Value-at-Risk time scaling for long-term risk estimation

TL;DR

This paper presents a methodology for accurately scaling Value-at-Risk over long horizons and extreme percentiles, considering non-normal P&L distributions and their impact on economic capital estimation.

Contribution

It introduces a convolution-based scaling approach for VaR that accounts for leptokurtic distributions, improving long-term risk estimation accuracy.

Findings

Convolution-based scaling can increase VaR estimates by up to four times compared to normal assumptions.

Different distribution fits significantly affect the long-term VaR calculation.

The choice of scaling method impacts the estimated Economic Capital substantially.

Abstract

In this paper we discuss a general methodology to compute the market risk measure over long time horizons and at extreme percentiles, which are the typical conditions needed for estimating Economic Capital. The proposed approach extends the usual market-risk measure, ie, Value-at-Risk (VaR) at a short-term horizon and 99% confidence level, by properly applying a scaling on the short-term Profit-and-Loss (P&L) distribution. Besides the standard square-root-of-time scaling, based on normality assumptions, we consider two leptokurtic probability density function classes for fitting empirical P&L datasets and derive accurately their scaling behaviour in light of the Central Limit Theorem, interpreting time scaling as a convolution problem. Our analyses result in a range of possible VaR-scaling approaches depending on the distribution providing the best fit to empirical data, the desired percentile level and the time horizon of the Economic Capital calculation. After assessing the different approaches on a test equity trading portfolio, it emerges that the choice of the VaR-scaling approach can affect substantially the Economic Capital calculation. In particular, the use of a convolution-based approach could lead to significantly larger risk measures (by up to a factor of four) than those calculated using Normal assumptions on the P&L distribution.