Bayesian Risk Management. Sekerke Matt

and especially Hayek. These pages are part of an ongoing process of coming to grips with the wealth of ideas to which Professor Hanke exposed me. Professor Hanke has also supported my writing efforts from the very beginning and continues to be a source of encouragement and wise counsel to me in virtually all matters of importance.

      Chris Culp has been incredibly supportive to me for nearly 15 years as a mentor and a colleague. His boundless productive energy and generosity of spirit have been an inspiration to me from the beginning. (“Ask Culp” was one of the more common prescriptions heard in Professor Hanke's office.) The insightful ways in which Chris connects problems in risk management with fundamental problems in economics and corporate finance were decisive in sparking my interest in the subject. More directly, without his introduction to Bill Falloon at Wiley, this project would have remained in the realm of wishful thinking.

      Bill Falloon has shown me a staggering degree of support with this book and more generally in developing as an author. I look forward to more projects with him and his fantastic team, especially Meg Freeborn, who kept my developing manuscript on the rails despite multiple interruptions and radical, wholesale revisions.

      Most important, I am grateful for the unflagging support of my incredible wife, Nancy. She kept me going on this project whenever the going got tough, and patiently auditioned my many attempts to distill my thesis to a simple and forthright message. Whatever clarity may be found in a book dense with mathematics and quantitative finance is probably due to her. All of the shortcomings of the book are, however, mine alone.

      Chapter 1

      Models for Discontinuous Markets

      The broadening and deepening of markets for risk transfer has marked the development of financial services perhaps more than any other trend. The past 30 years have witnessed the development of secondary markets for a wide variety of financial assets and the explosion of derivative instruments made possible by financial engineering. The expansion of risk transfer markets has liquefied and transformed the business of traditional financial firms such as banks, asset managers, and insurance companies. At the same time, markets for risk transfer have enabled nontraditional players to enter financial services businesses, invigorating competition, driving down prices, and confounding the efforts of regulators. Such specialist risk transfer firms occupy a number of niches in which they can outperform their more diversified counterparts in the regulated financial system by virtue of their specialized knowledge, transactional advantages, and superior risk management.

      For all firms operating in risk transfer markets, traditional and nontraditional alike, the ability to create, calibrate, deploy, and refine risk models is a core competency. No firm, however specialized, can afford to do without models that extract information from market prices, measure the sensitivity of asset values to any number of risk factors, or forecast the range of adverse outcomes that might impact the firm's financial position.

      The risk that a firm's models may fail to capture shifts in market pricing, risk sensitivities, or the mix of the firm's risk exposures is thus a central operational risk for any financial services business. Yet many, if not most, financial services firms lack insight into the probabilistic structure of risk models and the corresponding risk of model failures. My thesis is that most firms lack insight into model risk because of the way they practice statistical modeling. Because generally accepted statistical practice provides thin means for assessing model risk, alternative methods are needed to take model risk seriously. Bayesian methods allow firms to take model risk seriously – hence a book on Bayesian risk management.

      Risk Models and Model Risk

      Throughout this book, when I discuss risk models, I will be talking about parametric risk models. Parametric risk models are attempts to reduce the complexity inherent in large datasets to specific functional forms defined completely by a relatively low-dimensional set of numbers known as parameters. Nonparametric risk models, by contrast, rely exclusively on the resampling of empirical data, so no reduction of the data is attempted or accomplished. Such models ask: Given the risk exposures I have today, what is the distribution of outcomes I can expect if the future looks like a random draw from some history of market data? Nonparametric risk models lack model specification in the way we would normally understand it, so that there is no risk of misspecification or estimation error by construction. Are such models therefore superior? Not at all. A nonparametric risk model cannot represent any outcome different from what has happened, including any outcomes more extreme than what has already happened. Nor can it furnish any insight into the ultimate drivers of adverse risk outcomes. As a result, nonparametric risk models have limited use in forecasting, though they can be useful as a robustness check for a parametric risk model.

      Parametric risk models begin life as a probability distribution, which is a statement of the likelihood of seeing different values conditional only on the parameters of the distribution. Given the parameters and the form of the distribution, all possibilities are encompassed. More parameters create more flexibility: A Weibull distribution is more flexible than an exponential distribution. Many risk models rely heavily on normal and lognormal distributions, parameterized by the mean and variance, or the covariance matrix and mean vector in the multivariate case. A great deal has been written on the usefulness of heavier-tailed distributions for modeling financial data, going back to Mandelbrot (1963) and Fama (1965).

      Undoubtedly, the unconditional distributions of most financial returns have heavier tails than the normal distribution. But to solve the problem of heavy tails solely through the choice of a different family of probability distributions is to seek a solution at a very low level of complexity.

      More complex risk models project a chosen risk distribution onto a linear system of covariates that helps to articulate the target risk. Regression models such as these seek to describe the distribution of the target variable conditional on other available information. The functional form of the distribution is subsumed as an error term. Familiar examples include the following:

      • Linear regression with normally distributed errors, widely used in asset pricing theory and many other applications.

      • Probit and logit models, which parameterize the success probability in binomial distributions.

      • Proportional hazard models from insurance and credit risk modeling, which project a series of gamma or Weibull distributions onto a linear system of explanatory factors.

      Parameters are added corresponding to each of the factors included in the projection. The gain in power afforded by projection raises new questions about the adequacy of the system: Are the chosen factors sufficient? Unique? Structural? What is the joint distribution of the system parameters, and can that tell us anything about the choice of factors?

      It seems the pinnacle in financial risk modeling is achieved when parameters governing several variables – a yield curve, a forward curve, a volatility surface – may be estimated from several time series simultaneously, where functional forms are worked out from primitives about stochastic processes and arbitrage restrictions. Such models pass over from the physical probability measure P to the risk-neutral probability measure Q. In terms of the discussion above, such models may be seen as (possibly nonlinear) transformations of a small number of factors (or state variables) whose distributions are defined by the nature of the underlying stochastic process posited for the factors. When the number of time series is large relative to the parameters of the model the parameters are overidentified, permitting highly efficient inference from the data. Such models are the ultimate in powerful description, offering the means to capture the dynamics of dozens of interest rates or forward contracts with a limited number of factors and parameters.

      Our hierarchy of risk models thus includes as elements probability distributions, parameters, and functional forms, which may be linear or nonlinear, theoretically motivated or ad hoc. Each element of the description may not conform to reality, which is to say that each element is subject to error. An incorrect choice of distribution or functional form constitutes specification error on the part of the analyst. Errors in parameters arise from estimation error, but also collaterally from specification errors. The collection of all such opportunities for error in risk modeling is what I will call model risk.

      Time-Invariant Models