Estimating and backtesting risk under heavy tails

Marcin Pitera; Thorsten Schmidt

Home/Archive/arXiv

arXiv · 2201.10454

Estimating and backtesting risk under heavy tails

Marcin Pitera·Thorsten Schmidt

View original source ↗Read source document ↗Export metadata ↓

Abstract

While the estimation of risk is an important question in the daily business of banking and insurance, many existing plug-in estimation procedures suffer from an unnecessary bias. This often leads to the underestimation of risk and negatively impacts backtesting results, especially in small sample cases. In this article we show that the link between estimation bias and backtesting can be traced back to the dual relationship between risk measures and the corresponding performance measures, and discuss this in reference to value-at-risk, expected shortfall and expectile value-at-risk. Motivated by the consistent underestimation of risk by plug-in procedures, we propose a new algorithm for bias correction and show how to apply it for generalized Pareto distributions to the i.i.d.\ setting and to a GARCH(1,1) time series. In particular, we show that the application of our algorithm leads to gain in efficiency when heavy tails or heteroscedasticity exists in the data.

Explore related subjects

q-fin.RM

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Marcin Pitera, Thorsten Schmidt. 2022-01-26. Estimating and backtesting risk under heavy tails. https://arxiv.org/abs/2201.10454

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Simplifying Cyber Cat(astrophe)s with Cyber Kittens: Power Law Plausibility for Cyber Insurance Risks

Cyber insurance requires accurate modeling of worst-case catastrophic (cat) events, but the field lacks robust quantitative approaches for estimating upper-bound losses. Building on a recent dataset of 24 cyber cat events over 30 years, this work tests whether cyber economic losses follow a power law distribution. We analyze "cyber kittens" - sub-1B USD events distinguished from cat events (1B+ USD) only by magnitude - extracted via LLM from cyber insurance claims data (2020-2024). Using victim count (weighted by claim year) as a proxy for economic loss, we link kitten-sized events to known cat events to estimate losses. The kitten distribution proved consistent with the cat dataset, and power laws were statistically plausible: each order-of-magnitude increase in event size corresponds to a 5-7x drop in probability. Extrapolating, an event 100x the largest 2020-2024 cat event is expected roughly every 206 years, translating to 100-250B USD in losses - catastrophic, but not extraordinary relative to other insurance lines.

q-fin.RM↗

arXiv2026

Robust Asset-Liability Management

Financial institutions often cannot replicate long-dated liabilities with available bonds, especially when leverage and collateral constraints bind. We characterize the feasible portfolio that minimizes, to first order, the worst equity loss over a prespecified set of yield curve changes. The solution accommodates general deterministic liabilities and portfolio constraints and includes duration and convexity matching as special cases. The minimized loss measures the economic capital buffer needed to absorb the specified interest rate stress. Under an $\ell^2$ stress set, the portfolio is a constrained generalized least squares projection. A portfolio representation shows how additional stress directions can reduce reliance on unstable exact hedges. In U.S. Treasury applications, robust immunization delivers the best or nearly best funding performance for most liabilities and constraints. Simulated and historical dynamic exercises also show modest leverage, turnover, and transaction costs.

q-fin.RM↗

arXiv2026

Quantifying the degree of risk aversion of spectral risk measures

This paper introduces a quantitative notion of the degree of risk aversion of spectral risk measures. We define a family of degree functionals characterized by three axioms governing normalization, mixtures, and continuity. The resulting degrees admit representations in terms of both the Kusuoka measure and the dual utility function, leading to connections with generalized means, the Gini coefficient, and an Arrow-Pratt-type measure of curvature. We further relate the parameter of the degree functional to the tail behavior of losses through generalized Pareto distributions, which provides an interpretation of the parameter choice and a basis for calibrating spectral risk measures according to their desired treatment of different tail behaviors. The degree functional is consistent with several dominance relations between dual utility functions, and the full degree profile uniquely identifies a spectral risk measure. Finally, we extend the degree functional to law-invariant coherent risk measures through their minimal Kusuoka representations.

q-fin.RM↗