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Working Paper
Constant proportion debt obligations: a post-mortem analysis of rating models
In its complexity and its vulnerability to market volatility, the CPDO might be viewed as the poster child for the excesses of financial engineering in the credit market. This paper examines the CPDO as a case study in model risk in the rating of complex structured products. We demonstrate that the models used by S&P and Moody's would have assigned very low probability to the spread levels realized in the investment grade corporate credit default swap market in late 2007, even though these spread levels were comparable to those of 2002. The spread levels realized in the first quarter of 2008 ...
Working Paper
On the distribution of a discrete sample path of a square-root diffusion
We derive the multivariate moment generating function (mgf) for the stationary distribution of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). The form of the mgf establishes that the stationary joint distribution of (X(t(1)),...,X(t(n))) for any fixed vector of observation times (t(1),...,t(n)) is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. As a corollary, we obtain the mgf for the increment X(t+dt)-X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma ...
Working Paper
The Bank as Grim Reaper : Debt Composition and Bankruptcy Thresholds
We offer a model and evidence that private debtholders play a key role in setting the endogenous asset value threshold below which corporations declare bankruptcy. The model, in the spirit of Black and Cox (1976), implies that the recovery rate at emergence from bankruptcy on all of the firm's debt taken together is increasing in the pre-bankruptcy share of private debt in all debt. Empirical evidence supports this and other implications of the model. Indeed, debt composition has a more economically material empirical influence on recovery than all other variables we try taken together.
Working Paper
Expectations of functions of stochastic time with application to credit risk modeling
We develop two novel approaches to solving for the Laplace transform of a time-changed stochastic process. We discard the standard assumption that the background process (Xt) is Levy. Maintaining the assumption that the business clock (Tt) and the background process are independent, we develop two different series solutions for the Laplace transform of the time-changed process X-tildet=X(Tt). In fact, our methods apply not only to Laplace transforms, but more generically to expectations of smooth functions of random time. We apply the methods to introduce stochastic time change to the ...
Working Paper
A comparative anatomy of credit risk models
Within the past two years, important advances have been made in modeling credit risk at the portfolio level. Practitioners and policy makers have invested in implementing and exploring a variety of new models individually. Less progress has been made, however, with comparative analyses. Direct comparison often is not straightforward, because the different models may be presented within rather different mathematical frameworks. This paper offers a comparative anatomy of two especially influential benchmarks for credit risk models, J.P. Morgan's CreditMetrics and Credit Suisse Financial ...
Working Paper
Counterparty Risk and Counterparty Choice in the Credit Default Swap Market
We investigate how market participants price and manage counterparty risk in the post-crisis period using confidential trade repository data on single-name credit default swap (CDS) transactions. We find that counterparty risk has a modest impact on the pricing of CDS contracts, but a large impact on the choice of counterparties. We show that market participants are significantly less likely to trade with counterparties whose credit risk is highly correlated with the credit risk of the reference entities and with counterparties whose credit quality is relatively low. Furthermore, we examine ...
Working Paper
A risk-factor model foundation for ratings-based bank capital rules
When economic capital is calculated using a portfolio model of credit value-at-risk, the marginal capital requirement for an instrument depends, in general, on the properties of the portfolio in which it is held. By contrast, ratings-based capital rules, including both the current Basel Accord and its proposed revision, assign a capital charge to an instrument based only on its own characteristics. I demonstrate that ratings-based capital rules can be reconciled with the general class of credit VaR models. Contributions to VaR are portfolio-invariant only if (a) there is only a single ...
Working Paper
Computationally convenient distributional assumptions for common value auctions
Although the mathematical foundations of common value auctions have been well understood since Milgrom & Weber (1982), equilibrium bidding strategies are computationally complex. Very few calculated examples can be found in the literature, and only for highly specialized cases. This paper introduces two sets of distributional assumptions that are flexible enough for theoretical and empirical applications and yet permit straightforward calculation of equilibrium bidding strategies.
Working Paper
Granularity adjustment for mark-to-market credit risk models
The impact of undiversified idiosyncratic risk on value-at-risk and expected shortfall can be approximated analytically via a methodology known as granularity adjustment (GA). In principle, the GA methodology can be applied to any risk-factor model of portfolio risk. Thus far, however, analytical results have been derived only for simple models of actuarial loss, i.e., credit loss due to default. We demonstrate that the GA is entirely tractable for single-factor versions of a large class of models that includes all the commonly used mark-to-market approaches. Our approach covers both finite ...