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Abstract:
Risk-averse investors in credit sensitive securities such as equity and bonds require compensation for bearing exposure to non-diversifiable corporate default risk. One component of this compensation is an event premium for the abrupt changes in security prices that occur at default. While empirical research points to the significance of event premia in corporate bond and credit swap markets, the economic nature of the event premium is not fully understood. This paper uses a structural model of corporate default risk to show that informational asymmetries can induce an event premium. If public investors are unable to observe the threshold asset value at which firm management liquidates the firm, then they face instantaneous default risk as they cannot discern the firm's distance to default. Investors are taken by surprise when the firm reaches the default threshold, causing a sudden downward jump in the prices of securities issued by or referenced on the firm. The resulting event premium is governed by the degree of investors' aversion to the randomness in the location of the unobserved default threshold. Firm management has an incentive to improve the threshold transparency in order to reduce the credit premium required by investors, and therefore the cost to the firm of equity and debt financing.

risk premium, default event risk, jump risk, incomplete information, asymmetric information, measure change

Abstract:
A stochastic model of correlated event timing is at the center of applications in many areas, including finance, insurance, reliability, queuing and health care. The event dependence structure often involves feedback loops along with the influence of exogenous stochastic risk factors. This paper analyzes a family of multivariate point process models of correlated event timing whose arrival intensity is driven by an affine jump diffusion. The components of an affine point process are self- and cross-exciting, and facilitate the description of complex event dependence structures. Ordinary differential equations characterize the transform of an affine point process and the probability distribution of an integer-valued affine point process. The moments of an affine point process take a closed form. This guarantees a high degree of computational tractability in applications. We illustrate this in the context of portfolio credit risk, where the correlation of corporate defaults is the main issue. We consider the valuation of securities exposed to correlated default risk, and demonstrate the significance of our results in a market calibration exercise that addresses the turmoil in the credit markets during the fall of 2008.

Self-exciting point process, affine jump diffusion, Hawkes process, transform, portfolio credit derivative, correlated default, index and tranche swap

Abstract:
Extreme value statistics provides a practical, flexible, mathematically elegant framework in which to develop financial risk management tools that are consistent with empirical data. In this introductory survey, we discuss some of the basic tools including power law distributions, the peaks over thresholds estimation procedure and point processes.

Extreme events, normal distribution, extreme value distribution, power law, Pareto distribution, peaks over thresholds, tail index, shortfall risk, Hurst exponent, clustering, contagion, point process

Abstract:
Recent high-profile defaults of investment grade bond issuers have demonstrated the weakness of conventional ratings in rapidly changing circumstances. We propose a simple method to derive market-based ratings from spread data, and show that classifying bonds using such ratings provides a more reliable basis for modeling return relationships than does a classification driven by agency ratings.

Credit rating, risk, market implied ratings

Abstract:
We develop a portfolio risk model that uses high-frequency data to forecast the loss surface, which is the set of loss distributions at future time horizons. Our model uses a fully automated, semi-parametric fitting procedure that has its basis in extreme value statistics. We take account of distributional asymmetry, heavy tails, heteroscedasticity and serial correlation. Loss distributions are time aggregated by taking products of characteristic functions. We test loss-surface-implied forecasts of value at risk and expected shortfall out of sample on a diverse set of portfolios and we compare our forecasts to industry-standard risk forecasts that are based on asset and factor covariance matrices. The empirical results make a compelling case for the application and further development of our approach.

Extreme risk, loss surface, expected shortfall, peaks over thresholds, temporal risk aggregation

Abstract:
We develop a portfolio risk model that uses high-frequency data to forecast the loss surface, which is the set of loss distributions at future time horizons. Our model uses a fully automated, semi-parametric fitting procedure that has its basis in extreme value statistics. We take account of distributional asymmetry, heavy tails, heteroscedasticity and serial correlation. Loss distributions are time aggregated by taking products of characteristic functions. We test loss-surface-implied forecasts of value at risk and expected shortfall out of sample on a diverse set of portfolios and we compare our forecasts to industry-standard risk forecasts that are based on asset and factor covariance matrices. The empirical results make a compelling case for the application and further development of our approach.

value at risk, expected shortfall, loss surface, downside risk, tail risk, peaks over thresholds, semi-parametric distribution, Fourier transform, temporal dependence

Abstract:
Quantitative risk management relies on a constellation of tools that are used to analyze portfolio risk. We develop the standard toolkit, which includes betas, risk budgets and correlations, in a general, coherent, mnemonic framework centered around marginal risk contributions. We apply these tools to generate side-by-side analyses of volatility and expected shortfall, which is a measure of average portfolio excess of value-at-risk. We focus on two examples whose importance is highlighted by the current economic crisis. By examining downside protection provided by an out-of-the-money put option we show that the diversification benefit of the option is greater for a risk measure that is more highly concentrated in the tail of the distribution. By comparing two-asset portfolios that are distinguished only by the likelihood of coincident extreme events, we show that expected shortfall measures market contagion in a way that volatility cannot.

risk management, quantitative extreme

Abstract:
We give an empirical assessment of I^2, a structural credit model based on incomplete information. In this model, investors cannot observe a firm's default barrier. As a consequence, I^2 exhibits both the economic appeal of a structural model and the tractable pricing formulae and empirical plausibility of a reduced form model. We compare default probability and credit spread forecasts generated by I^2 and the well-known structural models of Merton (1974) and Black & Cox (1976). We find that I^2 reacts more quickly to new information and, unlike the other two models, it forecasts positive short term credit spreads.

credit risk, incomplete information, pricing trend, short spreads, default barrier

Abstract:
Given a collection of single-market covariance matrix forecasts for different markets, we describe how to embed them into a global forecast of total risk.

We do this by starting with any global covariance matrix forecast that contains information about cross-market correlations, and revising it to agree with the pre-specified submarket matrices, preserving the requirement that a covariance matrix be positive semi-definite.

We characterize the ways this can be done and address the resulting numerical optimization problem.

portfolio risk, total risk, optimization, positive definite

Abstract:
We describe a class of quantitative credit risk models that take account of the unavoidable gaps in investors' information. These incomplete information models are structural/reduced form hybrids. They combine the best features of both traditional approaches while avoiding many of their shortcomings.

Credit risk, incomplete information, default, recovery, risk premium, power curve

Abstract:
A multi-name credit derivative is a security that is tied to an underlying portfolio of corporate bonds and has payoffs that depend on the loss due to default in the portfolio. The value of a multi-name derivative depends on the distribution of portfolio loss at multiple horizons. Intensity-based models of the loss point process that are specified without reference to the portfolio constituents determine this distribution in terms of few economically meaningful parameters, and lead to computationally tractable derivatives valuation problems. However, these models are silent about the portfolio constituent risks. They cannot be used to address applications that are based on the relationship between portfolio and component risks, for example constituent risk hedging. This paper develops a method that extends the reach of these models to the constituents. We use random thinning to decompose the portfolio intensity into the sum of the constituent intensities. We show that a thinning process, which allocates the portfolio intensity to constituents, uniquely exists and is a probabilistic model for the next-to-default. We derive a formula for the constituent default probability in terms of the thinning process and the portfolio intensity, and develop a semi-analytical transform approach to evaluate it. The formula leads to a calibration scheme for the thinning processes, and an estimation scheme for constituent hedge sensitivities. Our empirical analysis for September 2008 shows that the constituent hedges generated by our method outperform the hedges prescribed by the Gaussian copula model, which is widely used in practice.

correlated defaults, point process, random thinning, single-name hedging, top-down model

Abstract:
We propose a multi-firm first-passage credit model in which investors have incomplete information. In this model, investors observe neither a firm's value nor its default barrier. The model accounts for the short term risk inherent in default events, the market-wide impact of defaults on security prices due to counterparty relations among firms, and the cyclical default correlation effects observed in credit markets. We explicitly calculate the pricing trend and the arrival intensity of the kth-to-default. These results furnish (1) tractable reduced form formulae for arrival probabilities of sequential dependent defaults and prices of multi-name credit derivatives and (2) an algorithm for the simulation of sequential unpredictable default times.

correlated defaults; incomplete information, pricing trend, intensity, simulation, first-to-default

Abstract:
We present a generalization of the two sample t-test for the equality of means to the case where the sample values have unequal weights. This is a natural situation in financial risk modelling where some samples are considered more reliable than others in predicting a common mean. We describe pooled and unpooled weighted t-tests, and show with an example of real credit data that using the standard unweighted t-test can lead to the wrong statistical conclusion.

Weighted t-statistic, mean, risk model

Abstract:
Risk analysis involves gaining deeper insight into the sources of risk, and evaluating whether these risks accurately reflect the views of the portfolio manager. In this paper, we show how to extend standard volatility analytics to shortfall, a measure of extreme risk. Using two examples, we show how shortfall provides a more complete and intuitive picture of risk than value at risk. In two subsequent examples we illustrate the additional perspective offered by analyzing shortfall and volatility in tandem.

extreme, risk, analysis, volatility, shortfall, different, risk, measures, standard, analytics

Abstract:
We examine the efficacy of the I-squared incomplete information credit model in a broad context that is relevant to fund and asset managers.

Using a rigorous statistical analysis, we show that I-squared is a powerful forecaster of the following events:

- Rating agency downgrades
- Investment grade to high yield downgrades
- High yield defaults

These statistical results translate directly into useful portfolio construction techniques. For example, we show that the number of high-yield defaults in a portfolio can be reduced dramatically by excluding a very small number of names.

default, downgrade, credit event, incomplete information credit model, power curve

Abstract:
An extended history of market returns reveals aspects of financial risk that are not evident over short timescales. The most enduring risk measure is variance, which quantifies short-term regularities in return dispersion. An alternative measure, shortfall, quantifies the risk of extreme market moves, and calls for a deep history to inform its forecasts. Both variance and shortfall are convex, meaning that they tend to promote diversification and can be used in optimization. By offering a long-view counterpart to variance, shortfall can significantly broaden an investor's risk perspective.

financial risk, extended history, market returns, variance, forecasts, shortfall, risk, perspective

Abstract:
We evaluate several long/short strategies for managing a portfolio of default swaps. The strategies are based on a ranking of credits by residuals, which are the differences between market spreads and spreads generated by the iSpread structural model. Investment grade portfolios for the U.S. and Europe earned an average of 70 basis points for each long dollar notional between January 2004 and December 2006. Non-investment grade portfolios earned 321 basis points averaged over the same regions and time period. Transaction cost estimates based on scenario analysis ranged from 19 to 27 basis points for investment grade and 26 to 54 basis points for non-investment grade portfolios. Strategies that aim to mitigate transaction cost by holding trades with little profit showed mixed results.

CDS, implied spread, structural model, relative value, credit risk, default swap, iSpread, rich-cheap strategy, carry-neutral strategy, beta-neutral strategy, transaction costs, information ratio, structural credit model

Abstract:
Portfolio risk forecasts are commonly evaluated using test statistics that are sums of random variables. We study the distributional properties of these test statistics for value at risk, expected shortfall, and volatility. For a diverse collection of 74 US equity portfolios, risk forecasts based on an extreme value theory model greatly outperform a conditional normal model with a 23-day halflife. On the other hand, we show that the common assumption of asymptotic normality in test statistics for these risk measures is not always satisfied, especially for test statistics related to volatility.

hyptothesis test, value at risk, expected shortfall

Abstract:
Factor models are standards in investment management. For decades, Barra factor models have provided valuable risk forecasts and inputs for the portfolio construction process. Most uses of factor models have targeted longer horizons of months or years. However, we demonstrate in this paper that factor models can also provide accurate risk forecasts for shorter horizons of one to ten days. Furthermore, factor models have the advantage of explaining risk sources and providing consistency in risk management processes across all time horizons.

We present a factor model with a methodology appropriately tailored to shorter horizons. Our basic approach is to retain the same common risk factors currently used in the Barra Integrated Model (BIM) and adopt a number of techniques that exploit daily data. As we show for different asset classes, markets, and sectors, this factor model approach yields similarly accurate shorter horizon risk forecasts compared to asset-by-asset approaches.

We specifically focus on the accuracy of Value-at-Risk (VaR) estimates.

modeling value, risk factors, investment, management, forecasts, portfolio, construction, shorter, longer, horizons, value-at-risk, VaR

Abstract:
An extended history of market returns reveals aspects of financial risk that are not evident over short timescales. The most enduring risk measure is variance, which quantifies short-term regularities in return dispersion. An alternative measure, shortfall, quantifies the risk of extreme market moves, and calls for a deep history to inform its forecasts. Both variance and shortfall are convex, meaning that they tend to promote diversification and can be used in optimization. By offering a long-view counterpart to variance, shortfall can significantly broaden an investor's risk perspective.

long view financial risk, market returns, risk measure variance, alternative, shortfall, extreme moves, convex

Abstract:
Systematic model bias has been implicated in the global recession that began in 2007, and this bias can be traced back to assumptions about the normality of data. Nonetheless, the normal distribution continues to play a foundational role in quantitative finance. One reason for this is that the normal often emerges, without prompting, as the distribution of sums or averages of large collections of random variables. Precise statements of this miracle are known as Central Limit Theorems, and they appear throughout the physical and social sciences. In this note, we review some of the most widely-used Central Limit Theorems. Subsequently, we explore the gap between the normal distribution and financial risk. This can be traced to a failure of the financial data to satisfy the assumptions of even the most liberal versions of the Central Limit Theorem.

Systematic model global recession quantitative finance random variables central limit theorem normal distribution financial risk

Abstract:
We analyze the default swap market with the two factor I2 structural model, which is driven by firm value and firm leverage. As we show empirically, the default swap market incorporates these risks differentially over time, by region, by industry, and by coarse quality. This leads us to pool firms with similar characteristics into calibration groups whose parameters are used to align model and market sensitivities to the risk factors. We include equity factor returns to account for contagion and momentum effects for industries or credits that have suffered recent downturns. The close alignment of our model spreads with the market enables us to extract systematic effects reflected in the dynamics of average levels of model inputs and outputs, and discern relative value among credits by analyzing model errors. Applications of our model include assessment of relative value, pricing of illiquid names, cross market hedging and monitoring credit portfolios. A rich-cheap portfolio construction strategy based on our model shows consistent profits in most calibration groups between January 2004 and May 2006.

Credit spread, structural model, risk factor, calibration parameter, firm value, firm leverage, distance to effective default, bounded influence estimation, inversion-minimization algorithm

Abstract:
Given a collection of single-market covariance matrix forecasts for different markets, we describe how to embed them into a global forecast of total risk. We do this by starting with any global covariance matrix forecast that contains information about cross-market correlations and revise it to agree with the pre-specified sub-market matrices, preserving the requirement that a covariance matrix be positive semi-definite. We characterize the ways this can be done and address the resulting numerical optimization problem.

single-market covariance matrix forecasts, global forecast, global covariance matrix forecast, cross-market correlations, positive semi-definite, numerical optimization problem

Abstract:
The Modigliani-Miller theorem describes conditions under which the value of a firm is independent of its leverage ratio. It is one of the cornerstones of finance. A history of this result along with a modern perspective on its derivation is given in Rubinstein (2003). We extend this history by examining the relationship between the Modigliani-Miller theorem and quantitative models of credit risk. In the first part of the paper, we sort out the role of the Modigliani-Miller theorem and Merton's classical structural model. This material may be familiar to some readers. Subsequently, we explore the relationship between the Modigliani-Miller theorem and I(2), which is a hybrid structural-reduced form model based on incomplete information, Goldberg (2004). The I(2) model is not consistent with the Modigliani-Miller theorem. It provides a new way to measure the deviation of the real markets from the idealized markets in which the Modigliani-Miller theorem holds.

Credit risk, leverage ratio, incomplete information model, Modigliani-Miller theorem, Merton model

Abstract:
A recent article of Flesaker and Hughston introduces a one factor interest rate model called the rational lognormal model. This model has a lot to recommend it including guaranteed finite positive interest rates and analytic tractability. Consequently, it has received a lot of attention among practitioners and academics alike. However, it turns out to have the undesirable feature of predicting that the asymptotic value of the short rate volatility is zero. This theoretical result is proved rigorously in this article. The outcome of an empirical study complementing the theoretical result is discussed at the end of the article. European call options are valued with the rational lognormal model and a comparably calibrated mean reverting Gaussian model. Unsurprisingly, rational lognormal option values are considerably lower than the analogous mean reverting Gaussian option values. In other words, the volatility in the rational lognormal model declines so quickly that options are severely undervalued.