# How big should a Stress Shock be?

## TL;DR

This paper determines the number of standard deviations needed for stress shocks to surpass historical observations considering kurtosis, validating stress test models and improving bounds over classical inequalities.

## Contribution

It introduces a method to calculate stress shock sizes accounting for kurtosis, enhancing the accuracy of stress testing and bounds on deviations.

## Key findings

- Stress shocks exceeding historical maxima can be quantified using kurtosis.
- Validation of the Brace-Lauer-Rado stress test model with new bounds.
- Tighter bounds than classical Chebyshev's Inequality extensions.

## Abstract

Stress shocks are often calculated as multiples of the standard deviation of a history set. This paper investigates how many standard deviations are required to guarantee that this shock exceeds any observation within the history set, given the additional constraint of kurtosis. The results of this analysis are then used to validate the shocks produced by some stress test models, in particular that of Brace-Lauer-Rado. A secondary application of our results is to investigate three known extensions of Chebyshev's Inequality where the kurtosis is known. It is found that our results give a tighter bound than the well-known inequalities.