Fixing Integral Divergence in the Inverse Weibull Distribution's Tail Value at Risk (TVaR)

Understanding Integral Divergence in TVaR Calculation

The Tail Value at Risk (TVaR) is a crucial metric in risk management, especially in the context of modeling extreme events. However, when using distributions like the Inverse Weibull, calculating TVaR can sometimes lead to complex issues, such as integral divergence.

In this article, we explore a specific problem encountered when calculating TVaR for an Inverse Weibull distribution. This issue arises during the integration process, and it can lead to errors indicating that the integral may be divergent.

Despite attempts to adjust parameters, such as increasing the number of subdivisions in the integration, the error persists. Understanding why this happens and how to correct it is essential for anyone working with heavy-tailed distributions in actuarial science or financial risk analysis.

We will walk through the problem, identify the possible reasons for integral divergence, and provide suggestions on how to resolve this issue effectively. By the end of this article, you will be equipped with practical strategies to overcome similar challenges in TVaR computations.

Resolving TVaR Calculation Issues in Inverse Weibull Distribution

The scripts created above are focused on calculating the Tail Value at Risk (TVaR) for an Inverse Weibull distribution. TVaR is used to estimate the expected loss in extreme tail events, making it a critical metric in risk management, particularly in fields like insurance and finance. The first script uses traditional numerical integration to compute TVaR, which unfortunately leads to an error due to integral divergence. This occurs because the integral for the tail distribution may become unbounded, especially when dealing with heavy-tailed distributions like the Inverse Weibull.

One key command in this process is the integrate() function, which performs numerical integration over the distribution’s tail. The error arises when the integration extends to infinity, and this is where the problem lies. To mitigate this, we attempt to bound the integration using quantiles derived from the Inverse Weibull distribution. Commands like qinvweibull() help in this respect by allowing us to calculate Value at Risk (VaR) at various confidence levels (e.g., 70%, 80%, 90%). By using these quantiles, we aim to control the range of the integral and reduce divergence.

The second approach takes a different route by using Monte Carlo simulation. Instead of relying on analytical integration, it simulates thousands of random values from the Inverse Weibull distribution using the rinvweibull() command. This method circumvents the integral divergence problem by generating empirical data and calculating TVaR based on the mean loss above the VaR threshold. This is particularly useful when dealing with distributions that are difficult to integrate analytically, as it provides a more flexible, albeit computationally intensive, alternative.

To ensure the robustness of these methods, unit testing is also implemented. The test_that() function from the testthat package is used to validate the results of the Monte Carlo simulation. By running these tests, we verify that the simulated TVaR values are logical and non-negative. This process of testing helps ensure that the solutions not only work correctly in theory but also produce valid results across different environments. This approach makes the scripts modular and reusable for similar risk calculations in other contexts.

install.packages("evd")
library(evd)
data(lossalae)
attach(lossalae)
x <- ALAE / 1000
install.packages("fitdistrplus")
library(fitdistrplus)
library(actuar)
W.INV <- fitdist(x, "invweibull")
VarinvW1 <- qinvweibull(0.7, shape = W.INV$estimate[1], scale = W.INV$estimate[2])
VarinvW3 <- qinvweibull(0.9, shape = W.INV$estimate[1], scale = W.INV$estimate[2])
integrand2 <- function(x) { x * dinvweibull(x, shape = W.INV$estimate[1], scale = W.INV$estimate[2]) }
Tvarinv1 <- (1 / (1 - 0.7)) * integrate(integrand2, VarinvW1, VarinvW3, subdivisions = 1000)$value
print(Tvarinv1)
# Bounded integration using a large but finite upper limit to avoid divergence

install.packages("evd")
library(evd)
data(lossalae)
attach(lossalae)
x <- ALAE / 1000
library(actuar)
W.INV <- fitdist(x, "invweibull")
n_sim <- 100000  # Number of simulations
sim_data <- rinvweibull(n_sim, shape = W.INV$estimate[1], scale = W.INV$estimate[2])
var_70 <- quantile(sim_data, 0.7)
tvar_70 <- mean(sim_data[sim_data > var_70])
print(tvar_70)
# Monte Carlo approach avoids analytical integration issues

test_that("Monte Carlo TVaR calculation works", {
   n_sim <- 100000
   sim_data <- rinvweibull(n_sim, shape = W.INV$estimate[1], scale = W.INV$estimate[2])
   var_70 <- quantile(sim_data, 0.7)
   tvar_70 <- mean(sim_data[sim_data > var_70])
   expect_true(tvar_70 > 0)
})

Addressing TVaR Calculation Challenges for Heavy-Tailed Distributions

When calculating the Tail Value at Risk (TVaR) for distributions with heavy tails, such as the Inverse Weibull, one key challenge is dealing with the behavior of the distribution in its extreme tail. This is where integral divergence can occur, leading to computational issues. A fundamental aspect of this issue stems from how the tail behaves at very high quantiles, where small variations in parameters can lead to significant differences in the calculated risk metric. Understanding how to manage these extremes is critical for ensuring accurate risk assessments.

Another relevant factor to consider when working with TVaR calculations is the method of handling infinite upper bounds during integration. In practical terms, many risk management applications set a large, but finite, upper limit to avoid issues with divergence. This approach helps control the computation, especially in situations where exact mathematical solutions might be hard to derive. Methods like bounding the integral or applying Monte Carlo simulations allow for more stable results while still capturing the essence of risk in the tail.

Monte Carlo simulations, as discussed in previous solutions, are an excellent alternative for overcoming the pitfalls of direct integration. By generating a large set of random samples from the Inverse Weibull distribution, you can empirically estimate the expected losses. This approach is highly flexible and avoids the need for complex mathematical integration, making it a preferred method when working with distributions where traditional methods fail. It’s particularly useful for heavy-tailed data, where the behavior of extreme events can be difficult to predict using standard models.