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Abstract:
We develop a model for pricing risky debt and valuing credit derivatives that is easily calibrated to existing variables. Our approach is based on expanding the Das and Sundaram (2000) extension of the Heath-Jarrow-Morton (1990) term-structure model to allow for multiple ratings classes of debt. The framework has two salient features: (i) it employs a ratings transition matrix as the driver or the default process, and (ii) the entire set of rating categories is calibrated jointly, allowing arbitrage-free restrictions across rating classes, as a bond migrates amongst them. We provide an illustration of the approach by applying it to price credit-sensitive notes that have coupon payments that are linked to the rating of the underlying credit.

Risky Debt, Rating Transitions, Credit Derivatives, Cresdit Senstive Note, HJM Model

Abstract:
We develop a model for pricing risky debt and valuing credit derivatives that is easily calibrated to existing variables. Our approach is based on expanding the Das and Sundaram (2000) extension of the Heath-Jarrow-Morton (1990) term-structure model to allow for multiple ratings classes of debt. The framework has two salient features: (i) it employs a ratings transition matrix as the driver or the default process, and (ii) the entire set of rating categories is calibrated jointly, allowing arbitrage-free restrictions across rating classes, as a bond migrates amongst them. We provide an illustration of the approach by applying it to price credit-sensitive notes that have coupon payments that are linked to the rating of the underlying credit.

Risky debt, Rating Transitions

Abstract:
We develop a model for pricing risky debt and valuing credit derivatives that is easily calibrated to existing variables. Our approach is based on expanding the Das and Sundaram (2000) extension of the Heath-Jarrow-Morton (1990) term-structure model to allow for multiple ratings classes of debt. The framework has two salient features: (i) it employs a ratings transition matrix as the driver or the default process, and (ii) the entire set of rating categories is calibrated jointly, allowing arbitrage-free restrictions across rating classes, as a bond migrates amongst them. We provide an illustration of the approach by applying it to price credit-sensitive notes that have coupon payments that are linked to the rating of the underlying credit.

Abstract:
We develop a model for pricing risky debt and valuing credit derivatives that is easily calibrated to existing variables. Our approach is based on expanding the Das and Sundaram (2000) extension of the Heath-Jarrow-Morton (1990) term-structure model to allow for multiple ratings classes of debt. The framework has two salient features: (i) it employs a ratings transition matrix as the driver or the default process, and (ii) the entire set of rating categories is calibrated jointly, allowing arbitrage-free restrictions across rating classes, as a bond migrates amongst them. We provide an illustration of the approach by applying it to price credit-sensitive notes that have coupon payments that are linked to the rating of the underlying credit.

Abstract:
We develop a model for pricing risky debt and valuing credit derivatives that is easily calibrated to existing variables. Our approach is based on expanding the Heath-Jarrow-Morton (1990) term-structure model and its extension, the Das-Sundaram (2000) model to allow for defaultable debt with rating transitions. The framework has two salient features, comprising extensions over the earlier work: (i) it employs a rating transition matrix as the driver for the default process, and (ii) the entire set of rating categories is calibrated jointly, allowing, with minimal assumptions, arbitrage-free restrictions across rating classes, as a bond migrates amongst them. We provide an illustration of the approach by applying it to price credit sensitive notes that have coupon payments that are linked to the rating of the underlying credit.

Risky debt, rating transitions, credit derivatives, credit sensitive note, HJM model

Abstract:
The pricing of credit derivatives is reaching some level of modeling maturity. In particular, "reduced form" models that directly specify the default process or the credit spread have resulted in successful conjoint implementations of term structure models with default models. We contribute to this literature by presenting a discrete-time reduced-form model for valuing risky debt based on the term-structure model of Heath, Jarrow, and Morton (1990).

We extend the HJM model to include risky debt by adding a "forward spread" process to the forward rate process for default risk-free bonds as in Das and Sundaram (2000). Instead of modeling the movement of the spread itself, the engineering of our model focuses on the stochastic process for inter-rating spreads. Working with inter-rating spreads provides any credit spread as the sum of higher rated inter-rating spreads. This approach offers analytical tractability. No restrictions are placed on the correlation between these stochastic processes. The probability of default at any point in time is allowed to depend on the entire history of the process to that point, and is determined from rating transition matrices, exogenously supplied. The model is flexible to incorporate any specification for the recovery process that is consistent with the default process and the spread processes.

In Das and Sundaram (2000), the pricing lattice was developed by computing a no-arbitrage tree embedding the riskless term structure and the term structure of credit spreads. While this tree considered the modeling of only a single rating category at a time, this paper extends that model by calibrating all rating classes jointly on the same pricing lattice. Embedding all rating categories on one pricing lattice requires a set of conditions ensuring consistency across all classes of debt. The additional information required to engineer this comes from the introduction of the rating transition matrix. Thus, in our model, we are now able to price credit derivatives based on multiple classes of debt, which was not possible using simpler models.

To understand the consistency conditions across rating classes, note that the credit rating of a corporate borrower can improve or deteriorate during the life of its issued debt. Thus, the credit spread on its debt contains valuable information about the future credit spreads on debt of all possible rating classes that the borrower could migrate to. This is true for a corporate borrower with any given rating at a point of time. This interdependence of spreads across rating classes immediately implies that calibration of the forward spread process for a given rating class must be undertaken simultaneously with the calibration of the forward spread processes for all other rating classes. Formalizing this interdependence and characterizing the joint calibration process (Proposition 3.2) is the primary contribution of this paper.

Our model requires as input the government yield curve. In addition, it also uses the term structures of credit spreads for each rating class, available from providers such as Bloomberg. The same source delivers required interest rate and spread volatilities. The model can be efficiently implemented and lends itself most appropriately to pricing of credit derivatives such as credit sensitive notes where the coupon payments are linked to credit quality of the underlying corporate borrower. We provide a numerical example to illustrate the calibration of the model and its use to price credit sensitive notes.

Risky Debt, Rating Transitions, Credit Derivatives, Credit Sensitive Note, HJM Model

Abstract:
We present a model for pricing risky debt and valuing credit derivatives that is easily calibrated to existing variables. Our approach expands a classical term-structure modal to allow for multiple rating classes of debt. The framework has two salient features: (1) it uses a rating-transition matrix as the driver for the default process, and (2) the entire set of rating categories is calibrated jointly, which allows arbitrage-free restrictions across rating classes as a bond migrates among them. We illustrate tha approach by applying it to price credit-sensitive notes that have coupon payments linked to the rating of the underlying credit.

Risky Debt, Rating Transitions, Credit Derivatives, Credit Sensitive Note, HJM Model

Abstract:
Little is known about the risk and return characteristics of private equity investments. In this paper we examine 52,322 financing rounds in 23,208 unique firms, over the period 1980 through 2000 by venture and buyouts funds and estimate the probability of exit, the exit multiples and the expected gains from private equity investments. We find that the gains from venture-backed investments depend upon the industry, the stage of the firm being financed, the financing amount, the valuation at the time of financing, and the prevailing market sentiment. The expected multiple ranges from a low of 1.12 for late-stage firms to a high of 5.12 for firms financed in their early stages. Our study is the first step in understanding the risk premium required for the valuation of private equity investments. It will be of particular interest to the VC community and valuation practitioners.

Private Equity Discount

Abstract:
Empirical anamolies in the Black-Scholes model have been widely documented in the Finance literature. Pattern in these anamolies (for instance, the behavior of the volatility smile or of unconditional returns at different maturities) have also been widely documented. Theoretical efforts in the literature at addressing these anamolies have largely centered around extensions of the basic Black-Scholes model. Two approaches have become especially popular in this context ' introducing jumps into the return process, and allowing volatility to be stochastic. This paper employs commonly used versions of these two classes of models to examine the extent to which the models are theoretically capable of resolving the observed anamolies. We focus especially on the possible 'term-structures' of skewness, kurtosis, and the implied volatility smile that can rise under each model. Our central finding is that each model exhibits moment patterns and implied volatility smiles that are consistent with some of the observed anamolies, but not with others. In sum, neither class of models constitutes and adequate explanation of the empirical evidence, although the stochastic volatility models fair better than jumps in this regard.

Abstract:
An extensive empirical literature in finance has documented not only the presence of anomalies in the Black-Scholes model, but also the "term-structures" of these anomalies (for instance, the behavior of the volatility smile or of unconditional returns at different maturities). Theoretical efforts in the literature at addressing these anomalies have largely focused on two extensions of the Black-Scholes model: introducing jumps into the return process, and allowing volatility to be stochastic. This paper employs commonly used versions of these two classes of models to examine the extent to which the models are theoretically capable of resolving the observed anomalies. We find that each model exhibits some "term-structure" patterns that are fundamentally inconsistent with those observed in the data. As a consequence, neither class of models constitutes an adequate explanation of the empirical evidence, although stochastic volatility models fare better than jumps in this regard.

Abstract:
Empirical anamolies in the Black-Scholes model have been widely documented in the Finance literature. Patterns in these anamolies (for instance, the behavior of the volatility smile or of unconditional returns at different maturities) have also been widely documented. Theoretical efforts in the literature at addressing these anamolies have largely centered around extensions of the basic Black--Scholes model. Two approaches have become especially popular in this context -- introducing jumps into the return process, and allowing volatility to be stochastic. This paper employs commonly-used versions of these two classes of models to examine the extent to which the models are theoretically capable of resolving the observed anamolies. We focus especially on the possible term-structures": of skewness, kurtosis, and the implied volatility smile that can arise under each model. Our central finding is that each model exhibits moment patterns and implied volatility smiles that are consistent with some of the observed anamolies, but not with others. In sum, neither class of models constitutes an adequate explanation of the empirical evidence, although stochastic volatility models fare better than jumps in this regard.

Abstract:
The internet has made it feasible to tap a continuous stream of public sentiment from the world wide web, quite literally permitting one to "feel the pulse" of any issue under consideration. We present a methodology for real time sentiment extraction in the domain of finance. With the advent of the web, there has been a sharp increase in the influence of individuals on the stock market via web-based trading and the posting of sentiment to stock message boards. While it is important
to capture this "sentiment" of small investors, as yet, no index of sentiment has been compiled. This paper comprises (a) a technology for extracting small investor sentiment from web sources to create an index, and (b) illustrative applications of the methodology. We make use of computerized natural language and statistical algorithms for the automated classification of messages posted on the web. We design a suite of classification algorithms, each of different theoretical content, with a view to characterizing the sentiment of any single posting to a message board. The use of multiple methods allows imposition of voting rules in the classification process. It also enables elimination of "fuzzy" messages which are better off uninterpreted. A majority rule across algorithms vastly improves classification accuracy, but also leads to a natural increase in the number of messages classified as "fuzzy". The classifier achieves an accuracy of 62% (versus a random classification accuracy of 33%), and compares favorably against human agreement on message classification, which was 72%. The technology is computationally efficient, allowing the access and interpretations of thousands of messages within minutes. Our illustrative applications show evidence of a strong link between market movements and sentiment. Based on approximately 25,000 messages for the last quarter of 2000, we found evidence that sentiment is based on stock movements.

Abstract:
In one-factor models, such as Cox, Ingersoll, and Ross (1985) or Vasicek (1977), the conditional mean of the instantaneous rate changes with its current level. This paper gathers evidence that the conditional mean of the one- month rate explains variations in bond yields of different maturities, even after controlling for the effect of the current level of the one-month rate. This suggests the presence of a second factor driving the conditional mean, other than the level of the one-month rate: we refer to this second factor as the central tendency. The above idea is captured in a two-factor model of the term structure where the instantaneous rate fluctuates around a stochastic central tendency. We then build a proxy for the central- tendency factor based on the information contained in the term structure of interest rates. We use the proxy to estimate the process for the one-month rate, and find the central-tendency proxy to be significant in explaining the conditional mean of the one-month rate.

Abstract:
This paper develops a model for pricing securities that may be a function of several different sources of risk, namely, equity, interest-rate, default and liquidity risks. The model is also useful for extracting probabilities of default (PDs) in a model with equity, interest rate and credit risk. The model is not based on the stochastic process for the value of the firm, but on the stochastic process for interest rates and the equity price, which are observable. The model comprises two components. First, a risk-neutral setting in which the joint process of interest rates and equity are modelled together with the boundary conditions for security payoffs. Second, the model is embedded on a recombining lattice generated using an approximation technique. This makes implementation of the pricing scheme feasible with polynomial complexity. We present a simple approach to calibration of the model to market observable data. The model is extensible to handling correlated default risk and may be used to value distressed convertible bonds, debt-equity swaps, and credit portfolio products such as CDOs.

risk-neutral, PDs, reduced-form models

Abstract:
Using a comprehensive and unique data set from Moody's, we examine correlations between default risk for over 7,000 U.S. public firms. This is the first paper to empirically document the correlation structure both in the time-series and in the cross-section across almost all U.S. non-financial firms. We find that default probabilities of issuers vary over time, and are positively correlated. Moreover, the correlations across firms also vary over time systematically, in a manner that is related to an economy-wide level of default risk. Joint default risk increases as the default risk in the economy increases. Our results also suggest that the magnitude of joint default depends on the quality of issuers; highest quality issuers have higher default correlations than medium grade firms.

Abstract:
We develop a model for pricing derivative and hybrid securities whose value may depend on different sources of risk, namely, equity, interest-rate, and default risks. In addition to valuing such securities the framework is also useful for extracting probabilities of default (PD) functions from market data. Our model is not based on the stochastic process for the value of the firm [which is unobservable], but on the stochastic process for interest rates and the equity price, which are observable. The model comprises a risk-neutral setting in which the joint process of interest rates and equity are modeled together with the default conditions for security payoffs. The model is embedded on a recombining lattice which makes implementation of the pricing scheme feasible with polynomial complexity. We present a simple approach for calibration of the model to market observable data. The framework is shown to nest many familiar models as special cases. The model is extensible to handling correlated default risk and may be used to value distressed convertible bonds, debt-equity swaps, and credit portfolio products such as CDOs. We present several numerical and calibration examples to demonstrate the applicability and implementation of our approach.

Risk-neutral, PDs, reduced-form models

Abstract:
We offer an alternative framework for the analysis of mutual funds and use it to examine the rationale behind existing regulations that require mutual fund advisor fees to be of the fulcrum' variety. We find little justification for the regulations. Indeed, we find that asymmetric incentive fees' in which the advisor receives a flat fee plus a bonus for exceeding a benchmark index provide Pareto-dominant outcomes with a lower level of equilibrium volatility. Our model also offers some insight into fee structures actually in use in the asset-management industry. We find that when leveraging is not permitted and the fee structure must be of the fulcrum variety, the equilibrium fee in our model is a flat fee with no performance component; if asymmetric incentive fees are allowed and leveraging is permitted the equilibrium fee is an incentive fee with a large performance component. These predictions match observed fee structures in the mutual fund industry and the hedge fund industry, respectively.

Abstract:
We offer an alternative framework for the analysis of mutual funds and use it to examine the rationale behind existing regulations that require mutual fund adviser fees to be of the "fulcrum" variety. We find little justification for the regulations. Indeed, we find that asymmetric "incentive fees" in which the adviser receives a flat fee plus a bonus for exceeding a benchmark index provide Pareto-dominant outcomes with a lower level of equilibrium volatility.

Our model also offers some insight into fee structures actually in use in the asset-management industry. We find that when leveraging is not permitted and the fee structure must be of the fulcrum variety, the equilibrium fee in our model is a flat fee with no performance component; while if asymmetric incentive fees are allowed and leveraging is permitted the equilibrium fee is an incentive fee with a large performance component. These predictions match observed fee structures in the mutual fund industry and the hedge fund industry, respectively.

Abstract:
Returns on international equities are characterized by jumps; moreover, these jumps tend to occur at the same time across countries leading to systemic risk. In this paper, we evaluate whether systemic risk reduces substantially the gains from international diversification. First, in order to capture these stylized facts, we develop a model of international equity returns using a multivariate system of jump-diffusion processes where the arrival of jumps is simultaneous across assets. Second, we determine an investor's optimal portfolio for this model of returns. Third, we show how one can estimate the model using the method of moments. Finally, we illustrate our portfolio optimization and estimation procedure by analyzing portfolio choice across a riskless asset, the US equity index, and five international indexes. Our main finding is that, while systemic risk affects the allocation of wealth between the riskless and risky assets, it has a small effect on the composition of the portfolio of only-risky assets, and reduces marginally the gains to a US investor from international diversification: For an investor with a relative risk aversion of 3 and a horizon of one year, the certainty-equivalent cost of ignoring systemic risk is of the order $1 for every $1000 of initial investment. These results are robust to whether the international indexes are for developed or emerging countries, to constraints on borrowing and shortselling, and to reasonable deviations in the value of the parameters around their point estimates; the cost increases with the investment horizon and decreases with risk aversion.

asset allocation, contagion, emerging markets, skewness, jump-diffusion processes

Abstract:
Returns on international equities are characterized by jumps; moreover, these jumps tend to occur at the same time across countries leading to systemic risk. In this Paper, we evaluate whether systemic risk reduces substantially the gains from international diversification. First, in order to capture these stylized facts, we develop a model of international equity returns using a multivariate system of jump-diffusion processes where the arrival of jumps is simultaneous across assets. Second, we determine an investor's optimal portfolio for this model of returns. Third, we show how one can estimate the model using the method of moments. Finally, we illustrate our portfolio optimization and estimation procedure by analysing portfolio choice across a riskless asset, the US equity index, and five international indexes. Our main finding is that, while systemic risk affects the allocation of wealth between the riskless and risky assets, it has a small effect on the composition of the portfolio of only-risky assets, and reduces marginally the gains to a US investor from international diversification - for an investor with a relative risk aversion of 3 and a horizon of one year, the certainty-equivalent cost of ignoring systemic risk is of the order $1 for every $1000 of initial investment. These results are robust to whether the international indexes are for developed or emerging countries, to constraints on borrowing and shortselling, and to reasonable deviations in the value of the parameters around their joint estimates; the cost increases with the investment horizon and decreases with risk aversion.

Asset allocation, contagion, emerging markets, skewness, jump-diffusion processes

Abstract:
In this paper, we develop a methodology to model, simulate and assess the joint default process of hundreds of issuers. Our study is based on a data set of default probabilities supplied by Moody's Risk Management Services. We undertake an empirical examination of the joint stochastic process of default risk over the period of 1987-2000 using copula functions. To determine the appropriate choice of the joint default process, we propose a new metric. This metric accounts for different aspects of default correction, namely (i) level, (ii) asymmetry and (iii) tail-dependence and extreme behavior.

Our model, based on estimating a joint system of over 600 issuers, is designed to replicate the empirical joint distribution of defaults. A comparison of a jump model and a regime-switching model shows that the latter provides a better representation of the properties of correlated default. We also find that the skewed double-exponential distribution is the best choice for the marginal distribution of each issuer's hazard rate process, and combines well with the normal, Gumbel, Clayton and students copulas in the joint dependence relationship amongst issuers. As a complement to the methodological innovation, we show that (a) appropriate choices of marginal distributions and copulas are essential in modeling correlated default, (b) accounting for regimes is an important aspect of joint specifications of default risk, and (c) misspecification of credit portfolio risk can occur easily if joint distributions are inappropriately chosen. The empirical evidence suggests that improvements are indeed possible over the standard Gaussian copula used in practice.

Correlated default, copulas, tail dependence

Abstract:
Modeling correlated default risk is a new phenomenon currently sweeping through the credit markets. In this paper, we develop a methodology to model, simulate and assess the joint default process of hundreds of issuers. Our study is based on a data set of default probabilities supplied by Moody's Risk Management Services. We undertake an empirical examination of the joint stochastic process of default risk over the period of 1987-2000 using copula functions. To determine the appropriate choice of the joint default process, we propose a new metric. This metric accounts for different aspects of default correlation, namely (i) level, (ii) asymmetry and (iii) tail-dependence and extreme behavior. Our model, based on estimating a joint system of over 600 issuers, is designed to replicate the empirical joint distribution of defaults. A comparison of a jump model and a regime-switching model shows that the latter provides a better representation of the properties of correlated default. We also find that the skewed double-exponential distribution is the best choice for the marginal distribution of each issuer's hazard rate process, and combines well with the normal, Gumbel, Clayton and student's t copulas in the joint dependence relationship amongst issuers. As a complement to the methodological innovation, we show that (a) appropriate choices of marginal distributions and copulas are essential in modeling correlated default, (b) accounting for regimes is an important aspect of joint specifications of default risk, and (c) misspecification of credit portfolio risk can occur easily if joint distributions are inappropriately chosen. The empirical evidence suggests that improvements are indeed possible over the standard Gaussian copula used in practice.

copula, tail-dependence, default

Abstract:
We develop, and apply to data on U.S. corporations from 1979-2004, tests of the standard doubly-stochastic assumption under which firms'default times are correlated only as implied by the correlation of factors determining their default intensities. This assumption is violated in the presence of contagion or "frailty" (unobservable explanatory variables that are correlated across firms). Our tests do not depend on the time-series properties of default intensities. The data do not support the joint hypothesis of well specified default intensities and the doubly-stochastic assumption. There is also some evidence of default clustering in excess of that implied by the doubly-stochastic model with the given intensities.

Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Abstract:
We develop, and apply to data on U.S. corporations from 1979-2004, tests of the standard doubly-stochastic assumption under which firms' default times are correlated only as implied by the correlation of factors determining their default intensities. This assumption is violated in the presence of contagion or "frailty" (unobservable explanatory variables that are correlated across firms). Our tests do not depend on the time-series properties of default intensities. The data do not support the joint hypothesis of well specified default intensities and the doubly-stochastic assumption. There is also some evidence of default clustering in excess of that implied by the doubly-stochastic model with the given intensities.

Correlated default, doubly stochastic, contagion, frailty

Abstract:
The Investment Advisors Act of 1940 (as amended in 1970) prohibits mutual funds in the US from offering their advisers asymmetric incentive fee' contracts in which the advisers are rewarded for superior performance via-a-vis a chosen index but are not correspondingly penalized for underperforming it. The rationale offered in defense of the regulation by both the SEC and Congress is that incentive fee structures of this sort encourage excessive' risk-taking by advisers. This paper uses an adverse selection model with multiple funds and multiple risky securities to study this issue. We find that incentive funds do, as alleged, lead to more (and suboptimal) risk-taking than do symmetric fulcrum fees.' Nevertheless, from the more important welfare angle, we find that investors may be strictly better off under asymmetric incentive fee structures. Thus, there appears to be little justification for this legislation.

Abstract:
The Investment Advisers Act of 1940 (as amended in 1970) prohibits mutual funds in the US from offering their advisers asymmetric "incentive fee" contracts in which the advisers are rewarded for superior performance via-a-vis a chosen index but are not correspondingly penalized for underforming it. The rationale offered in defense of the regulation by both the SEC and Congress is that incentive fee structures of this sort encourage "excessive" risk-taking by advisers.

Apart from affecting portfolio selection incentives, however, the fee structure also influences equilibrium welfare levels in two other important ways: (a) through its risk-sharing properties, and (b) through its potential at conveying information about the adviser's abilities. This paper examines a signaling model with multiple funds and multiple risky securities in which all of these effects are present. We find that incentive fees do, as alleged, lead to more (and suboptimal) risk taking than do symmetric "fulcrum fees." Nonetheless, taking into account the other roles of the fee structure, we find under robust conditions that investors may actually be strictly better off from a welfare stand point under asymmetric incentive fee structures. In summary, we do not find much justification for the regulation.

Abstract:
The relevance of accounting data to providers of capital has been strongly debated. In this paper we provide compelling evidence that accounting metrics are important to providers of debt capital. Models of firm distress are mostly either purely accounting-based (e.g. Altman, 1968; Ohlson, 1980) or purely market-based (e.g. Merton, 1974). We examine the information content of accounting-based and market-based metrics in pricing firm distress using a sample of Credit Default Swap (CDS) spreads. Credit Default Swaps are derivatives that offer protection from the event a given firm defaults on its obligations. CDS spreads provide a clean measure of default risk as they are the compensation that market participants require for bearing that risk. Using a sample of 2,860 quarterly CDS spreads available over the period 2001-2005 we find that a model of distress which is entirely composed of accounting-based metrics performs comparably, if not better, than market-based structural models of default. Furthermore, we find that both sources of information (accounting- and market-based) are complementary in pricing distress. These results support the notion that accounting metrics have direct value- or valuation-relevance to debt holders and holders of credit derivatives.

credit default swap, credit risk, bankruptcy prediction

Abstract:
In the absence of forward-looking models for recovery rates, market participants tend to use exogenously assumed constant recovery rates in pricing models. We develop a exible jump-to-default model that uses observables: the stock price and stock volatility in conjunction with credit spreads to identify implied, endogenous, dynamic functions of the recovery rate and default probability. The model in this paper is parsimonious and requires the calibration of only three parameters, enabling the identication of the risk-neutral term structures of forward default probabilities and recovery rates. Empirical application of the model shows that it is consistent with stylized features of recovery rates in the literature. The model is exible, i.e., it may be used with di_erent state variables, alternate recovery functional forms, and calibrated to multiple debt tranches of the same issuer. The model is robust, i.e., evidences parameter stability over time, is stable to changes in inputs, and provides similar recovery term structures for di_erent functional speci_cations. Given that the model is easy to understand and calibrate, it may be used to further the development of credit derivatives indexed to recovery rates, such as recovery swaps and digital default swaps, as well as provide recovery rate inputs for the implementation of Basel II.

Implied, Recovery, Loss-Given-Default, Credit Default Swaps

Abstract:
Credit default swap (CDS) spreads are directly related to equity market liquidity in the Merton (1974) model via hedging. This relationship is monotone increasing when credit quality worsens. Empirical tests confirm this relationship. We theorize and confirm this new channel by means of which liquidity costs are embedded in CDS spreads.

Credit Default Swaps, Liquidity

Abstract:
Whereas many studies in finance have examined and established a strong link between stock returns and information, the physical mechanics of this link have been relatively unexplored. With the advent of stock message boards, it has become feasible to look more closely at the group process by which information impacts prices and vice versa. This paper utilizes a large universe of messages posted to stock market discussion forums to understand how opinions are linked across tickers during small investor discussion. We define a collective information unit, the financial community. These are clusters of tickers sharing and accessing the same information generators. Graph theoretic techniques are used to detect financial communities and to summarize their properties. Community stocks display connectedness, and we find that the greater the connectedness in a financial community, the greater the covariance of returns within the community as opposed to that amongst stocks that are not part of a major financial community. Highly connected stocks, on average, have lower return variance and higher mean returns. Using eigenvector techniques, we detect stocks that are hubs for information flow, using a measure known as centrality. We find that stocks with high centrality scores tend to have greater average covariance with other stocks than those with low scores. Our analysis of connectedness and centrality establishes a link between one arena of the information generation process and stock return correlations.

Graph theory, connectedness, centrality

Abstract:
How good are people at interpreting numerical information presented in graphical form? Are they better at doing so when the stakes are high? There is evidence for pessimism on both counts. This paper takes an information processing view of the manner in which people process a large amount of numerical information. We propose that people sample points in a numerical series chosen due to their salience. To the extent the choice of sample points does not accurately represent the entire population of data from which they are chosen this process systematically distorts perceptions of the statistical properties of the series. This paper examines a specific bias in the presentation of graphical numerical data: the run length of a stock series. Run length is the number of consecutive periods over which the stock price moves up or down. The increasing use of graphical data in financial decisionmaking implies that visual biases in data interpretation are of growing economic importance. A primary and robust effect across three experimental studies is that stocks with longer run lengths are perceived as riskier than stocks with shorter runs, leading to a preference for the latter, despite controlling for the first four moments of the stock return paths. The effect of run length on preference for stocks appears to be driven by perceptions of risk rather than perceptions of return and are exacerbated when the stakes of the decision maker are higher. Results are robust to sample characteristics such as gender and financial experience. They are also robust to contextual differences in presentation format. Results are explained in terms of series with higher run lengths being associated with higher local maxima and minima, with these points in the series being more likely to be sampled due to their higher salience. Theoretical implications for the processing of numerical information, graphical information, and financial information are discussed.

Information processing, runlength, salience, bias

Abstract:
We develop a theoretical framework of equity returns to hypothesize that average run lengths are related to common measures of liquidity such as trading volume and trade price-impact. This relationship holds irrespective of the observation frequency in the computation of run lengths. Thus, liquidity can be detected by examining a stock's run length signature. Tests using daily equity return data for all stocks over the period 1962-2005 find that run lengths are decreasing in turnover, and increasing with bid-ask spreads, and price-impact. We develop a market-wide illiquidity factor based on run lengths and find that it is priced using standard asset-pricing specifications.

run length, liquidity, liquidity risk, asset pricing

Abstract:
We develop a model for pricing derivative and hybrid securities whose value may depend on different sources of risk, namely, equity, interest-rate, and default risks. In addition to valuing such securities the framework is also useful for extracting probabilities of default (PD) functions from market data. Our model is not based on the stochastic process for the value of the firm [which is unobservable], but on the stochastic process for interest rates and the equity price, which are observable. The model comprises a risk-neutral setting in which the joint process of interest rates and equity are modeled together with the default conditions for security payoffs. The model is embedded on a recombining lattice which makes implementation of the pricing scheme feasible with polynomial complexity. We present a simple approach to calibration of the model to market observable data. The framework is shown to nest many familiar models as special cases. The model is extensible to handling correlated default risk and may be used to value distressed convertible bonds, debt-equity swaps, and credit portfolio products such as CDOs. We present several numerical and calibration examples to demonstrate the applicability and implementation of our approach.

Abstract:
That interest rates move in a discontinuous manner is no surprise to participants in the bond markets. This paper proposes and estimates a class of Poisson-Gaussian processes that allow for jumps in interest rates. Estimation is undertaken using exact continuous-time and discrete-time estimators. Analytical derivations of the characteristic functions, moments and density functions of jump-diffusion stochastic process are developed and employed in empirical estimation. These derivations are general enough to accommodate any jump distribution. We find that jump processes capture empirical features of the data which would not be captured by diffusion models. The models in the paper enable an assessment of the impact of Fed activity and day-of-week effects on the stochastic process for interest rates. There is strong evidence that existing diffusion models would be well-enhanced by jump processes.

Abstract:
We assume the short-term rate to revert towards a central tendency which in, turn, is stochastically changing over time. We impose minimal restrictions on the joint behavior of the short-term rate and the central-tendency factor, and derive implications for the term structure of interest rates. The analysis suggests a proxy for the central tendency which is then used to estimate the short-term rate process. Our model captures variations in the short-term rate better than the Vasicek (1977) and Cox, Ingersoll and Ross (1985) models, where the central tendency is assumed to be constant. Also, the central-tendency proxy explains the conditional volatility of the short-term rate better than the short-term rate itself.

term structure

Abstract:
We assume that the instantaneous riskless rate reverts towards a central tendency which, in turn, is changing stochastically over time, and we derive a model of the term structure of interest rates. Our term-structure model implies that a linear combination of any two rates can be used as a proxy of the central tendency. Based on the central-tendency proxy, we estimate a model of the one-month rate which performs better than models which assume the central tendency to be constant.

term structure

Abstract:
We assume that the instantaneous riskless rate reverts toward a central tendency which, in turn, is changing stochastically over time. As a result, current short-term rates are not sufficient to predict future short-term rates movements, as it would be the case if the central tendency was constant. However, since longer-maturity bond prices incorporate information about the central tendency, longer-maturity bond yields can be used to predict future short-term rate movements. We develop a two-factor model of the term-structure which implies that a linear combination of any two rates can be used as a proxy for the central tendency. Based on this central-tendency proxy, we estimate a model of the one-month rate which performs better than models which assume the central tendency to be constant.

term structure

Abstract:
We assume that the instantaneous riskless rate reverts toward a central tendency which, in turn, is changing stochastically over time. As a result, current short-term rates are not sufficient to predict future short-term rates movements, as it would be the case if the central tendency was constant. However, since longer-maturity bond prices incorporate information about the central tendency, longer-maturity bond yields can be used to predict future short-term rate movements. We develop a two-factor model of the term-structure which implies that a linear combination of any two rates can be used as a proxy for the central tendency. Based on this central-tendency proxy, we estimate a model of the one-month rate which performs better than models which assume the central tendency to be constant.

term structure

Abstract:
We assume that the instantaneous riskless rate reverts towards a central tendency which in turn, is changing stochastically over time. As a result, current short-term rates are not" sufficient to predict future short-term rates movements, as would be the case if the central" tendency was constant. However, since longer-maturity bond prices incorporate information" about the central tendency, longer-maturity bond yields can be used to predict future short-term" rate movements. We develop a two-factor model of the term-structure which implies that a" linear combination of any two rates can be used as a proxy for the central tendency. Based on" this central-tendency proxy, we estimate a model of the one-month rate which performs better" than models which assume the central tendency to be constant.

Abstract:
This paper develops a model for the pricing of credit derivatives using observables. The model (i) is arbitrage-free, (ii) accommodates path-dependence, and (iii) handles a range of securities, even with American features. The computer implementation uses a recursive scheme that is convenient and seamlessly processes forward induction and backward recursion, needed to compute more complicated derivative securities.

Abstract:
It is widely acknowledged that many financial markets exhibit a considerably greater degree of kurtosis (and sometimes also skewness) than is consistent with the Geometric Brownian Motion model of Black and Scholes (1973). Among the many alternative models that have been proposed in this context, two have become especially popular in recent years: models of jump-diffusions, and models of stochastic volatility. This paper explores the statistical properties of these models with a view to identifying simple criteria for judging the consistency of either model with data from a given market; our specific focus is on the patterns of skewness and kurtosis that arise in each case as the length of the interval of observations changes. We find that, regardless of the precise parameterization employed, these patterns are strikingly similar within each class of models, enabling a simple consistency test along the desired lines. As an added bonus, we find that for most parameterizations, the set of possible patterns differs sharply across the two models, so that data from a given market will typically not be consistent with both models. However, there exist exceptional parameter configurations under which skewness and kurtosis in the two models exhibit remarkably similar behavior from a qualitative standpoint. The results herein will be useful to empiricists, theorists and practitioners looking for parsimonious models of asset prices.

Abstract:
This review paper describes basic auction concepts, and provides a summary of the theory in this area, particularly as it relates to Treasury auctions.

Abstract:
This paper develops a framework for modelling risky debt and valuing credit derivatives that is exible and simple to implement, and that is, to the maximum extent possible, based on observables. Ourapproach is based on expanding the Heath-Jarrow-Morton term-structure model to allow for defaultable debt. We do not follow the procedure of implying out the behavior of spreads from assumptions concerning the default process, instead working directly with the evolution of spreads. We show thatrisk-neutral drifts in the resulting model possess a recursive representation that particularly facilitates implementation and makes it possible to handle path-dependence and early exercise features without difficulty. The framework permits embedding a variety of specifications for default; we present an empirical example of a default structure which provides promising calibration results.

Abstract:
Term structure models employing Poisson-Gaussian processes may be used to accommodate the observed skewness and kurtosis of interest rates. This paper extends the discrete-time, pure-Gaussian version of the Heath-Jarrow-Morton model to the pricing" of American-type bond options when the underlying term structure of interest rates follows a Poisson-Gaussian process. The Poisson-Gaussian process is specified using a hexanomial tree (six nodes emanating from each node), and the tree is shown to be recombining. The scheme is parsimonious and convergent. This model extends the class of HJM models by (i) introducing a more generalized volatility specification than has been used so far, and (ii) inducting jumps, yet retaining lattice recombination, thus making the model useful for practical applications.

Abstract:
This article develops a simple approach to solving continuous-time portfolio choice problems. Portfolio problems for which no closed-form solutions are available may be handled by this technique, which substitutes the numerical solution of partial differential equations with a non-linear numerical algorithm approximating the solution. This paper complements the wide literature in economics on the solution of dynamic problems in dicrete time. The algorithm is parismonious, and is illustrated by solving two examples, one, the standard Merton problem, and two, a jump-diffusion problem.

Abstract:
We develop analytic pricing models for options on averages by means of a state-space expansion method. These models augment the class of Asian options to markets where the underlying traded variable follows a mean-reverting process. The approach builds from the digital Asian option on the average and enables pricing of standard Asian calls and puts, caps and floors, as well as other exotica. The models may be used (i) to hedge long period interest rate risk cheaply, (ii) to hedge event risk (regime based risk), (iii) to manage long term foreign exchange risk by hedging through the average interest differential, (iv) managing credit risk exposures, and (v) for pricing specialized options like range-Asians. The techniques in the paper provide several advantages over existing numerical approaches.

Abstract:
The Investment Advisers Act of 1940 (as amended in 1970) prohibits mutual funds in the US from offering their advisers asymmetric "incentive fee" contracts in which the advises are rewarded for superior performance via-a-vis a chosen index but are not correspondingly penalized for underforming it. The rationale offered in defense of the regulation by both the SEC and Congress is that incentive fee structures of this sort encourage "excessive" risk-taking by advisers. Apart from affecting portfolio selection incentives, however, the fee structure also influences equilibrium welfare levels in two other important ways: (a) through its risk-sharing properties, and (b) through its potential at conveying information about the adviser's abilities. This paper examines a signalling model with multiple funds and multiple risky securities in which all of these effects are present. We find the incentives fees do, as alleged, lead to more (and suboptimal) risk-taking than do symmetric "fulcrum fees." Nonetheless, taking into account the other roles of the fee structure, we find under robust conditions that investors may actually be strictly better off from a welfare standpoint under asymmetric incentives fee structures. In summary, we do not find much justification for the regulation.

Abstract:
The fee structure used to compensate investment advisers is central to the study of fund design, and affects investor welfare in at least three ways: (i) by influencing the portfolio-selection incentives of the adviser, (ii) by affecting risk-sharing between adviser and investor, and (iii) through its use as a signal of quality by superior investment advisers. In this paper, we describe a model in which all of these features are present, and use it to compare two popular and contrasting forms of fee contracts, the "fulcrum" and the "incentive" types, from the standpoint of investor welfare. While the former has some undeniably attractive features (that have, in particular, been used by regulators to justify its mandatory use in a mutual fund context), we find surprisingly that it is the latter that is often more attractive from the standpoint of investor welfare. Our model is a flexible one; our conclusions are shown to be robust to many extensions of interest. The results are also extended to consider unrestricted fee structures and competitive markets for fund managers.

Abstract:
Existing regulations require fee structures used to compensate advisers in the mutual fund industry to be the "fulcrum" variety, decreasing for underperforming a given index in the same way in which they increase for outperforming it. In this paper, we offer a new model for analysing the mutual fund industry, and use this model to examine the impact of restricting the fee structures that may be employed. We find little justification for existing regulations. Indeed, we find that "incentive fees" in which the advisor receives a flat fee plus a bonus for exceeding a benchmark index provide Pareto-dominant outcomes with a lower level of equilibrium volatility.

Abstract:
We examine the information flow for four stocks over seven months to trace the relationship between on-line discussion, news activity, and stock price movements. On-line discussions support numerous unsubstantiated rumors, substantial on-point exchanges, and quick dissemination of imminent and recently released information. Applying language-processing routines to message board postings and news, we create sentiment and disagreement measures or eInformation. We analyze the determinants of sentiment and disagreement, and trace links between news, eInformation, and stock returns. This intensive clinical study of on-line discussions suggests mechanisms individual investors and groups can use to analyze and digest company information.

Abstract:
We present a scheme for pricing derivatives on M assets on K-factor recombining trees with N periods. The computational complexity of these trees is O(NK 1), i.e. polynomial in N, making it possible to price a wide range of derivatives without resorting to Monte Carlo simulation. Numerical implementation examples are provided, along with a discussion of the issues that arise when these models are implemented on multicore processors. A calibration example is provided that shows how individual assets may be embedded on a multi-factor tree.

High-dimension; multi-factor trees; multi-threading

Abstract:
We integrate appealing features of Markowitz's mean-variance portfolio theory (MVT) and Shefrin and Statman's behavioral portfolio theory (BPT) into a new mental accounting (MA) framework. Features of the MA framework include a mental accounting structure of portfolios, a definition of risk as the probability of failing to reach the threshold level in each mental account, and attitudes toward risk that vary by account. We demonstrate a mathematical equivalence between MVT, MA and risk management using VaR. The aggregate allocation across MA sub-portfolios is mean-variance efficient with short-selling. Short-selling constraints on mental accounts impose very minor reductions in certainty equivalents, only if binding for the aggregate portfolio, or setting utility losses from errors in specifying risk aversion coefficients in MVT applications. These generalizations of MVT and BPT via a united MA framework result in a fruitful connection between investor consumption goals and portfolio production.

Abstract:
Derivative security pricing and risk measurement relies increasingly on lattice representations of stochastic processes, which are a discrete approximation of the movement of the underlying securities. Pricing is undertaken by summation of node values on the lattice. When the lattice is large (which is the case when high accuracy is required), exhaustive enumeration of the nodes becomes prohibitively costly. Instead, Monte Carlo simulation is used to estimate the lattice value by sampling appropriately from the nodes. Most sampling methods become extremely error-prone in situations where the node values vary widely. This paper presents a Markov chain Monte Carlo scheme, adapted from Sinclair and Jerrum (Information and Computation 82 (1989)), that is able to overcome this problem, provided some partial (possibly very inaccurate) information about the lattice sum is available. This partial information is used to direct the sampling, in similar fashion to traditional importance sampling methods. The key difference is that the algorithm allows backtracking on the lattice, which acts in a "self-correcting" manner to minimize the bias in the importance sampling.

Risk management

Abstract:
In this paper we examine 52,322 financing rounds in 23,208 unique firms, over the period 1980 through 2000 by venture and buyouts funds and estimate the probability of exit, time to exit, exit multiples and the expected gains from private equity investments. The expected multiple (after accounting for dilution and the probability of exit) ranges from a low of 1.12 for late-stage firms to a high of 5.12 for firms financed in their early stages. We find that the gains from venture-backed investments depend upon the industry, the stage of the firm being financed, the valuation at the time of financing, and the prevailing market sentiment. Our study is a first step in understanding the risk premium required for the valuation of private equity investments.

Private Equity

Abstract:
This article models the pricing of derivatives on credit risk, instruments proposed in 1992 by the International Swap Dealers Association that have started attracting market attention. The exact structure of the instruments continues to evolve today.We develop a framework to understand the key features of this class of products. It is shown that the price of the credit risk option is the expected forward value of a put option on a risky bond with a credit level-adjusted exercise price. Stripping of credit risk from the total risk of the bond is enabled by employing a stochastic strike price for the credit risk option. The article provides a framework for trading and hedging credit risk.

Abstract:
This paper provides a methodology for pricing American type interest rate contingent claims for jump-diffusion processes. The method enhances the standard finite- differencing approach to deal with partial differential- difference equations derived in a jump-diffusion world. The numerical stability and convergence of the scheme is also proved. Numerical illustrations compare jump-diffusion and pure-diffusion models. Whereas the existence of jumps affects call options on bonds very much like those on stocks, this is not the case for puts which are affected by the asymmetric convexity of the bond pricing functions. Early exercise behavior is also analyzed.

Abstract:
This paper develops models of the term structure when the short rate follows a jump-diffusion process. An empirical implementation demonstrates that jump-diffusions better explain interest rate behavior than pure diffusion models. The fit is shown to be improved by an augmented jump-diffusion time varying volatility model proposed here. The effect of skewness and kurtosis on the term structure of interest rates is analyzed. The economic implications of jump activity are explored with the analysis of changes in Federal Reserve target rates and their relationship to the term structure.

Abstract:
We propose a new framework for the analysis of mutual funds and use it to examine the rationale behind existing regulations that require mutual fund adviser fees to be of the "fulcrum" variety. We find little justification for the regulations. Indeed, we find that asymmetric "incentive fees" in which the adviser receives a flat fee plus a bonus for exceeding a benchmark index provide Pareto-dominant outcomes with a lower level of equilibrium volatility.

Our model also offers some insight into fee structures actually in use in the asset-management industry. We find that when leveraging is not permitted and the fee structure must be of the fulcrum variety, the equilibrium fee in our model is a flat fee with no performance component; while if asymmetric incentive fees are allowed and leveraging is permitted the equilibrium fee is an incentive fee with a large performance component. These predictions match observed fee structures in the mutual fund industry and the hedge fund industry, respectively.

Abstract:
A variety of realistic economic considerations make jump- diffusion models of interest rate dynamics an appealing modeling choice to price interest rate contingent claims. However, exact closed form solutions for bond prices when interest rates follow a mixed jump-diffusion process have proved very hard to derive. This paper puts forward two new models of interest rate dynamics which combine infrequent, discrete changes in the interest rate level, modeled as a jump process, with short lived, mean reverting shocks, modeled as a diffusion process. The two models differ in the way jumps affect the central tendency of interest rates; in one case shocks are temporary, in the other shocks are permanent. We derive exact closed form solutions for the price of a discount bond, and computationally tractable schemes to price bond options.

Abstract:
This article develops a model for the pricing of credit-sensitive debt contracts. Over the past two decades, the debt markets have seen a proliferation of contracts designed to reapportion interest rate and credit risks between issuer and investors. Contracts including credit-sensitive notes (CSNs), spread adjusted notes (SPANs), and floating rate notes (FRNs) adjust investors' exposures to three risks: interest rate risk, changes in credit risk caused by changes in the credit rating of the issuer of the debt, and changes in credit risk caused by changes in spreads on the debt, even when ratings have not changed. In this article, we develop a pricing model incorporating all three risks, with special emphasis on credit risks. The model incorporates a decomposition of credit spreads into two stochastic elements: the default process and the recovery process in the event of default. The model is easily implementable as it uses observable inputs. By using a discrete time formulation, the model is numerically easy to employ and also permits the pricing of debt with embedded American-type options. It also allows for pricing contracts between parties with varying credit ratings (such as swaps) where each counterparty may have different credit quality. These features impart a degree of generality and practicality to the model, which should make it attractive to academics and practitioners alike.