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

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

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.

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

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.

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

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.

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

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.

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

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.

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)

Arguments

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)

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)

Arguments

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')

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, ...)

Arguments

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

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)

Arguments

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)

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)

Arguments

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)

Profit & Loss operator function

Description

Calculates portfolio returns or losses by assigning weights

Usage

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

Arguments

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))

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, ...)

Arguments

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, ...)

Arguments

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(),
  ...
)

Arguments

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')

Application for downloading data from Yahoo Finance

Description

Application for downloading data from Yahoo Finance

Usage

runFTSdata()

Value

None

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)

Arguments

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)

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, ...)

Arguments

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