A novel scaling approach for unbiased adjustment of risk estimators

Paper: arXiv 2312.05655

Abstract

The assessment of risk based on historical data faces many challenges, in particular due to the limited amount of available data, lack of stationarity, and heavy tails. While estimation on a short-term horizon for less extreme percentiles tends to be reasonably accurate, extending it to longer time horizons or extreme percentiles poses significant difficulties. The application of theoretical risk scaling laws to address this issue has been extensively explored in the literature. This paper presents a novel approach to scaling a given risk estimator, ensuring that the estimated capital reserve is robust and conservatively estimates the risk. We develop a simple statistical framework that allows efficient risk scaling and has a direct link to backtesting performance. Our method allows time scaling beyond the conventional square-root-of-time rule, enables risk transfers, such as those involved in economic capital allocation, and could be used for unbiased risk estimation in small sample settings. To demonstrate the effectiveness of our approach, we provide various examples related to the estimation of value-at-risk and expected shortfall together with a short empirical study analysing the impact of our method.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a dense theoretical framework involving advanced statistical theory, risk measures, and probability, resulting in high math complexity. The empirical section is described as a ‘short empirical study’ without detailed backtests, datasets, or code, placing it in the lower range for empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Unbiased Risk Scaling for VaR/ES"] --> B["Methodology<br>Novel Statistical Scaling Framework"]
  
  B --> C["Inputs<br>Historical Risk Data"]
  C --> D["Compute<br>Scaled Risk Estimators<br>with Backtesting Link"]
  
  D --> E{"Validation"}
  E -->|Time Scaling| F["Outcomes<br>Robust Capital Reserves"]
  E -->|Risk Transfer| F
  E -->|Small Samples| F
  
  F --> G["Empirical Results<br>Improved Accuracy & Conservatism"]

Cited References (246)

1. Strategic long-term financial risks: Single risk factors (2005)
2. Time-scaling of value-at-risk in GARCH(1,1) and AR(1)-GARCH(1,1) processes (2007)
3. Performance of Scaling by the Square-root-of-time rule for FRTB Internal Model Expected Shortfall (2022)
4. On the time scaling of value-at-risk with trading (2011)
5. On the statistics of scaling exponents and the multiscaling value at risk (2021)
6. Long-term value at risk (2004)
7. A simple method for time scaling value-at-risk: Let the data speak for themselves (2010)
8. Value-at-Risk and Extreme Returns (2000)

0 of 246 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.