Measuring the unmeasurable: an application of uncertainty quantification to Treasury bond portfolios

Abstract

We extract from the yield curve a new measure of fundamental economic uncertainty, based on McDiarmid’s diameter and related methods for optimal uncertainty quantification (OUQ). OUQ seeks analytical bounds on a system’s behaviour, even where aspects of the underlying data-generating process and system response function are not completely known. We use OUQ to stress test a simple fixed-income portfolio, certifying its safety—i.e. that potential losses will be ‘small’ in an appropriate sense. The results give explicit tradeoffs between: scenario count, maximum loss, test horizon, and confidence level. Unfortunately, uncertainty peaks in late 2008, weakening certification assurances just when they are needed most.

Keywords:

• Uncertainty
• Yield curve
• Optimal uncertainty quantification
• Model risk
• Stress testing

Acknowledgements

The authors thank Paul Glasserman, Greg Feldberg, Greg Duffee, and conference and seminar participants at the OFR, the 2015 meeting of the Financial Engineering and Banking Society (FEBS) at Audencia Nantes University, the 2015 IMS-FIPS Workshop at Rutgers University, the December 2015 meeting of the Consortium for Systemic Risk Analytics (CSRA) at the Massachusetts Institute of Technology, and the 2015 RiskLab/Bank of Finland/ESRB Conference on Systemic Risk Analytics at Arcada University for numerous helpful comments. Research support from the OFR is gratefully acknowledged. This work was initiated while Rich Sowers was a visiting Research Principal with the Office of Financial Research (OFR), Washington, DC. Some of this work was completed during the ‘Broad Perspectives and New Directions Financial Mathematics’ program at the Institute for Pure and Applied Mathematics (IPAM) at the University of California at Los Angeles; the authors would like to thank IPAM for its hospitality. Any remaining errors or omissions are the responsibility of the authors alone. Comments and suggestions are welcome and should be directed to the authors.

Notes

No potential conflict of interest was reported by the authors.

Views and opinions expressed are those of the authors and do not necessarily represent official OFR or Treasury positions or policies, or those of any of the authors’ affiliates or employers.

1 For an example of the procedure, see the fault tree analysis in the ARP4761 airline safety standard (Society of Automotive Engineers 1996, appendix D). The maximum acceptable probability (Owhadi et al. 2013) of a catastrophic event per flight hour in civil aviation is . Actuarial data on system failures are often unavailable, regardless of the industry. The Operational Riskdata eXchange (Operational Riskdata eXchange Association 2012) for financial institutions is an exception, accumulating data on common operational events. Nonetheless, probabilistic assessments of failure are a key input to important decisions, such as whether to fly the plane or close the bank.

2 For a simple example of a constraint, one might know the mean of a distribution from the law of large numbers, but not its variance or any other moments. Owhadi et al. (2013) work through an illustration in which knowledge of the mean constrains the set of admissible scenarios in a certification exercise.

3 Practical discussions of financial model risk abound (Rebonato 2002, Derman 2011, Hénaff and Martini 2011, Morini 2011, Crouhy et al. 2014, ch. 15). Derman’s early paper (Derman 1996) categorizes financial models as fundamental, phenomenological, or statistical and discusses the model risks inherent in each group.

4 David Viniar of Goldman Sachs famously described a 2007 stress episode as ‘25 standard-deviation moves, several days in a row,’ in effect condemning the accuracy of his own models (Dowd et al. 2008).

5 Portfolio managers, of course, must balance many other considerations besides the possibility of extreme losses, such as the frequency of drawdowns, correlations with other assets, rebalancing costs, etc. Our application of OUQ does not address this full set of risk management issues, but rather offers a novel metric for a risk characteristic of particular concern.

6 Even ‘homogeneous’ portfolios can present difficulties for the OUQ techniques. For example, the problem of correlated defaults is a well-known challenge for the management of loan portfolios. Such correlated tail events would violate the independence requirements of the OUQ approach. Our example sidesteps these issues by considering only portfolios of Treasury bonds. Possible extensions to other portfolio types remain an issue for future research.

7 Banking supervisors announced their intention to perform the SCAP stress test in February 2009 (Board of Governors of the Federal Reserve, Federal Deposit Insurance Corporation 2009). They published the SCAP results in May 2009 (Board of Governors of the Federal Reserve 2009), which included calls for new capital injections for 10 of the 19 bank holding companies tested. See also Hirtle et al. (2009) and Tarullo (2014).

8 A contemporaneous proposed statement of policy (Board of Governors of the Federal Reserve 2013, p. 2) similarly noted that, ‘... these stress tests provided valuable information to market participants, reduced uncertainty about the financial condition of the participating bank holding companies under a scenario that was more adverse than that which was anticipated to occur at the time, and had an overall stabilizing effect.’

9 For an overview of the current state of supervisory stress testing, see Foglia (2009) and Bookstaber et al. (2014). The latter also offer suggestions for a research agenda. Blaschke et al. (2001) describe the FSAP methodology. Pattison (2014) and Flood et al. (2014) discuss the increasing reliance on data-driven methods generally in financial supervision.

10 Glasserman and Xu (2014) adopt an alternative approach to the imperfect knowledge of G and , treating this as a problem of model risk and robust control (Hansen and Sargent 2007) and applying techniques of relative entropy to measure and bound possible model errors.

11 Leyendecker et al. (2010) report that this strategy has been assessed for a robot-arm manoeuvre where measured performance is the placement accuracy of the arm tip.

12 For additional background on concentration inequalities, see Dubhashi and Panconesi (2012), Ledoux (2001), and Boucheron et al. (2004).

13 One might also compute a number of alternative statistics, such as the expected maximum loss over a horizon or the probability that the loss exceeds a certain level; the calculations would be similar.

14 Glasserman et al. (2015, sections 7 and 8) discuss techniques for optimal use of conditioning information to identify internally coherent stress scenarios in the multidimensional tail.

15 A comparison of our OUQ-based uncertainty measure with Duffee’s 2011 hidden factor(s) is an interesting question for future research.

16 The standard reference for this decomposition is Litterman et al. (1991). Alexander (2008a, p. 56), shows that the first three principal components from the correlation matrix of daily spot rates for UK government debt account for over 99% of total variation. Intuitively, there is a great deal of common variation across the various tenors in the term structure, because bonds of nearby maturities are close economic substitutes whose comovement is encouraged by arbitrage.

17 Because is orthonormal, its inverse and transpose are equal: .

18 Of course, another byproduct is that the empirical covariance matrix of is in fact diagonal, meaning that the factors are uncorrelated. The calculations of section 3.3 take us even further than absence of correlation, and allow us to (non-linearly) identify independent risk factors.

19 The eigenvectors here are based on the covariance of prices over time, not yields; this is a departure from typical practice in the literature (Litterman and Scheinkman 1991, Litterman et al. 1991). The typical approach supports a clear economic interpretation of the first three principal components as the level, slope and curvature of the yield curve. The conversion from yields to prices disrupts this interpretation of the factors, but it is a mechanical transformation that does not alter the dominant role of the first three eigenvectors in explaining the variation. For our purposes, the key point is not the interpretation, but that the first two principal components continue to capture nearly all of the variation; this permits us to operate with confidence in a much more tractable two-dimensional space. Alexander (2008a, example II.2.1) applies PCA to the UK Treasury yield curve, for which the first two principal components explain 97.99% of the variance, and later Alexander (2008b) extends the analysis to a stress-testing context. Similarly, when we apply PCA to the daily bond returns in our laddered portfolio (i.e. not the Treasury yield curve itself, but the exponential function given by (3(3) ), we find that, once again, the first two components explain more than 99.9% of the total variation (more precisely, 99.9390%).

20 These are the autocorrelations of the first two principle components, as defined by equation (7(7) ). As a robustness check, we also calculated the autocorrelation of the independent factors, as defined in section 3.3. The results are qualitatively very similar.

21 Sklar’s theorem is a foundation for the application of copulas to multivariate distributions. The original paper Sklar (1959) can be difficult to find; Sklar (1996) reprises the analytic proof. Embrechts and Hofert (2014) discuss Sklar’s theorem in the context of financial risk management. McNeil et al. (2015, ch. 7) provide a textbook introduction.

22 The assumption that the market is a fully exogenous forcing function is typical in microprudential stress testing. It is also a weakness of the current procedure. Bookstaber et al. (2014) propose that stress testing should move to a ‘version 3.0’ that explicitly considers feedback effects. Such a shift would effectively expand the scope of analysis from firm level to full financial system or subsystem.

23 Given our standing assumption that market price fluctuations are exogenous, this is an inconsequential rescaling to simplify the notation.

24 Cherny and Madan (2009) explore the implications of arbitrary portfolio holdings, and derive formal restrictions on acceptability.

25 is identically the vector of 1s in , so the inner product is the sum of the components of across all maturities.

26 The extension to higher dimensions would necessitate more complicated expressions for . We leave that to the reader.

27 Bookstaber et al. (2014) argue that the absence of feedback effects is a significant limitation of the current generation of stress tests (which they refer to as ‘version 2.0’ stress testing), especially if the goals are macroprudential.

28 See, for example, Bloom (2009) and the references therein.

30 The computational time is 121.7 s to find the global optimal solution on a Scientific Linux 6.4 operating system with a duo-core 3.10 GHz processor.

Additional information

Funding

Richard Sowers was supported during part of the preparation of this work by NSF DMS [1312071].