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Portfolio Stress Testing and Value at Risk (VaR) Incorporating Current Market Conditions

Krishan Mohan Nagpal

Abstract

Value at Risk (VaR) and stress testing are two of the most widely used approaches in portfolio risk management to estimate potential market value losses under adverse market moves. VaR quantifies potential loss in value over a specified horizon (such as one day or ten days) at a desired confidence level (such as 95'th percentile). In scenario design and stress testing, the goal is to construct extreme market scenarios such as those involving severe recession or a specific event of concern (such as a rapid increase in rates or a geopolitical event), and quantify potential impact of such scenarios on the portfolio. The goal of this paper is to propose an approach for incorporating prevailing market conditions in stress scenario design and estimation of VaR so that they provide more accurate and realistic insights about portfolio risk over the near term. The proposed approach is based on historical data where historical observations of market changes are given more weight if a certain period in history is "more similar" to the prevailing market conditions. Clusters of market conditions are identified using a Machine Learning approach called Variational Inference (VI) where for each cluster future changes in portfolio value are similar. VI based algorithm uses optimization techniques to obtain analytical approximations of the posterior probability density of cluster assignments (market regimes) and probabilities of different outcomes for changes in portfolio value. Covid related volatile period around the year 2020 is used to illustrate the performance of the proposed approach and in particular show how VaR and stress scenarios adapt quickly to changing market conditions. Another advantage of the proposed approach is that classification of market conditions into clusters can provide useful insights about portfolio performance under different market conditions.

Portfolio Stress Testing and Value at Risk (VaR) Incorporating Current Market Conditions

Abstract

Value at Risk (VaR) and stress testing are two of the most widely used approaches in portfolio risk management to estimate potential market value losses under adverse market moves. VaR quantifies potential loss in value over a specified horizon (such as one day or ten days) at a desired confidence level (such as 95'th percentile). In scenario design and stress testing, the goal is to construct extreme market scenarios such as those involving severe recession or a specific event of concern (such as a rapid increase in rates or a geopolitical event), and quantify potential impact of such scenarios on the portfolio. The goal of this paper is to propose an approach for incorporating prevailing market conditions in stress scenario design and estimation of VaR so that they provide more accurate and realistic insights about portfolio risk over the near term. The proposed approach is based on historical data where historical observations of market changes are given more weight if a certain period in history is "more similar" to the prevailing market conditions. Clusters of market conditions are identified using a Machine Learning approach called Variational Inference (VI) where for each cluster future changes in portfolio value are similar. VI based algorithm uses optimization techniques to obtain analytical approximations of the posterior probability density of cluster assignments (market regimes) and probabilities of different outcomes for changes in portfolio value. Covid related volatile period around the year 2020 is used to illustrate the performance of the proposed approach and in particular show how VaR and stress scenarios adapt quickly to changing market conditions. Another advantage of the proposed approach is that classification of market conditions into clusters can provide useful insights about portfolio performance under different market conditions.
Paper Structure (14 sections, 1 theorem, 55 equations, 9 figures, 1 algorithm)

This paper contains 14 sections, 1 theorem, 55 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Brief Introduction to Variational Inference
  3. VaR Incorporating Current Market Conditions
  4. Underlying Model for Data Generation
  5. Variational Inference Calibration for VaR
  6. Determining distribution of $D$ day P&L and estimating VaR
  7. Illustrative example of computing VaR for a portfolio of equities and bonds
  8. Stress Scenario Design Incorporating Current Market Conditions
  9. Underlying Model for Data Generation
  10. Variational Inference Calibration for Stress Scenario Design
  11. Stress Scenario Design : determining target scenario loss and corresponding risk factor shifts
  12. Scenario Design for Specific Market Conditions
  13. Illustrative example of stress scenario design
  14. Summary

Key Result

Proposition 4.1

Let $X$ and $Y$ be Gaussian random variables with the following mean and variances: Then the conditional estimate of $Y$ given $X$ is

Figures (9)

  • Figure 1: Illustrative example of modeling framework based on three clusters ($K=3$) and three categories of portfolio value changes from each cluster ($J=3$). Variational Inference is used to identify clusters and estimate parameters of Dirichlet distribution for different categories of outcomes (portfolio value changes) from each cluster. Distribution of portfolio returns within each of the $J$ categories is obtained from historical data of risk factor changes.
  • Figure 2: Comparison of $95\%$ One-Day P&L and VaR for the year $2020$ represented in terms of $z$-scores. VaR derived using the proposed VI approach is compared to that estimated using historical simulation and Gaussian distribution calibrated to historical simulation. VaR estimate in all cases is based on the same historical data of prior $250$ observations of daily risk factor changes. Both Historical Simulation and Gaussian approaches underestimate VaR in February and March relative to VI. When volatility subsides after May $2020$, VI VaR estimate decreases towards normal levels as it adapts to market conditions while Historical Simulation and Gaussian VaR remain at elevated levels until the volatile historical data is no longer included in the look-back period for VaR estimation.
  • Figure 3: Comparison of $97.5\%$ One-Day P&L and VaR for the year $2020$ represented in terms of $z$-scores. VaR derived using the proposed VI approach is compared to that estimated using historical simulation and Gaussian distribution calibrated to historical simulation. VaR estimate in all cases is based on the same historical data of prior $250$ observations of daily risk factor changes. Both Historical Simulation and Gaussian approaches underestimate VaR in February and March relative to VI. Gaussian distribution based VaR is lower than Historical Simulation VaR as Gaussian distribution does not capture the heavy tails of the distribution. When volatility subsides after May $2020$, VI VaR estimate decreases towards normal levels as it adapts to market conditions while Historical Simulation based VaR remains at elevated levels until the volatile historical data is no longer included in the look-back period for VaR estimation.
  • Figure 4: Comparison of $95\%$ Ten-Day P&L and VaR for the year $2020$ represented in terms of $z$-scores. VaR derived using the proposed VI approach is compared to that estimated using historical simulation and Gaussian distribution calibrated to historical simulation. VaR estimate in all cases is based on the same historical data of prior $250$ observations of daily risk factor changes. Both Historical Simulation and Gaussian approaches underestimate VaR in February and March relative to VI. Gaussian distribution based VaR is lower than Historical Simulation VaR as Gaussian distribution does not capture the heavy tails of the distribution. When volatility subsides after May $2020$, VI VaR estimate decreases towards normal levels as it adapts to market conditions while Historical Simulation based VaR remains at elevated levels until the volatile historical data is no longer included in the look-back period for VaR estimation.
  • Figure 5: Comparison of $97.5\%$ Ten-Day P&L and VaR for the year $2020$ represented in terms of $z$-scores. VaR derived using the proposed VI approach is compared to that estimated using historical simulation and Gaussian distribution calibrated to historical simulation. VaR estimate in all cases is based on the same historical data of prior $250$ observations of daily risk factor changes. Both Historical Simulation and Gaussian approaches underestimate VaR in February and March relative to VI. Gaussian distribution based VaR is lower than Historical Simulation VaR as Gaussian distribution does not capture the heavy tails of the distribution. When volatility subsides after May $2020$, VI VaR estimate decreases towards normal levels as it adapts to market conditions while Historical Simulation based VaR remains at elevated levels until the volatile historical data is no longer included in the look-back period for VaR estimation.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Proposition 4.1