Darrell Duffie

AN OVERVIEW OF VALUE AT RISK

This article gives a broad and accessible overview o f models o f value at risk (WR), a popular measure o f the market risk of afinancialfirm’s “book,” the list ofpositions in various instruments that expose the firm to financial risk. Roughly speaking, the value at risk o f a portjolio is the loss in market value over a given time period, such as one day or two weeks, that is exceeded with a small probability, such as 1%. We focus narrowly on the market risk associated with changes in the prices or rates of underlying traded instruments. Traditionally, this would include such aspects o f credit risk as the risk o f changes in the spreads ofpublicly traded corporate and sovereign bonds. In order to maintain a narrow focus, however, I/aR does not traditionally include, and we do not review here, the risk ofdefdult on long-term derivative contracts. We begin with models o f the distribution ofunderlying market returns and ofvolatility, emphasizing the roles o f price jumps and o f stochastic volatility in determining the ‘ffatness” o f the tails o f the distributions o f returns in various markets. We then turn to methods for approximating the value at risk o f derivatives, using numerical approximations based on delta andgamma. Estimation o f the value at risk o f a portjolio ofpositions is then discussed, beginning with the choice of which risk factors to include. A n extensive numerical example illustrates the accuracy of various alternative methods, allowing for correlated jumps in the underlying market prices. Finally, we review “scenario analysis,” which treats the potential losses associated with particular scenarios, such as a given parallel shiji o f a yield curve or a given change in volatility.

Term Structures of Credit Spreads with Incomplete Accounting Information

We study the implications of imperfect information for term structures of credit spreads on corporate bonds. We suppose that bond investors cannot observe the issuers assets directly, and receive instead only periodic and imperfect accounting reports. For a setting in which the assets of the firm are a geometric Brownian motion until informed equityholders optimally liquidate, we derive the conditional distribution of the assets, given accounting data and survivorship. Contrary to the perfect-information case, there exists a default-arrival intensity process. That intensity is calculated in terms of the conditional distribution of assets. Credit yield spreads are characterized in terms of accounting information. Generalizations are provided.

Simulated Moments Estimation of Markov Models of Asset Prices

This paper provides a simulated moments estimator (SME) of the parameters of dynamic models in which the state vector follows a time-homogeneous Markov process. Conditions are provided for both weak and strong consistency as well as asymptotic normality. Various tradeoffs among the regularity conditions underlying the large sample properties of the SME are discussed in the context of an asset pricing model.

Multi-Period Corporate Default Prediction with Stochastic Covariates

We provide maximum likelihood estimators of term structures of conditional probabilities of bankruptcy over relatively long time horizons, incorporating the dynamics of firm-specific and macroeconomic covariates. We find evidence in the U.S. industrial machinery and instruments sector, based on over 28,000 firm-quarters of data spanning 1971 to 2001, of significant dependence of the level and shape of the term structure of conditional future bankruptcy probabilities on a firms distance to default (a volatility-adjusted measure of leverage) and on U.S. personal income growth, among other covariates. Variation in a firms distance to default has a greater relative effect on the term structure of future failure hazard rates than does a comparatively sized change in U.S. personal income growth, especially at dates more than a year into the future.

A Liquidity Based Model of Security Design

The authors consider the problem of design and sale of a security backed by specified assets. Given access to higher-return investments, the issuer has an incentive to raise capital by securitizing part of these assets. At the time the security is issued, the issuers or underwriters private information regarding the payoff of the security may cause illiquidity in the form of a downward-sloping demand curve for the security. The authors characterize the optimal security design in several cases. They also demonstrate circumstances under which standard debt is optimal and show that the riskiness of the debt is increasing in the issuers retention costs for assets.

Risk and Valuation of Collateralized Debt Obligations

In this discussion of risk analysis and market valuation of collateralized debt obligations, we illustrate the effects of correlation and prioritization on valuation and discuss the “diversity score” (a measure of the risk of the CDO collateral pool that has been used for CDO risk analysis by rating agencies) in a simple jump diffusion setting for correlated default intensities. A collateralized debt obligation (CDO) is an asset-backed security whose underlying collateral is typically a portfolio of (corporate or sovereign) bonds or bank loans. A CDO cash flow structure allocates interest income and principal repayments from a collateral pool of different debt instruments to prioritized CDO securities (tranches). A standard prioritization scheme is simple subordination: Senior CDO notes are paid before mezzanine and lower subordinated notes are paid, and any residual cash flow is paid to an equity piece. CDOs form an increasingly large and important class of fixed-income securities. Our analysis may provide useful approaches to valuation and diagnostic measures of risk. We concentrate on cash flow CDOs-those for which the collateral portfolio is not subjected to active trading by the CDO manager. The implication of this characteristic is that the uncertainty regarding interest and principal payments to the CDO tranches is determined mainly by the number and timing of defaults of the collateral securities. We do not analyze market-value CDOs, those in which the CDO tranches receive payments based essentially on the marked-to-market returns of the collateral pool as determined largely by the trading performance of the CDO manager. In our analysis of the risk and market valuation of cash flow CDOs, we illustrate the effects of correlation and prioritization for the market valuation, “diversity score” (a measure of the risk of the CDO collateral pool that has been used for CDO risk analysis by rating agencies), and risk of CDOs in a simple jump diffusion setting for correlated default intensities. The main issue is the impact of the joint distribution of default risk of the underlying collateral securities on the risk and valuation of the CDO tranches. We also address the efficacy of alternative computational methods and the role of diversity scores. We show that default-time correlation has a significant impact on the market values of individual tranches. The priority of the senior tranche, by which it is effectively “short a call option” on the performance of the underlying collateral pool, causes its market value to decrease with the risk-neutral default-time correlation. The value of the equity piece, which resembles a call option, increases with correlation. Optionality has no clear effect on intermediate tranches. With sufficient overcollateralization, the option “written” (to the lower tranches) dominates, but it is the other way around for sufficiently low levels of overcollateralization. Spreads, at least for mezzanine and senior tranches, are not especially sensitive to the “lumpiness” of the arrival of information about credit quality, in that replacing the contribution of diffusion with jump risks (of various types), while holding constant the degree of mean reversion and the term structure of credit spreads, plays a relatively small role in determining the spreads. Regarding alternative computational methods, we show that if (risk-neutral) diversity scores can be evaluated accurately, which is computationally simple in the framework we propose, these scores can be used to obtain good approximate market valuations for reasonably well-collateralized tranches.

Securities Lending, Shorting, and Pricing

We present a model of asset valuation in which short-selling is achieved by searching for security lenders and by bargaining over the terms of the lending fee. If lendable securities are di cult to locate, then the price of the security is initially elevated, and expected to decline over time. This price decline is to be anticipated, for example,after an initial public o ering (IPO), among other cases, and is increasing in the degree of heterogeneity of beliefs of investors about the likely future value of the security. The initial price of a securitymay be above even the most optimistic buyers valuation of the securitys future dividends, because of the additional prospect of lendingfees for owners.

Measuring Default Risk Premia from Default Swap Rates and EDFs

This paper estimates recent default risk premia for U.S. corporate debt, based on a close relationship between default probabilities, as estimated by Moodys KMV EDFs, and default swap (CDS) market rates. The default-swap data, obtained through CIBC from 22 banks and specialty dealers, allow us to establish a strong link between actual and risk-neutral default probabilities for the 69 firms in the three sectors that we analyze: broadcasting and entertainment, healthcare, and oil and gas. We find dramatic variation over time in risk premia, from peaks in the third quarter of 2002, dropping by roughly 50% to late 2003.

Security Markets, Stochastic Models.

Static Market Concepts: The Geometry of Choices and Prices. Preferences. Market Equilibrium. First Probability Concepts. Expected Utility. Special Choice Spaces. Portfolios. Optimization Principles. Second Probability Concepts. Risk Aversion. Equilibrium in Static Markets under Uncertainty. Stochastic Economies: Event Tree Economies. A Dynamic Theory of the Firm. Stochastic Processes. Stochastic Integrals and Gains from Security Trade. Stochastic Equilibria. Transformations to Martingale Gains From Trade. Discrete-Time Asset Pricing: Markov Processes and Markov Asset Valuation. Discrete-Time Markov Control. Discrete-Time Equilibrium Pricing. Continuous-Time Asset Pricing: An Overview of the Ito Calculus. The Black--Scholes Model of Security Valuation. An Introduction to the Control of Ito Processes. Consumption and Portfolio Demand with I.I.D. Returns. Continuous-Time Equilibrium Asset Pricing. Bibliography. Index. Glossary.

Equilibrium in incomplete markets: I: A basic model of generic existence

Abstract This paper demonstrates the generic existence of general equilibria in incomplete markets. Our economy is a model of two periods, with uncertainty over the state of nature to be revealed in the second period. Securities are claims to commodity bundles in the second period that are contingent on the state of nature, and are insufficient in number to span all state contingent claims to value, regardless of the announced spot commodity prices. Under smooth preference assumptions, equilibria exist except for an exceptional set of endowments and securities, a closed set of measure zero. The paper includes partial results for fixed securities, showing the existence of equilibria except for an exceptional set of endowments.