[Paper Review] New Bayesian method for estimation of Value at Risk and Conditional Value at Risk

J. Martı́n, M. Isabel Parra+2 more|arXiv (Cornell University)|Jun 21, 2023

Financial Risk and Volatility Modeling4 citations

TL;DR

This paper proposes a novel Bayesian method, IPBMH, for estimating Value at Risk (VaR) and Conditional Value at Risk (CVaR) by leveraging highly informative priors that incorporate full dataset information, especially tail behavior via Generalized Pareto Distribution (GPD) relationships. The method outperforms standard Metropolis-Hastings (MH) and Bayesian methods with non-informative priors in accuracy, precision, and stability—especially with small samples—demonstrated through simulations and real financial data from the IBEX35 index.

ABSTRACT

Value at Risk (VaR) and Conditional Value at Risk (CVaR) have become the most popular measures of market risk in Financial and Insurance fields. However, the estimation of both risk measures is challenging, because it requires the knowledge of the tail of the distribution. Therefore, tools from Extreme Value Theory are usually employed, considering that the tail data follow a Generalized Pareto distribution (GPD). Using the existing relations from the parameters of the baseline distribution and the limit GPD's parameters, we define highly informative priors that incorporate all the information available for the whole set of observations. We show how to perform Metropolis-Hastings (MH) algorithm to estimate VaR and CVaR employing the highly informative priors, in the case of exponential, stable and Gamma distributions. Afterwards, we perform a thorough simulation study to compare the accuracy and precision provided by three different methods. Finally, data from a real example is analyzed to show the practical application of the methods.

Motivation & Objective

• To address the challenge of estimating VaR and CVaR when tail distribution parameters are unknown and data are scarce.
• To improve estimation accuracy and precision in risk modeling by incorporating all available data, not just extreme values above a threshold.
• To develop a Bayesian framework that leverages known relationships between baseline distribution parameters and GPD parameters for more informative inference.
• To evaluate the performance of the new method (IPBMH) against standard MH and Bayesian methods under varying sample sizes and distributions.
• To demonstrate practical applicability using real financial returns data from the IBEX35 index during 2020 and 2022.

Proposed method

• Proposes a new Bayesian method, IPBMH, using highly informative priors derived from the full dataset, including relationships between baseline distribution parameters and GPD parameters.
• Applies the Metropolis-Hastings (MH) algorithm to estimate VaR and CVaR under exponential, stable (Cauchy, Normal), and Gamma distributions.
• Constructs informative priors that weight tail observations more heavily by embedding known asymptotic relationships between baseline and GPD parameters.
• Employs the Peaks-over-Threshold (POT) method, modeling excesses over a high threshold $u$ using a Generalized Pareto Distribution (GPD).
• Uses historical simulation as a benchmark for real-data comparison, computing VaR and CVaR from past returns $R_t = \ln(P_t / P_{t-1})$.
• Performs simulation studies across multiple sample sizes and parameter settings to compare accuracy, precision, and bias of IPBMH, MH, and BMH methods.

Experimental results

Research questions

• RQ1 How does the proposed IPBMH method compare in accuracy and precision to standard MH and Bayesian methods when estimating VaR and CVaR?
• RQ2 To what extent does using informative priors based on full dataset information improve estimation performance, especially with small samples?
• RQ3 How well does the IPBMH method perform on real financial data, particularly during periods of market stress like the COVID-19 pandemic?
• RQ4 What is the impact of the baseline distribution (exponential, Gamma, stable) on the performance of the IPBMH method?
• RQ5 Can the IPBMH method provide more stable and less dispersed estimates than MH when the true distribution is unknown?

Key findings

• IPBMH consistently provided the most accurate and precise estimates of VaR and CVaR across all simulated distributions and sample sizes, especially when data were scarce.
• For small sample sizes, IPBMH showed significantly lower bias and dispersion compared to MH and BMH, with the advantage diminishing as sample size increased.
• In the real-data application using IBEX35 daily logarithmic returns, IPBMH produced more stable estimates with lower variability than MH, as shown in Figure 8.
• The green dashed lines in Figure 8, representing the true risk measures, were consistently closer to the IPBMH (blue) estimates than to the MH (red) estimates.
• BMH and IPBMH showed similar performance for stable distributions (Cauchy and Normal), regardless of parameter values or sample size, indicating robustness in heavy-tailed settings.
• The method’s reliance on full-data informative priors enabled better tail modeling than MH, which only uses data above a threshold, especially when the baseline distribution is unknown.