A financially justifiable and practically implementable approach to coherent stress testing

Abstract

We present an approach to stress testing that is both practically implementable and solidly rooted in well-established financial theory. We present our results in a Bayesian-net context, but the approach can be extended to different settings. We show (i) how the consistency and continuity conditions are satisfied; (ii) how the result of a scenario can be consistently cascaded from a small number of macrofinancial variables to the constituents of a granular portfolio; and (iii) how an approximate but robust estimate of the likelihood of a given scenario can be estimated. This is particularly important for regulatory and capital-adequacy applications.

Keywords:

• Stress testing
• Scenario analysis
• Risk management
• Bayesian nets
• Factor models

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

† For a survey of the stress testing-related regulatory initiatives of the post-Basel-II period, see, e.g. Cannata and Quagliarello (2011), Chapter 6 and Part III in particular.

† Breuer et al. (2009) clearly highlight this aspect of model risk—also raised in Berkowitz (1999)—for their approach, which is based on the Mahalanobis distance: ‘We measure plausibility of scenarios by its Mahalanobis distance from the expectation. This measure of distance involves a fixed distribution of risk factors, which is estimated from historical data ’ (p. 333).

‡ See Diebold and Rudebusch (2013) for a discussion of the topic, and Joslin et al. (2013) for the conditions under which non-statistical information improves the predicting power of probabilistic models.

† Purely historical scenarios are immune to this criticism. Useful as they are, their shortcomings are well known.

† p. 8: ‘A stress test is commonly described as the evaluation of the financial position of a bank under a severe but plausible scenario to assist in decision making within the bank’.

‡ p. 13.

§ The stress-testing approach we propose is built on causal foundations. For a thorough discussion of how a causal (as opposed to associative, as embodied in statistical correlations) organization of information leads much more directly to intervention, see Pearl (2009) and Pearl and Mackenzie (2018), Chapter 7 in particular.

† For instance, in markets where options are traded a continuum of strikes gives access to the state price densities (in the pricing measure). See, Breeden and Litzenberger (1978). Market-implied volatilities of options have been shown to be quickly responsive to changing market conditions (but they also include a component due to the changing market price of volatility risk). There is very little information, however, about market-implied correlations.

‡ For a detailed treatment of Bayesian nets, see, e.g. Jensen and Nielsen (2007) and Pearl (2009) for the advantages of causally based Bayesian nets.

† The second cornerstone of Bayesian Net construction is the No-Constraint theorem, which says that if one deals only with ‘canonical’ conditional probabilities—i.e. exactly the conditional probabilities required by the conditional probability tables—one can assign to them any number between 0 and 1, and rest assured that the resulting Bayesian Net will be mathematically consistent. See Rebonato and Denev (2013) for a proof of the theorem and for the definition of canonical conditional probabilities. See Rebonato (2010) for the problems than can arise if one assigns non-canonical conditional probabilities.

† To avoid confusion, we only use Greek letters with different symbols from the corresponding Latin ones.

‡ Theoretically, we could accommodate as many states as required; the limit of three is imposed for pragmatic reasons.

† For a practitioner-friendly introduction to the Arbitrage Pricing Theory and the construction of factor-mimicking portfolios, see Burnmeister et al. (1994).

‡ The exact equality sign in Equation (10(10) $E\left[r_i\right]={\lambda }_0+{\lambda }_{\mathrm{\Omega }}{\beta }_{i\mathrm{\Omega }}+{\lambda }_{\mathrm{\Gamma }}{\beta }_{i\mathrm{\Gamma }}+{\eta }_i.$(10) ) applies in the case of well-diversified factor portfolios. We assume that this is always the case.

† If we so wished, we could also change its volatility. We do not pursue this angle, but the generalization presents no conceptual difficulties.

† One could argue that during a severe stress the actions of the arbitrageurs may be seriously impeded, and that therefore imposing no-arbitrage in a stress condition is either unnecessary or, perhaps, even counterproductive if the stress test is used for prudential purposes. We believe that even in distressed market conditions no-arbitrage remains a useful reference point, and that temporary deviations from perfect absence of arbitrage can be modeled by adding uncorrelated shocks, ${ϵ}_i\left(t\right)$.

‡ Since portfolio payoffs, not factors, covary with consumption, and risk premia come from the covariance between a payoff and consumption, using portfolio allows a more direct interpretation of the risk premia. However, as long as the mapping between the factors and the associated portfolios is linear (as it is invariably the case), a translation from one set of ‘coordinates’ to the other can always be accomplished.

§ The assumption is invoked when using the normal-time correlation matrix between the granular and representative market indices needed to calculate the principal components.

† Sections 2.1 and 2.2.