loss_functions,fEGarch_risk-method {fEGarch} | R Documentation |
Loss Function Calculation
Description
Compute loss function values given log-returns and corresponding value at risk (VaR) and expected shortfall (ES) series.
Usage
## S4 method for signature 'fEGarch_risk'
loss_functions(object, penalty = 1e-04, ...)
Arguments
object |
an object of class |
penalty |
the penalty term to use in the opportunity cost terms. |
... |
currently without use. |
Details
Let n \in \mathbb{N}
be the number of observations of a (log-)return
series \{r_t\}
, t=1,\dots,n
, and let \text{VaR}_t
and
\text{ES}_t
be the
estimated or forecasted VaR and ES (at some confidence level \alpha
) at time
t
, respectively. Such series are included in an object of class
"fEGarch_risk"
. In the following, a risk measure at time t
is
simply denoted by \text{RM}_t
and can either mean
\text{VaR}_t
or
\text{ES}_t
.
Based on a calculated VaR and / or expected shortfall (ES), capital needs
to be held back following regulatory rules. Commonly, among many models
used for forecasting risk measures that fulfill regulatory conditions,
loss functions are computed that also consider opportunity costs in to
assess, what model that fulfills regulatory rules minimizes such loss
functions. Let \Omega \geq 0
be the penalty term.
For all loss functions we have
\text{LF}_i = \sum_{t=1}^{n} l_{t,i}, \hspace{3mm} i = 0,1,2,3,
as the loss function with
l_{t,i} = (\text{RM}_t - r_t)^2, \hspace{3mm} i = 0,1,2,3,
for r_t < \text{RM}_t
. They differ in how the case
r_t \geq \text{RM}_t
is treated.
The regulatory loss function (rlf
) uses l_{t,0} = 0
.
The firm's loss function (Sarma et al., 2003) (flf
) considers
l_{t,1}=\Omega |\text{RM}_t|
.
The adjusted loss function (Abad et al., 2015) (alf
) makes use
of l_{t,2} = \Omega |\text{RM}_t - r_t|
.
The corrected loss function (Feng, forthcoming) (clf
) has
l_{t,3} = \Omega \text{min}\left(|\text{RM}_t - r_t|, |\text{RM}_t|\right)
.
Value
Returns a list with the four elements rlf
, flf
,
alf
and clf
, each lists with numeric vector
elements VaR
and ES
. The four elements correspond
to the regulatory loss function, the firm's loss function,
the adjusted loss function and the corrected loss function.
References
Abad, P., Muela, S. B., & MartÃn, C. L. (2015). The role of the loss function in value-at-risk comparisons. The Journal of Risk Model Validation, 9(1): 1-19. DOI: 10.21314/JRMV.2015.132.
Sarma, M., Thomas, S., & Shah, A. (2003). Selection of Value-at-Risk models. Journal of Forecasting, 22(4): 337-358. DOI: 10.1002/for.868.
Examples
window.zoo <- get("window.zoo", envir = asNamespace("zoo"))
rt <- window.zoo(SP500, end = "2002-12-31")
model <- fEGarch(egarch_spec(), rt, n_test = 250)
fcast <- predict_roll(model)
risk <- measure_risk(fcast, measure = c("VaR", "ES"), level = c(0.95, 0.975, 0.99))
loss_functions(risk)