lossfunc function

Loss Functions

This functions allows for the calculation of loss functions for the selection of models.

lossfunc(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 varcast function can be directly passed to lossfunc.

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

Returns

an S3 class object, which is a list of

• loss.func1: Regulatory loss function.
• loss.func2: Firm's loss function following Sarma et al. (2003).
• loss.func3: Loss function following Abad et al. (2015).
• loss.func4: Feng's loss function. A compromise of regulatory and firm's loss function.

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.

Let $K$ be the number of observations and $r_t$ the observed return series. Following Sarma et al. (2003)

$$l_{t,1} = \{\widehat{ES}_t (\alpha) + r_t \}^2,$$

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

$$l_{t,1} = \beta * \widehat{ES}_t (\alpha),$$

otherwise,

is a suitable loss function (firm's loss function), where $\beta$ is the opportunity cost of capital. The regulatory loss function is identical to the firm's loss function with the exception of $l_{t,1} = 0$ for $-r_t \leq \widehat{ES}_t (\alpha)$.

Abad et al. (2015) proposed another loss function

$$l_{t,a} = \{\widehat{ES}_t(\alpha) + r_t\}^2,$$

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

$$l_{t,a} = \beta * (\widehat{ES}_t (\alpha) + r_t),$$

otherwise,

that, however, also considers opportunity costs for $r_t > 0$. An adjustment has been proposed by Feng. Following his idea,

$$l_{t,2} = \{\widehat{ES}_t(\alpha) + r_t\}^2,$$

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

$$l_{t,2} = \beta * \min\{\widehat{ES}_t(\alpha) + r_t, \widehat{ES}_t(\alpha)\},$$

otherwise,

should be considered as a compromise of the regulatory and the firm's loss functions. Note that instead of the ES, also a series of Value-at-Risk values can be inserted for the argument obj$ES. However this is not possible if a list returned by the varcast function is directly passed to lossfunc.

Examples

# Example for Walmart Inc. (WMT)
prices <- WMT$price.close
output <- varcast(prices)
Loss <- -output$ret.out
ES <- output$ES
loss.data <- list(Loss = Loss, ES = ES)
lossfunc(loss.data)

# directly passing an output object of 'varcast()' to 'lossfunc()'
x <- WMT$price.close
output <- varcast(prices)
lossfunc(output)

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.

Author(s)

• Sebastian Letmathe (Scientific Employee) (Department of Economics, Paderborn University)
• Dominik Schulz (Scientific Employee) (Department of Economics, Paderborn University),