#### Measuring The Unmeasurable: An Application Of Uncertainty Quantification To Financial Portfolios by OFR

**Abstract**

We extract from the yield curve a new measure of fundamental economic uncertainty, based on McDiarmid’s distance and related methods for optimal uncertainty quantification (OUQ). OUQ seeks analytical bounds on a system’s behavior, even where the underlying data-generating process and system response function are incompletely specified. 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.

### Measuring The Unmeasurable: An Application Of Uncertainty Quantification To Financial Portfolios – Introduction

This paper extracts a new measure of fundamental economic uncertainty, based on McDiarmid’s distance, from the Treasury yield curve. McDiarmid’s distance is a centerpiece of a set of methods for optimal uncertainty quantification (OUQ) recently developed by the engineering community. The OUQ approach seeks analytical bounds on a system’s behavior, even where the underlying data-generating process and system response function are not fully specified. We adapt the methods to the problem of stress testing financial portfolios.

Uncertainty plays an important role in economics and finance, where the term traditionally refers to the statistically unmeasurable situation of Knightian uncertainty, where the event space is known but probabilities are not [18]. Full ignorance of the underlying probability structure is often an unrealistically extreme assumption, however. Uncertainty quantification originated in the engineering domain, which requires precise assurances about the probability of system failures, such as an airplane crash or bridge collapse, despite partial ignorance or ambiguity about the data-generating process. In engineered systems, failure events for the individual component parts can be rendered statistically independent by physically segregating the components. The probability of overall system failure then becomes an intricate, but straightforward, structured calculation of Bernoulli probabilities of subsystem and component failures.

OUQ, described in Section 2.2, generalizes the approach to a broad class of problems in this middle ground, where the full probability law remains unknown, but can nonetheless be constrained in important ways. OUQ requires that the system have a bounded response to each element in a collection of exogenous forcing factors. For example, in the context of a bond portfolio, McDiarmid’s distance reveals the possible variation of portfolio losses (system response) resulting from a given set of interest-rate shock scenarios (input impulses). In addition, OUQ requires that the exogenous forcing factors be statistically independent. With these assumptions, the McDiarmid mathematics of concentration of measure optimally aggregate the individual bounds to produce a quantitative upper bound on the overall likelihood of extreme outcomes. Intuitively, the bounds define univariate worst-case scenarios for the response function, while the independence assumption implies that it is improbable that multiple factors will achieve their worst case bounds simultaneously.

As a stress-testing exercise, OUQ techniques allow one to “certify” portfolio soundness. More specifically, certification means asserting, with a specific confidence level, that a particular real-valued response function| the quantity of interest|will not exceed a predefined safety threshold. For example, a portfolio manager might wish to assert to clients that a drawdown during the next 12 months exceeding 25 percent of the portfolio’s current market value will occur with at most a 0.1 percent probability. OUQ is a toolkit for supporting such certifications.

As described in Section 2, uncertainty played a key role in the recent rise to prominence of financial stress testing, especially as a supervisory exercise. Nonetheless, stress testing still lacks a rigorous theoretical foundation. Such an overarching framework could help identify shortcomings in current practice and help guide new policy and implementation choices. Financial stress testing instead has evolved as a practical technique for addressing the important problem of rare but severe events that can overwhelm individual financial firms and entire financial systems. This is inherently a modeling exercise, because it involves assessing the behavior of the system when extrapolated to the unusual counterfactual conditions defined by the stress scenario(s). Done carefully, stress testing can also act as a technical audit, revealing shortcomings in the design and implementation of risk models. It can also deliver an attribution analysis identifying the segments of a large portfolio most vulnerable to particular stressors.

We illustrate the use of OUQ in the case of a simple but realistic portfolio of Treasury bonds. The example demonstrates the basic feasibility of uncertainty quantification in a financial context. An important qualifier is the tightness of the bounds one can assert regarding portfolio response. Although these bounds may seem quite loose (100 percent loss of principal), we show that OUQ techniques generate significant gains in certification confidence over the more familiar approach of Chebychev’s inequality, which uses an assumption of finite variance to bound uncertainty.

See full PDF below.