Default Time Research Articles

Asymmetric Defaultable Interest Rate Swap Valuation and Bilateral Credit Value Adjustment

The unilateral defaultable claim valuation problems have been studied extensively, but the valuation of a bilateral contingent claim with asymmetric credit qualities is still lacking convincing mechanism. This paper presents an analytical model for valuing contingent claims, e.g., interest rate swaps, subject to default by both counterparties. The default-free interest rates are modeled by the Market Models, while the default time is modeled by the reduced-form model as the first jump of a time-inhomogeneous Poisson process. All quantities modeled are market-observable. The closed-form solution gives us a better understanding of the impact of the credit asymmetry on swap value, credit value adjustment, swap rate and swap spread.

A copula approach to credit valuation adjustment for swaps under wrong-way risk

This paper deals with the credit valuation adjustment (CVA) of interest rate swap (IRS) contracts in the presence of an adverse dependence between the default time and interest rates: so-called wrong-way risk (WWR). It compares the upper Frechet bound approach introduced in a 2013 paper by Umberto Cherubini with a new semi-analytical IRS–CVA formula that we are proposing as a modification of Cherubini’s approach. The approaches are compared via a numerical study, in which we find that our semianalytical formula (the modified approach) provides more precise IRS–CVA valuation results.

Optimal Inventory Decisions When Offering Layaway

This paper presents an inventory management policy for a retailer offering a layaway program. Layaway is a service provided by retailers that allows budget constrained consumers who have sufficiently high valuations to pay for a product in several installments rather than at once and obtain the product that has been reserved for them at the end of the payment period. If a consumer defaults on payments, then the reserved item is released back into store inventory. In this paper, we first determine the retailer's optimal order decisions when layaway is offered. We find that the order quantity under a layaway program decreases with the likelihood of consumers not finishing their layaway plans and that it is not always profitable for a retailer to offer a layaway program. We then identify the market conditions under which the retailer would benefit from a layaway program. Lastly, we consider an extension to capture the influence of the timing of consumer defaults.

Firm's default ? new methodological approach and preliminary evidence from Poland

Research background: Bankruptcy literature is populated with scores of (econometric) models ranging from Altman?s Z-score, Ohlson?s O-score, Zmijewski?s probit model to k-nearest neighbors, classification trees, support vector machines, mathematical programming, evolutionary algorithms or neural networks, all designed to predict financial distress with highest precision. We believe corporate default is also an important research topic to be identified with the prediction accuracy only. Despite the wealth of modelling effort, a unified theory of default is yet to be proposed.
 Purpose of the article: Due to the disagreement both on the definition and hence the timing of default, as well as on the measurement of prediction accuracy, the comparison (of predictive power) of various models can be seriously misleading. The purpose of the article is to argue for the shift in research focus from maximizing accuracy to the analysis of the information capacity of predictors. By doing this, we may yet come closer to understanding default itself.
 Methods: We critically appraise the bankruptcy research literature for its methodological variety and empirical findings. Default definitions, sampling procedures, in and out-of-sample testing and accuracy measurement are all scrutinized. In an empirical part, we use a double stochastic Poisson process with multi-period prediction horizon and a comprehensive database of some 15,000 Polish non-listed companies to illustrate the merits of our new approach to default modelling.
 Findings & Value added: In the theoretical part, we call for the construction of a single unified default forecasting platform estimated for the largest dataset of firms possible to allow testing the utility of various sources of micro, mezzo, and macro information. Our preliminary empirical evidence is encouraging. The accuracy ratio amounts to 0.92 for t = 0 and drops to 0.81 two years ahead of default. We point to the pivotal role played by the information on firm?s liquidity (alternatively in profitability) and ? in contrast to Altman?s tradition ? hardly any contribution to predictive power of other financial ratios. Macro data is shown to be critical. It adds, on average, more than 10 p.p. to accuracy ratio. In the future, we hope to integrate listed and non-listed firms data into one model, ideally at higher frequency than annual, and include the information on firm's competitiveness position.

Writing off sovereign debt: Default and recovery rates over the cycle

This paper studies the joint determination of sovereign borrowing, default and debt restructuring outcomes. In the data, low debt recovery rates are associated with deep recessions in defaulting countries, high indebtedness at the time of default, and high borrowing costs post-default. I develop a dynamic model of sovereign debt to account for these facts. Recovery rates in the model are determined as the result of two countervailing forces: Cyclical conditions which reduce recovery rates in recessions, and procyclical borrowing which has the opposite effect. The former needs to be sufficiently strong for the model to match the data, and I present empirical evidence and a theoretical rationale for such excess sensitivity of restructuring outcomes to cyclical conditions in the form of countercyclical bargaining power of the sovereign. In the calibrated model, I show that accounting for the cyclicality of recoveries is important for correctly predicting the timing of default events. Procyclical and low recovery rates are detrimental for welfare, but the gains from eliminating the cyclicality are more than twice as high as those from raising average recovery rates.

An enlargement of filtration formula with applications to multiple non-ordered default times

In this work, for a reference filtration $\mathbb {F}$ , we develop a method for computing the semimartingale decomposition of $\mathbb {F}$ -martingales in a specific type of enlargement of $\mathbb {F}$ . As an application, we study the progressive enlargement of $\mathbb {F}$ with a sequence of non-ordered default times and show how to deduce results concerning the first-to-default, $k$ th-to-default, k-out-of-n-to-default or all-to-default events. In particular, using this method, we compute explicitly the semimartingale decomposition of $\mathbb {F}$ -martingales under the absolute continuity condition of Jacod.

General dynamic term structures under default risk

We consider the problem of modelling the term structure of defaultable bonds, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term in the forward rate approach by Heath et al. (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability. In this framework, we derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes.

Indifference pricing with counterparty risk

Abstract We present counterparty risk by a jump in the underlying price and a structural change of the price process after the default of the counterparty. The default time is modeled by a default-density approach. Then we study an exponential utility-indifference price of an European option whose underlying asset is exposed to this counterparty risk. Utility-indifference pricing method normally consists in solving two optimization problems. However, by using the minimal entropy martingale measure, we reduce to solving just one optimal control problem. In addition, to overcome the incompleteness obstacle generated by the possible jump and the change in structure of the price process, we employ the BSDE-decomposition approach in order to decompose the problem into a global-before-default optimal control problem and an after-default one. Each problem works in its own complete framework. We demonstrate the result by numerical simulation of an European option price under the impact of jump’s size, intensity of the default, absolute risk aversion and change in the underlying volatility.

Why Is 10 Past 10 the Default Setting for Clocks and Watches in Advertisements? A Psychological Experiment.

Have you ever noticed that in watch advertisements the time is usually set at 10:10? The reasons and psychological effects of this default time setting are elusive. In Experiment 1, we hypothesized that watches showing a time setting resembling a smiling face (10:10) would enhance emotional valence and intention to buy compared to a neutral time setting (11:30), whereas a time setting resembling a sad face (8:20) would have the opposite effect. Moreover, we investigated a possible interaction effect with the gender of the participants. In Experiment 2, we directly tested the hypotheses that watches set at 10:10 resemble a smiling face, whereas watches set at 8:20 resemble a sad face. The data of the first experiment reveal that watches set at 10:10 showed a significant positive effect on the emotion of the observer and the intention to buy. However, watches set at 8:20 did not show any effect on the emotion or the intention to buy. Moreover, watches set at 10:10 induced in women significantly stronger ratings of pleasure than in men. The data of the second experiment show that participants consistently perceive high resemblance between watches set at 10:10 and a smiling face as well as high resemblance between watches set at 8:20 and a sad face. This study provides for the first time empirical evidence for the notion that using watches with a time setting resembling a smiling face (like 10:10) can positively affect the emotional response of the observers and their evaluation of a seen watch, even though they are not aware of the fact that the shown time setting is inducing this effect. Practical implications of the observed findings and alternative explanations are discussed.

A reduced-form model with default intensities containing contagion and regime-switching Vasicek processes

The contagion credit risk model is used to describe the contagion effect among different financial institutions. Under such a model, the default intensities are driven not only by the common risk factors, but also by the defaults of other considered firms. In this paper, we consider a two-dimensional credit risk model with contagion and regime-switching. We assume that the default intensity of one firm will jump when the other firm defaults and that the intensity is controlled by a Vasicek model with the coefficients allowed to switch in different regimes before the default of other firm. By changing measure, we derive the marginal distributions and the joint distribution for default times. We obtain some closed form results for pricing the fair spreads of the first and the second to default credit default swaps (CDSs). Numerical results are presented to show the impacts of the model parameters on the fair spreads.

A Theoretical and Empirical Analysis of Alternative Discount Rate Concepts for Computing LGDs Using Historical Bank Workout Data

The paper summarizes the discussions of the Global Credit Data (GCD) Discount Rate Work Group (WG). The WG does not provide a “correct” number for the discount rate but derives guiding principles for Loss rate Given Default (LGD) discount rate approaches. The WG has analyzed five main discount rate approaches: 1) Contract rate at origination (Contract): contractual interest rate of a loan, which is the combination of a base rate plus a risk premium at loan origination; 2) Loan weighted average cost of capital (WACC): bank funding costs, which is the combination of the capital ratio weighted cost of equity funding and the debt ratio weighted cost of debt funding. The capital ratio may be based on the loan‐specific regulatory capital requirement; 3) Return on equity (ROE): cost of equity funding; 4) Market return on defaulted debt: return on the bond price at resolution (i.e., the realized recovery at resolution) relative to the bond price at default (i.e., the expected recovery at default); 5) Market equilibrium return: based on the combination of a base rate plus a premium for systematic risk. The risk premium is based on the product of the systematic risk sensitivity and the equity risk premium. The key theoretical findings of the WG are: a) Appropriate discount rates are based on a combination of the risk free rate and risk premium for systematic risk at the time of default; b) Discount rates should not compensate for realized risk; c) Approaches may be separated into contract specific, comparable and equilibrium approaches; d) WACC and market equilibrium return have a slight preference in the Working Group; e) Contract rates are popular in practice as they have practical advantages: they adjust automatically for changing supply/demand and risk aversion in the economy and align with accounting requirements. However, contract rates have shortcomings: i) Rates are based on the origination time: this is a violation of the principle that discount rates should relate to the default time; ii) Rates include the expected loss in relation to default risk: this is a violation of the principle that discount rates should not compensate for realized risk. The key empirical findings of the WG are: Declining risk free rates are in part offset by increasing market risk premiums in the period 2000‐2013; Common empirical discount rates are found to be between the risk free rate and the return on equity; The variation of empirical LGDs is moderate for the various discount rate approaches; GCD may consider providing simple measures for systematic risk for homogenous risk segments to facilitate the appropriate reflection of systematic risk in the discount rates. Risk measures may include asset correlations and variation coefficients.

From the decompositions of a stopping time to risk premium decompositions

The occurrence of some events can impact asset prices and produce losses. The amplitude of these losses are partly determined by the degree of predictability of those events by the market investors, as risk premiums build up in an asset price as a compensation of the anticipated losses. The aim of this paper is to propose a general framework where these phenomena can be properly defined and quantified.Our focus are the default events and the defaultable assets, but the framework could apply to any event whose occurrence impacts some asset prices.We provide the general construction of a default time under the so called (H) hypothesis, which reveals a useful way in which default models can be built, using both market factors and idiosyncratic factors. All the relevant characteristics of a default time (i.e. the Azema supermartingale and its Doob-Meyer decomposition) are explicitly computed given the information about these factors.We then define the default event risk premiums and the default adjusted probability measure. These concepts are useful for pricing defaultable claims in a framework that includes possible economic shocks, such as jumps of the recovery process or of some default-free assets at the default time. These formulas are not classic and we point out that the knowledge of the default compensator (or the intensity process when the default time is totally inaccessible) is not a sufficient quantity for finding explicit prices; the Azema supermartingale and its Doob-Meyer decomposition are needed. The progressive enlargement of a filtration framework is the right tool for pricing defaultable claims in non standard frameworks where non defaultable assets or recovery processes may react at the default event.

Invariance Properties in the Dynamic Gaussian Copula Model

Based on Gaussian tail distribution estimates of independent interest, we study the mathematical properties of the default times (or any of their minima) in the dynamic Gaussian copula model. In particular, depending on the value of the correlation parameter ϱ in the model, the so-called invariance property of CrepeySong15c may be satisfied or not. This gives together an example of a model where the invariance property is satisfied but immersion does not hold, for small ϱ and, for larger ϱ an example of a model where the invariance property may not be satisfied.

Default and liquidation timing under asymmetric information

We consider a dynamic model in which shareholders delegate the running of a firm to a manager, who observes private information about the firm’s running and liquidation costs. We analytically derive the optimal compensation scheme contingent on the firm’s cost structure. Information asymmetries can change the high-cost firm’s bankruptcy choice and timing. Most notably, even if the liquidation value is higher than the face value of debt, the high-cost firm can choose default rather than liquidation to reduce the manager’s information rent. This can lead to the counterintuitive result that the debt value increases beyond the face value as the firm approaches default, and that it jumps upward at the default time. Information asymmetries accelerate negative liquidation and delay positive liquidation, whereas they accelerate default. The optimal leverage ratio of the asymmetric information case becomes higher than that of the symmetric information case because more debt mitigates the loss due to information asymmetries. Our results can potentially account for many empirical findings.

Reading between the ratings: Modeling residual credit risk and yield overlap

Abstract Credit ratings group firms by risk, yet yields are shown to overlap between firms of adjacent ratings. We model this by considering the residual risk arising from differences in the parameters of each firm's value process for firms with the same rating. To do so, our framework simultaneously incorporates jump default with Markov-governed likelihoods and continuous defaults in a default-barrier framework. We provide closed-form approximations for expected default time and tail probabilities, and empirically fit the S-shaped yield curve, intra-rating spread, and inter-rating overlap. Results are robust to time period, rating system, sub-rating, and common characteristics such as liquidity.

Macro-Economic Factors in Credit Risk Calculations: Including Time-Varying Covariates in Mixture Cure Models

The prediction of the time of default in a credit risk setting via survival analysis needs to take a high censoring rate into account. This rate is because default does not occur for the majority of debtors. Mixture cure models allow the part of the loan population that is unsusceptible to default to be modeled, distinct from time of default for the susceptible population. In this article, we extend the mixture cure model to include time-varying covariates. We illustrate the method via simulations and by incorporating macro-economic factors as predictors for an actual bank dataset.

Depression in Sardinian immigrants in Argentina and residents in Sardinia at the time of the Argentinian default (2001) and the Great Recession in Italy (2015)

BackgroundThe aim of this study is to measure in two samples of Sardinian immigrants in Buenos Aires and representatives of the population in Sardinia the prevalence of depressive symptoms at the time of an economic crisis in Sardinia and to compare these results with those collected at the time of a similar crisis in Argentina more than 10 years before.MethodsObservational study. The associations of Sardinian immigrants in Buenos Aires provided the lists of families of Sardinian origin. A random sample of one fifth of registered families was selected. The sample of a study carried out in Sardinia was used as the control. The results were compared with those of the previous study performed in 2001–2002. The Patient Health Questionnaire (PHQ9) was used for the screening of depression.ResultsThe Sardinian immigrants show a lower rate of scoring positively on PHQ9 (i.e. less risk of being depressed) and reach statistical significance after standardization (8.7% vs. 13.1%, P = 0.046). Young women (≤40) are at higher risk. On the contrary, the risk of depression was higher in Sardinian immigrants in Argentina during the 2001–2002 crises.ConclusionThe study indicates a risk for depressive episodes linked to the fallout of the economic crisis (in Argentina in 2001–2002, in Sardinia in 2015) and specifically more in females than in males. Due to the associated socio-demographic risk factors, these results could be interpreted as due to an increase in non-bipolar depression.

Credit default prediction and parabolic potential theory

We consider an approach to credit risk in which the information about the time of bankruptcy is modelled using a Brownian bridge that starts at zero and is conditioned to equal zero when the default occurs. This raises the question whether the default can be foreseen by observing the evolution of the bridge process. Unlike in most standard models for credit risk, we allow the distribution of the default time to be singular. Using a well known fact from parabolic potential theory, we provide a sufficient condition for its predictability.

CDO and HAC

Modelling portfolio credit risk is one of the crucial challenges faced by financial services industry in the last few years. We propose the valuation model of collateralized debt obligations (CDO) based on copula functions with up to three parameters, with default intensities estimated from market data and with a random loss given default that is correlated with default times. The methods presented are used to reproduce the spreads of the iTraxx Europe tranches. We apply hierarchical Archimedean copulae (HAC) whose construction allows for the fact that the risky assets of the CDO pool are chosen from six different industry sectors. The dependence among the assets from the same group is specified with the higher value of the copula parameter, otherwise the lower value of the parameter is ascribed. The copula with two and three parameters models the relation between the loss given default and the default times. Our approach describes the market prices better than the standard pricing procedure based on the Gaussian distribution.

No-arbitrage for informational discrete time market models

This paper focuses on the stability of no-arbitrage, for discrete time market models, under additional uncertainty generated by a random time . At the practical level, this random time represents the death time, the default time of a firm, or any occurrence time of an event that might affect the market somehow. We address the no-arbitrage issue for the resulting new flow of information (filtration) which makes the random time either a nontrivial stopping time (progressive enlargement) or a known time from the beginning (initial enlargement). Our main conclusions are twofold. On the one hand, for a fixed initial market S, we completely and precisely characterize the interplay between S and such that the no-arbitrage is preserved for the new market model. On the other hand, we give the necessary and sufficient conditions on to ensure the preservation of the no-arbitrage under the additional uncertainty of for any market. Two concrete examples are presented to illustrate the results.