Estimating Extreme Quantiles of Compound Frequency Distributions Using a Multiplier Method

Enhancing Operational Risk reserves: A Novel approach to Quantile estimation

By Archyde | October 27, 2023

Financial institutions frequently enough rely on the loss distribution approach to calculate operational risk reserves. The precision of these estimations is directly tied to how accurately the extreme quantiles of aggregate loss distributions are persistent.

Various methods have been developed to estimate these critical extreme quantiles. Many of these rely on approximations, such as those based on single-loss and perturbative models.These, in turn, require estimating even more extreme quantiles from the underlying severity distributions.

However,a significant challenge arises when fitting parametric severity distributions. While these models can accurately represent the bulk of the data, they often falter in capturing the tail characteristics that are crucial for robust risk assessment.

To overcome this, a novel approach proposes a more pragmatic solution. Instead of directly estimating the most extreme quantiles, it suggests a nonparametric estimation of a less extreme, lower quantile of the severity distribution. This is expected to yield greater accuracy.

This lower quantile is then multiplied by a specific factor to arrive at an estimate for the required extreme quantile of the compound distribution.This crucial multiplier is derived using principles of extreme value theory and single-loss and perturbative approximations.

A simulation

What are the limitations of using the expected value of the claim frequency (E[N]) as the effective frequency (λ) in the multiplier method, and how can a truncated expectation improve the estimation?

Compound frequency distributions – arising from the combination of severity and frequency distributions – are prevalent in actuarial science, risk management, and operational risk modeling. Accurately estimating extreme quantiles (Value at Risk or VaR, Expected Shortfall) for these distributions is crucial for capital allocation, regulatory compliance, and informed decision-making. Conventional methods often struggle with the computational complexity and accuracy when dealing with extreme tail events. This article details the multiplier method, a powerful technique for estimating these critical quantiles.

Understanding the Challenge: Compound Distributions & Tail Estimation

Many real-world risks aren’t governed by a single distribution. Instead, they’re compound. Consider insurance losses: the frequency of claims (number of claims within a period) follows one distribution (e.g., Poisson), while the severity of each claim (amount of each individual loss) follows another (e.g., Gamma, Lognormal).The overall distribution of aggregate losses is the convolution of these two.

Estimating extreme quantiles – say,the 99.9th percentile – presents several challenges:

Computational Burden: Direct convolution and quantile estimation can be computationally intensive, especially for high-frequency distributions.

Model Risk: Inaccurate assumptions about the underlying frequency or severity distributions can substantially impact quantile estimates.

Tail Dependence: The tails of the frequency and severity distributions interact, potentially leading to heavier tails in the aggregate distribution than anticipated. This is particularly relevant for extreme value theory applications.

Non-Standard Distributions: Compound distributions often lack closed-form expressions, making analytical quantile calculations unachievable.

The Multiplier Method: A Detailed Explanation

The multiplier method offers a computationally efficient and relatively accurate approach to estimating extreme quantiles of compound distributions. It leverages the concept of a “multiplier” to approximate the tail behavior.

Core Principle

The method relies on the following idea: for a large enough quantile level (e.g., 99.9%),the aggregate loss is dominated by a small number of large claims. The multiplier essentially scales the severity distribution to account for the expected number of claims contributing to the tail.

Steps Involved

1. Define Distributions: Clearly define the frequency distribution F(x) (with probability mass function p(x)) and the severity distribution S(y) (with probability density function s(y)).
2. Determine the Quantile level (q): Specify the desired quantile level (e.g., q = 0.999 for the 99.9th percentile).
3. Calculate the Mean Severity (μ): Compute the expected value of the severity distribution: μ = E[S] = ∫ y s(y) dy.
4. Estimate the Effective Frequency (λ): This is the crucial step. The effective frequency represents the expected number of claims contributing to the extreme tail. A common approximation is λ = E[N] where N is the number of claims. More complex approaches involve truncating the frequency distribution at a certain level.
5. Apply the Multiplier: The estimated quantile (VaR_q) is than approximated as: VaR_q ≈ λ Quantile_q(S), where Quantile_q(S) is the q-th quantile of the severity distribution.
6. Refinement (Optional): For improved accuracy, iterative refinement can be applied. The initial quantile estimate can be used to adjust the effective frequency, and the process repeated until convergence.

Practical Considerations & Enhancements

Frequency Distribution Choice: The Poisson distribution is frequently used for claim frequency, but negative binomial or other discrete distributions may be more appropriate depending on the data.

Severity Distribution Choice: Common choices include Gamma, Lognormal, Pareto, and Generalized Pareto distributions. Goodness-of-fit tests are essential to validate the chosen distribution.

effective Frequency Refinement: Instead of simply using E[N], consider a truncated expectation: λ = Σ[x[x p(x)]for x > threshold. The threshold can