Aug 07

Value at Risk (VaR): 2007 FRM. Part 6 - Stress Testing

by David Harper, CFA, FRM, CIPM

FRM | Risk |

var_correlationSpike_intro

2007 FRM Learning Outcomes

  • LO 7.13: Discuss the implications of correlation breakdown for scenario analysis.
  • LO 7.14: Describe the primary approaches to stress testing and the advantages and disadvantages of each approach.
  • LO 7.15: Describe the worst case scenario measure as an extension to VAR.

Correlation breakdown

For FRM candidates, a really important "object" is the covariance matrix. This is a square matrix that contains, as elements, the covariances between pairs of asset returns. Recall a basic relationship for the simple two-asset portfolio: covariance = [correlation between A,B][standard deviation A][standard deviation B]. Note how the correlation is embedded inside the covariance. It is the same with a covariance matrix: it contains an embedded correlation matrix. In either case, whether we refer to a covariance matrix or a correlation matrix, we refer to the object that quantifies the diversification benefit "into" the value at risk (VaR) calculation.

So, whether we are performing a delta-Normal (parametric) VaR or a structured Monte Carlo, the correlation matrix contains the critical assumption about the co-movement relationships. As below on the left, we have a correlation matrix (note: it must have ones on the diagonal, as a variable is perfectly correlated with itself!).

A thematic idea in the FRM is that correlations tend to breakdown during crisis. Or, as Abnormal Returns says, correlations tend toward one during "negative market extremes."

correlationmatrix2

This is the doubly-whammy fear of contagion: not only that systematic exposures suffer their their beta-based loss, but that portfolio diversification is temporarily erased. This is alarming: you don't need diversification on the upside; you need it on the downside. This is an important theme because many of the models (i.e., that we study in the FRM) contain an "embedded" correlation matrix.

Stress Testing

Stress testing of the portfolio can include either (i) imagined scenarios or (ii) historical scenarios. The latter takes the current portfolio and runs it through a historical scenario. For advantages and disadvantages, we have:

  Advantages Disadvantages

Structured Monte Carlo (from last post)
  • Generate correlated scenarios based on a statistical distribution
  • But may not be relevant going forward



Stress testing
  • Illuminates portfolio exposure to risk factors
  • Can focus on the extreme loss tails
  • Complements VaR
  • May generate unwarranted red flags
  • Highly subjective (can be hard to imagine catastrophes)

 

Worst-case Scenario (WCS)

You know by know that VaR is probabilistic. At 95% confidence, it is the loss that we expect to not exceed 95 out of 100 times. If we consider a 100% VaR, then we have the worst-case scenario. It is the worst possible loss, beyond the VaR, at the "tip of the loss tail." It points to a weakness in VaR: for any given VaR, there can be any number of loss distributions "below the VaR" (or "to the left of"). For example, if daily VaR at 95% is -$1 million, that doesn't tell you whether the worst-case scenario is -$3 million or -$50 million!

So the worst-case scenario complements (extends) VaR. For the learning outcome, there are three points:

  1. The WCS assumes the firm increases its level of investment when gains are realized; i.e., that the firm is "capital efficient."
  2. The effects of time-varying volatility are ignored
  3. There is still the extreme tail issue: it is still possible to underestimate the likelihood of extreme left-tail losses

Takeaways

  • Correlations tend toward one during market stress or crisis (and exacerbate the contagion effect)
  • Both stress testing and worst-case scenario (WCS) complement, do not necessarily replace, value at risk.
  • Stress testing is considered a vital approach to understanding the nuances of portfolio risk
  • The worst-case scenario (WCS) helps "plug the hole" in VaR; i.e., that VaR does not say anything about the loss distribution in excess of VaR. (Note: so does extreme value theory).

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