambda = 0.94,
            nboot = 10000)
ewma
# volatility weighting via GARCH
garch <- fhs(x = returns, p = 0.975, model = "GARCH", variance.model =
list(model = "sGARCH"), nboot = 10000)
garch

Financial Times Stock Exchange Index (FTSE) Financial Time Series Data

Description

A dataset that contains the daily financial data of the FTSE from 2000 to December 2023 (currency in EUR).

Usage

FTSE100
FTSE100

Format

A data frame with 6060 rows and 11 variables:

ticker: id of the stock
ref_date: date in format YY-MM-DD
price_open: opening price (daily)
price_high: highest price (daily)
price_low: lowest price (daily)
price_close: closing price (daily)
volume: trading volume
price_adjusted: adjusted closing price (daily)
ret_adjusted_prices: returns obtained from the adj. closing prices
ret_closing_prices: returns obtained from the closing prices
cumret_adjusted_prices: accumulated arithmetic/log return for the period

Source

The data was obtained from Yahoo Finance.

Nonparametric calculation of univariate Value at Risk and Expected Shortfall

Description

Computes Value at Risk and Expected Shortfall (Conditional Value at Risk) by means of plain and age-weighted historical simulation.

Usage

hs(x, p = 0.975, method = c("age", "plain"), lambda = 0.98)
hs(x, p = 0.975, method = c("age", "plain"), lambda = 0.98)

Arguments

`x`	a numeric vector of asset returns
`p`	confidence level for VaR calculation; default is `0.975`
`method`	method to be used for calculation; default is `'plain'`
`lambda`	decay factor for the calculation of weights; default is `0.98`

Value

Returns a list with the following elements:

VaR: Calculated Value at Risk
ES: Calculated Expected Shortfall (Conditional Value at Risk)
p: Confidence level for VaR calculation

Examples

prices <- DAX$price_close
returns <- diff(log(prices))
hs(x = returns, p = 0.975, method = 'plain')
hs(x = returns, p = 0.975, method = 'age', lambda = 0.98)
prices <- DAX$price_close
returns <- diff(log(prices))
hs(x = returns, p = 0.975, method = 'plain')
hs(x = returns, p = 0.975, method = 'age', lambda = 0.98)

Hang Seng Index (HSI) Financial Time Series Data

Description

A dataset that contains the daily financial data of the HSI from 2000 to December 2023.

Usage

HSI
HSI

Format

A data frame with 5914 rows and 11 variables:

ticker: id of the stock
ref_date: date in format YY-MM-DD
price_open: opening price (daily)
price_high: highest price (daily)
price_low: lowest price (daily)
price_close: closing price (daily)
volume: trading volume
price_adjusted: adjusted closing price (daily)
ret_adjusted_prices: returns obtained from the adj. closing prices
ret_closing_prices: returns obtained from the closing prices
cumret_adjusted_prices: accumulated arithmetic/log return for the period

Source

The data was obtained from Yahoo Finance.

Loss Functions

Description

This functions allows for the calculation of loss functions in order to assess the performance of models in regard to forecasting ES.

Usage

lossfun(obj = list(loss = NULL, ES = NULL), beta = 1e-04)
lossfun(obj = list(loss = NULL, ES = NULL), beta = 1e-04)

Arguments

obj

a list that contains the following elements:

loss: a numeric vector that contains the values of a loss series ordered from past to present; is set to NULL by default
ES: a numeric vector that contains the estimated values of the ES for the same time points of the loss series loss; is set to NULL by default

Please note that a list returned by the rollcast function can be directly passed to lossfun.

beta

a single numeric value; a measure for the opportunity cost of capital; default is 1e-04.

Details

Given a negative return series obj$loss, the corresponding Expected Shortfall (ES) estimates obj$ES and a parameter beta that defines the opportunity cost of capital, four different definitions of loss functions are considered.

Value

an S3 class object, which is a list of

loss.fun1: regulatory loss function
loss.fun2: firm's loss function following Sarma et al. (2003)
loss.fun3: loss function following Abad et al. (2015)
loss.fun4: Feng's loss function; a compromise of regulatory and firm's loss function

References

Abad, P., Muela, S. B., & Martín, C. L. (2015). The role of the loss function in value-at-risk comparisons. The Journal of Risk Model Validation, 9(1), 1-19.

Sarma, M., Thomas, S., & Shah, A. (2003). Selection of Value-at-Risk models. Journal of Forecasting, 22(4), 337-358.

Examples


prices <- DAX$price_close
returns <- diff(log(prices))
n <- length(returns)
nout <- 250 # number of obs. for out-of-sample forecasting
nwin <- 500 # window size for rolling forecasts
results <- rollcast(x = returns, p = 0.975, method = 'age', nout = nout,
                     nwin = nwin)
loss <- -results$xout
ES <- results$ES
loss.data <- list(loss = loss, ES = ES)
lossfun(loss.data)

# directly passing the output object of 'rollcast()' to 'lossfun()'
lossfun(results)

prices <- DAX$price_close
returns <- diff(log(prices))
n <- length(returns)
nout <- 250 # number of obs. for out-of-sample forecasting
nwin <- 500 # window size for rolling forecasts
results <- rollcast(x = returns, p = 0.975, method = 'age', nout = nout,
                     nwin = nwin)
loss <- -results$xout
ES <- results$ES
loss.data <- list(loss = loss, ES = ES)
lossfun(loss.data)

# directly passing the output object of 'rollcast()' to 'lossfun()'
lossfun(results)

Nikkei Heikin Kabuka Index (NIK) Financial Time Series Data

Description

A dataset that contains the daily financial data of the NIK from 2000 to December 2023.

Usage

NIK225
NIK225

Format

A data frame with 5881 rows and 11 variables:

ticker: id of the stock
ref_date: date in format YY-MM-DD
price_open: opening price (daily)
price_high: highest price (daily)
price_low: lowest price (daily)
price_close: closing price (daily)
volume: trading volume
price_adjusted: adjusted closing price (daily)
ret_adjusted_prices: returns obtained from the adj. closing prices
ret_closing_prices: returns obtained from the closing prices
cumret_adjusted_prices: accumulated arithmetic/log return for the period

Source

The data was obtained from Yahoo Finance.

Profit & Loss operator function

Description

Calculates portfolio returns or losses by assigning weights

Usage

plop(x, wts = NULL, approxim = c(0, 1))
plop(x, wts = NULL, approxim = c(0, 1))

Arguments

`x`	a numeric matrix of asset returns or losses
`wts`	a numeric vector or matrix containing the portfolio weights; portfolio value is standardized to 1 on any observation unit; sum of weights should not exceed 1 (row-wise for matrices); by default the portfolio is equally weighted over time and across all assets; if a vector is passed to `wts` the portfolio is equally weighted over time
`approxim`	controls if a first-order approximation for the calculation of returns or losses is used; default is `1` (first-order approximation is employed)

Value

Returns a list with the following elements:

pl: Weighted portfolio returns or losses
wts: Portfolio weights

Examples

# creating portfolio
portfol <- cbind(SP500$price_close, DJI$price_close)
returns <- apply(portfol, 2, function(x) diff(log(x)))
# defining weights and applying the P&L operator function
wts <- c(0.4, 0.6)
portret <- plop(returns, wts = wts, approxim = 1)
portloss <- plop(-returns, wts = wts, approxim = 1)
plot.ts(cbind(portret$pl, portloss$pl))

# creating portfolio
portfol <- cbind(SP500$price_close, DJI$price_close)
returns <- apply(portfol, 2, function(x) diff(log(x)))
# defining weights and applying the P&L operator function
wts <- c(0.4, 0.6)
portret <- plop(returns, wts = wts, approxim = 1)
portloss <- plop(-returns, wts = wts, approxim = 1)
plot.ts(cbind(portret$pl, portloss$pl))

Plot Method for the Package 'quarks'

Description

This function regulates how objects created by the package quarks are plotted.

Usage

## S3 method for class 'quarks'
plot(x, ...)
## S3 method for class 'quarks'
plot(x, ...)

Arguments

`x`	an input object of class `quarks`.
`...`	additional arguments of the standard plot method.

Value

None

Print Method for the Package 'quarks'

Description

This function regulates how objects created by the package quarks are printed.

Usage

## S3 method for class 'quarks'
print(x, ...)
## S3 method for class 'quarks'
print(x, ...)

Arguments

`x`	an input object of class `quarks`.
`...`	included for compatibility; additional arguments will however not affect the output.

Value

None

Rolling one-step ahead forecasts of Value at Risk and Expected Shortfall

Description

Computes rolling one-step ahead forecasts of Value at Risk and Expected Shortfall (Conditional Value at Risk) by means of plain historical simulation age- and volatility-weighted historical simulation as well as filtered historical simulation.

Usage

rollcast(
  x,
  p = 0.975,
  model = c("EWMA", "GARCH"),
  method = c("plain", "age", "vwhs", "fhs"),
  lambda = c(0.94, 0.98),
  nout = NULL,
  nwin = NULL,
  nboot = NULL,
  smoothscale = c("none", "lpr", "auto"),
  smoothopts = list(),
  ...
)
rollcast(
  x,
  p = 0.975,
  model = c("EWMA", "GARCH"),
  method = c("plain", "age", "vwhs", "fhs"),
  lambda = c(0.94, 0.98),
  nout = NULL,
  nwin = NULL,
  nboot = NULL,
  smoothscale = c("none", "lpr", "auto"),
  smoothopts = list(),
  ...
)

Arguments

`x`	a numeric vector of asset returns
`p`	confidence level for VaR calculation; default is `0.975`
`model`	model for estimating conditional volatility; options are `'EWMA'` and `'GARCH'`; if `model = 'GARCH'`, additional arguments can be adjusted via `...`; default is `'EWMA'`
`method`	method to be used for calculation; default is `'plain'`
`lambda`	decay factor for the calculation of weights; default is `0.98` for `method = 'age'` and `0.94` for `method = 'vwhs'` or `method = 'fhs'`
`nout`	number of out-of-sample observations; most recent observations are used; default is `NULL`
`nwin`	window size for rolling one-step forecasting; most recent observations before out-of-sample are used; default is `NULL`
`nboot`	size of bootstrap sample; must be a single non-NA integer value with `nboot > 0`; default is `NULL`
`smoothscale`	a character object; defines the smoothing approach for the unconditional variance from the logarithm of the squared centralized returns; for `smoothscale = 'lpr'`, the unconditional variance is smoothed via the `smoots::gsmooth()` function from the `smoots` package; the bandwidth has to be chosen manually; otherwise the default is used; if `smoothscale = 'auto'`, the function `smoots::msmooth()` is employed and the bandwidth is chosen automatically (data-driven); see the documentation of the `smoots` package for more information; is set to `smoothscale = 'none'` by default
`smoothopts`	additional arguments of `smoots::gsmooth()` and `smoots::msmooth()`; see the documentation of the `smoots` package for more information; is set to customized default settings
`...`	additional arguments of the `ugarchspec` function from the `rugarch`-package; only applied if `model = 'GARCH'`; default settings for the arguments `variance.model` and `mean.model` are: `variance.model` = `list(model = 'sGARCH', garchOrder = c(1, 1))` `mean.model` = `list(armaOrder = c(0, 0))`

Value

Returns a list with the following elements:

VaR: Numerical vector containing out-of-sample forecasts of Value at Risk
ES: Numerical vector containing out-of-sample forecasts of Expected Shortfall (Conditional Value at Risk)
xout: Numerical vector containing out-of-sample returns
p: Confidence level for VaR calculation
model: Model for estimating conditional volatility
method: Method to be used for calculation
nout: Number of out-of-sample observations
nwin: Window size for rolling one-step forecasting
nboot: Size of bootstrap sample

Examples


prices <- DAX$price_close
returns <- diff(log(prices))
n <- length(returns)
nout <- 250 # number of obs. for out-of-sample forecasting
nwin <- 1000 # window size for rolling forecasts


### Example 1 - plain historical simulation
results1 <- rollcast(x = returns, p = 0.975, method = 'plain', nout = nout,
                     nwin = nwin)
matplot(1:nout, cbind(-results1$xout, results1$VaR, results1$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Plain HS - 97.5% VaR and ES for the DAX30 return series')

### Example 2 - age weighted historical simulation
results2 <- rollcast(x = returns, p = 0.975, method = 'age', nout = nout,
                     nwin = nwin)
matplot(1:nout, cbind(-results2$xout, results2$VaR, results2$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Age weighted HS - 97.5% VaR and ES for the DAX30 return series')

### Example 3 - volatility weighted historical simulation - EWMA
results3 <- rollcast(x = returns, p = 0.975, model = 'EWMA',
                     method = 'vwhs', nout = nout, nwin = nwin)
matplot(1:nout, cbind(-results3$xout, results3$VaR, results3$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Vol. weighted HS (EWMA) - 97.5% VaR and ES for the DAX30 return
  series')

### Example 4 - volatility weighted historical simulation - GARCH
results4 <- rollcast(x = returns, p = 0.975, model = 'GARCH',
                     method = 'vwhs', nout = nout, nwin = nwin)
matplot(1:nout, cbind(-results4$xout, results4$VaR, results4$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Vol. weighted HS (GARCH) - 97.5% VaR and ES for the DAX30 return
  series')

### Example 5 - filtered historical simulation - EWMA
results5 <- rollcast(x = returns, p = 0.975, model = 'EWMA',
                     method = 'fhs', nout = nout, nwin = nwin, nboot = 10000)
matplot(1:nout, cbind(-results5$xout, results5$VaR, results5$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Filtered HS (EWMA) - 97.5% VaR and ES for the DAX30 return
  series')

### Example 6 - filtered historical simulation - GARCH
results6 <- rollcast(x = returns, p = 0.975, model = 'GARCH',
                     method = 'fhs', nout = nout, nwin = nwin, nboot = 10000)
matplot(1:nout, cbind(-results6$xout, results6$VaR, results6$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Filtered HS (GARCH) - 97.5% VaR and ES for the DAX30 return
  series')

prices <- DAX$price_close
returns <- diff(log(prices))
n <- length(returns)
nout <- 250 # number of obs. for out-of-sample forecasting
nwin <- 1000 # window size for rolling forecasts


### Example 1 - plain historical simulation
results1 <- rollcast(x = returns, p = 0.975, method = 'plain', nout = nout,
                     nwin = nwin)
matplot(1:nout, cbind(-results1$xout, results1$VaR, results1$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Plain HS - 97.5% VaR and ES for the DAX30 return series')

### Example 2 - age weighted historical simulation
results2 <- rollcast(x = returns, p = 0.975, method = 'age', nout = nout,
                     nwin = nwin)
matplot(1:nout, cbind(-results2$xout, results2$VaR, results2$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Age weighted HS - 97.5% VaR and ES for the DAX30 return series')

### Example 3 - volatility weighted historical simulation - EWMA
results3 <- rollcast(x = returns, p = 0.975, model = 'EWMA',
                     method = 'vwhs', nout = nout, nwin = nwin)
matplot(1:nout, cbind(-results3$xout, results3$VaR, results3$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Vol. weighted HS (EWMA) - 97.5% VaR and ES for the DAX30 return
  series')

### Example 4 - volatility weighted historical simulation - GARCH
results4 <- rollcast(x = returns, p = 0.975, model = 'GARCH',
                     method = 'vwhs', nout = nout, nwin = nwin)
matplot(1:nout, cbind(-results4$xout, results4$VaR, results4$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Vol. weighted HS (GARCH) - 97.5% VaR and ES for the DAX30 return
  series')

### Example 5 - filtered historical simulation - EWMA
results5 <- rollcast(x = returns, p = 0.975, model = 'EWMA',
                     method = 'fhs', nout = nout, nwin = nwin, nboot = 10000)
matplot(1:nout, cbind(-results5$xout, results5$VaR, results5$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Filtered HS (EWMA) - 97.5% VaR and ES for the DAX30 return
  series')

### Example 6 - filtered historical simulation - GARCH
results6 <- rollcast(x = returns, p = 0.975, model = 'GARCH',
                     method = 'fhs', nout = nout, nwin = nwin, nboot = 10000)
matplot(1:nout, cbind(-results6$xout, results6$VaR, results6$ES),
  type = 'hll',
  xlab = 'number of out-of-sample obs.', ylab = 'losses, VaR and ES',
  main = 'Filtered HS (GARCH) - 97.5% VaR and ES for the DAX30 return
  series')

Application for downloading data from Yahoo Finance

Description

Application for downloading data from Yahoo Finance

Usage

runFTSdata()
runFTSdata()

Value

None

Standard and Poor's (SP500) Financial Time Series Data

Description

A dataset that contains the daily financial data of the SP500 from 2000 to December 2023 (currency in USD).

Usage

SP500
SP500

Format

A data frame with 6037 rows and 11 variables:

ticker: id of the stock
ref_date: date in format YY-MM-DD
price_open: opening price (daily)
price_high: highest price (daily)
price_low: lowest price (daily)
price_close: closing price (daily)
volume: trading volume
price_adjusted: adjusted closing price (daily)
ret_adjusted_prices: returns obtained from the adj. closing prices
ret_closing_prices: returns obtained from the closing prices
cumret_adjusted_prices: accumulated arithmetic/log return for the period

Source

The data was obtained from Yahoo Finance.

Backtesting of Value-at-Risk via Traffic Light Test

Description

The Traffic Light Test, is applied to previously calculated Value-at-Risk series.

Usage

trftest(obj)
trftest(obj)

Arguments

obj

A list returned by the rollcast function, that contains a Value-at-Risk series; any other list that follows the name conventions of the rollcast function can be used as well.

Details

This function uses an object returned by the rollcast function of the quarks package as an input for the function argument obj. A list with different elements, such as the cumulative probabilities for the VaR series within obj, is returned. Instead of the list, only the traffic light backtesting results are printed to the R console.

Value

A list of class quarks is returned with the following elements.

model: selected model for estimation
method: selected method for estimation
p_VaR: cumulative probability of observing the number of breaches or fewer for (1 - p)100%-VaR
pot_VaR: number of exceedances for (1 - p)100%-VaR
p: coverage level for (1-p)100% VaR

Examples


prices <- DAX$price_close
returns <- diff(log(prices))
n <- length(returns)
nout <- 250 # number of obs. for out-of-sample forecasting
nwin <- 500 # window size for rolling forecasts
results <- rollcast(x = returns, p = 0.975, method = 'age', nout = nout,
                     nwin = nwin)
trftest(results)


prices <- DAX$price_close
returns <- diff(log(prices))
n <- length(returns)
nout <- 250 # number of obs. for out-of-sample forecasting
nwin <- 500 # window size for rolling forecasts
results <- rollcast(x = returns, p = 0.975, method = 'age', nout = nout,
                     nwin = nwin)
trftest(results)

Volatility weighted historical simulation

Description

Calculates univariate Value at Risk and Expected Shortfall (Conditional Value at Risk) by means of volatility weighted historical simulation. Volatility can be estimated with an exponentially weighted moving average or a GARCH-type model.

Usage

vwhs(x, p = 0.975, model = c("EWMA", "GARCH"), lambda = 0.94, ...)
vwhs(x, p = 0.975, model = c("EWMA", "GARCH"), lambda = 0.94, ...)

Arguments

`x`	a numeric vector of asset returns
`p`	confidence level for VaR calculation; default is `0.975`
`model`	model for estimating conditional volatility; default is `'EWMA'`
`lambda`	decay factor for the calculation of weights; default is `0.94`
`...`	additional arguments of the `ugarchspec` function from the `rugarch`-package; the default settings for the arguments `variance.model` and `mean.model` are `list(model = 'sGARCH', garchOrder = c(1, 1))` and `list(armaOrder = c(0, 0))`, respectively

Value

Returns a list with the following elements:

VaR: Calculated Value at Risk
ES: Calculated Expected Shortfall (Conditional Value at Risk)
p: Confidence level for VaR calculation
garchmod: The model fit. Is the respective GARCH fit for model = 'GARCH' (see rugarch documentation) and 'EWMA' for model = 'EWMA'

Examples

prices <- DAX$price_close
returns <- diff(log(prices))
# volatility weighting via EWMA
ewma <- vwhs(x = returns, p = 0.975, model = "EWMA", lambda = 0.94)
ewma
# volatility weighting via GARCH
garch <- vwhs(x = returns, p = 0.975, model = "GARCH", variance.model =
list(model = "sGARCH"))
garch
prices <- DAX$price_close
returns <- diff(log(prices))
# volatility weighting via EWMA
ewma <- vwhs(x = returns, p = 0.975, model = "EWMA", lambda = 0.94)
ewma
# volatility weighting via GARCH
garch <- vwhs(x = returns, p = 0.975, model = "GARCH", variance.model =
list(model = "sGARCH"))
garch

Package 'quarks'

Help Index

Unconditional and Conditional Coverage Tests, Independence Test

Description

Usage

Arguments

Details

Value

References

Examples

German Stock Market Index (DAX) Financial Time Series Data

Description

Usage

Format

Source

Dow Jones Industrial Average (DJI) Financial Time Series Data

Description

Usage

Format

Source

Exponentially weighted moving average

Description

Usage

Arguments

Value

Examples

Filtered historical simulation

Description

Usage

Arguments

Value

Examples

Financial Times Stock Exchange Index (FTSE) Financial Time Series Data

Description

Usage

Format

Source

Nonparametric calculation of univariate Value at Risk and Expected Shortfall

Description

Usage

Arguments

Value

Examples

Hang Seng Index (HSI) Financial Time Series Data

Description

Usage

Format

Source

Loss Functions

Description

Usage

Arguments

Details

Value

References

Examples

Nikkei Heikin Kabuka Index (NIK) Financial Time Series Data

Description

Usage

Format

Source

Profit & Loss operator function

Description

Usage

Arguments

Value

Examples

Plot Method for the Package 'quarks'

Description

Usage

Arguments

Value

Print Method for the Package 'quarks'

Description

Usage

Arguments

Value

Rolling one-step ahead forecasts of Value at Risk and Expected Shortfall

Description

Usage