Time to default in credit scoring using survival analysis: a benchmark study

We investigate the performance of various survival analysis techniques applied to ten actual credit data sets from Belgian and UK financial institutions. In the comparison we consider classical survival analysis techniques, namely the accelerated failure time models and Cox proportional hazards regression models, as well as Cox proportional hazards regression models with splines in the hazard function. Mixture cure models for single and multiple events were more recently introduced in the credit risk context. The performance of these models is evaluated using both a statistical evaluation and an economic approach through the use of annuity theory. It is found that spline-based methods and the single event mixture cure model perform well in the credit risk context.     

Warranty servicing strategies to improve customer satisfaction

Customer satisfaction with a purchased product depends on itsperformance under warranty and during the remainder of its usefullife. Dissatisfaction with an item is important to a manufacturersince it can lead to the loss of potential customers throughthe negative word-of-mouth effect as well as existing customersswitching to a competitor. In this paper, we define satisfactionin terms of the likelihood of a customer not switching to adifferent manufacturer when a new item needs to be purchased. Manufacturers can use specific servicing strategies to reducewarranty costs and this topic has already been addressed inthe literature without considering the effect of customer dissatisfaction.In this paper, we propose particular strategies that will increasecustomer satisfaction and we discuss methods for obtaining theoptimal parameters of these strategies.     

A tractable LIBOR model with default risk

We develop a model for the dynamic evolution of default-free and defaultable interest rates in a LIBOR framework. Utilizing the class of affine processes, this model produces positive LIBOR rates and spreads, while the dynamics are analytically tractable under defaultable forward measures. This leads to explicit formulas for CDS spreads, while semi-analytical formulas are derived for other credit derivatives. Finally, we give an application to counterparty risk.     

On the total operating costs up to default in a renewal risk model

The paper proposes a new approach to study a general class of ruin-related quantities in the context of a renewal risk model. While the classical approaches in Sparre Andersen models have their own merits, the approach presented in this paper has its advantages from the following perspectives. (1) The underlying surplus process has the flexibility to reflect a broad range of scenarios for surplus growth including dividend policies and interest returns. (2) The solution method provides a general framework to unify a great variety of existing ruin-related quantities such as Gerber–Shiu functions and the expected present value of dividends paid up to ruin, and facilitates derivations of new ruin-related quantities such as the expected present value of total claim costs up to ruin, etc. In the end, many specific examples are explored to demonstrate its application in renewal risk models.     

Calibration of the default probability model

In this paper, we study the calibration problem for the Merton–Vasicek default probability model [Robert Merton, On the pricing of corporate debt: the risk structure of interest rate, Journal of Finance 29 (1974) 449–470]. We derive conditions that guarantee existence and uniqueness of the solution. Using analytical properties of the model, we propose a fast calibration procedure for the conditional default probability model in the integrated market and credit risk framework. Our solution allows one to avoid numerical integration problems as well as problems related to the numerical solution of the nonlinear equations.     

A fractional credit model with long range dependent default rate

Motivated by empirical evidence of long range dependence in macroeconomic variables like interest rates we propose a fractional Brownian motion driven model to describe the dynamics of the short and the default rate in a bond market. Aiming at results analogous to those for affine models we start with a bivariate fractional Vasicek model for short and default rate, which allows for fairly explicit calculations. We calculate the prices of corresponding defaultable zero-coupon bonds by invoking Wick calculus. Applying a Girsanov theorem we derive today’s prices of European calls and compare our results to the classical Brownian model.     

A multinomial tree model for pricing credit default swap options

Among the traded credit derivatives, the market interest in credit default swap options (CDSwaptions) is enormous. We propose a multinomial tree model to price Bermudan CDSwaptions. Our basic rationale is that we distribute the occurring probability for each node in a branch proportional to the probability density function of the assumed (normal) distribution. Through this approach, without the need of solving a large number of equations simultaneously, only the first four moments are required to build an arbitrarily large N-branches tree. We also demonstrate the detailed model implementation procedure including the valuation and the estimation of critical prices through an empirical example in Tucker and Wei (J Fixed Income 15(1):88–95, 2005). Numerical results show that, in the valuation, the proposed multinomial tree model is accurate and can significantly save pricing time under the same degree of accuracy as the binomial tree model. In the estimation of critical prices, the results are less accurate than those in the valuation, but the relative errors are acceptable.     

Pricing basket default swaps in a tractable shot noise model

We value CDS spreads and kth-to-default swap spreads in a tractable shot noise model. The default dependence is modelled by letting the individual jumps of the default intensity be driven by a common latent factor. The arrival of the jumps is driven by a Poisson process. By using conditional independence and properties of the shot noise processes we derive tractable closed form expressions for the default distribution and the ordered survival distributions. These quantities are then used to price kth-to-default swap spreads. We calibrate a homogeneous version of the model to the term structure on market data from the iTraxx Europe index series sampled during the period 2008-01-14 to 2010-02-11. We perform 435 calibrations in this turbulent period and almost all calibrations yield very good fits. Finally we study kth-to-default spreads in the calibrated model.     

Pricing model of interest rate swap with a bilateral default risk

Under the foundation of Duffie & Huang (1996) [7], this paper integrates the reduced form model and the structure model for a default risk measure, giving rise to a new pricing model of interest rate swap with a bilateral default risk. This model avoids the shortcomings of ignoring the dynamic movements of the firm’s assets of the reduced form model but adds only a little complexity and simplifies the pricing formula significantly when compared with Li (1998) [10]. With the help of the Crank-Nicholson difference method, we give the numerical solutions of the new model to study the default risk effects on the swap rate. We find that for a one year interest rate swap with the coupon paid per quarter, the variance of the default fixed rate payer decreases from 0.1 to 0.01 only causing about a 1.35%’s increase in the swap rate. This is consistent with previous results.     

A comparison of computational strategies for geometric programs

Numerous algorithms for the solution of geometric programs have been reported in the literature. Nearly all are based on the use of conventional programming techniques specialized to exploit the characteristic structure of either the primal or the dual or a transformed primal problem. This paper attempts to elucidate, via computational comparisons, whether a primal, a dual, or a transformed primal solution approach is to be preferred.The authors wish to thank Captain P. A. Beck and Dr. R. S. Dembo for making available their codes. This research was supported in part under ONR Contract No. N00014-76-C-0551 with Purdue University.     

Equilibrium customer strategies in a single server Markovian queue with setup times

We consider a single server Markovian queue with setup times. Whenever this system becomes empty, the server is turned off. Whenever a customer arrives to an empty system, the server begins an exponential setup time to start service again. We assume that arriving customers decide whether to enter the system or balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine customer behavior under various levels of information regarding the system state. Specifically, before making the decision, a customer may or may not know the state of the server and/or the number of present customers. We derive equilibrium strategies for the customers under the various levels of information and analyze the stationary behavior of the system under these strategies. We also illustrate further effects of the information level on the equilibrium behavior via numerical experiments.      

Analysis of a finite-source customer assignment model with no state information

In this paper we analyse a closed queueing network in which customers have to be assigned to parallel queues. The routing decision may not depend on the numbers of customers in the queues. We present an algorithm and we show that it computes an average optimal policy in case of exponential service times. The algorithm also works for non-exponential service times, in which case periodic policies are found.The research of this author has been supported by the Netherlands Organization for Scientific Research (N.W.O.) and was carried out at the University of Leiden.     

Analysis of a chance-constrained new product risk model with multiple customer classes

We consider the non-convex problem of minimizing a linear deterministic cost objective subject to a probabilistic requirement on a nonlinear multivariate stochastic expression attaining, or exceeding a given threshold. The stochastic expression represents the output of a noisy system featuring the product of mutually-independent, uniform random parameters each raised to a linear function of one of the decision vector’s constituent variables. We prove a connection to (i) the probability measure on the superposition of a finite collection of uncorrelated exponential random variables, and (ii) an entropy-like affine function. Then, we determine special cases for which the optimal solution exists in closed-form, or is accessible via sequential linear programming. These special cases inspire the design of a gradient-based heuristic procedure that guarantees a feasible solution for instances failing to meet any of the special case conditions. The application motivating our study is a consumer goods firm seeking to cost-effectively manage a certain aspect of its new product risk. We test our heuristic on a real problem and compare its overall performance to that of an asymptotically optimal Monte-Carlo-based method called sample average approximation. Numerical experimentation on synthetic problem instances sheds light on the interplay between the optimal cost and various parameters including the probabilistic requirement and the required threshold.     

A simulation model for determining variable worker requirements in a service operation with time-dependent customer demand

In a service operation where worker requirements have to be determined for short scheduling time periods with nonstationary customer demand, the assumptions necessary for applying steady-state solutions to elementary queueing models are usually violated. This paper describes a simulation study of the behavior of such a service operation. The results are compared with the steady-state solutions to a queueing model where individual scheduling time periods are assumed to be independent. It is found that if the system utilization is below a derived maximum value (based on a service level criterion), then the steady-state solutions are robust enough to explain the behavior of the system and can be used to schedule worker requirements.     

An explicit model of default time with given survival probability

For a given filtered probability space (Ω,F,P), an F-adapted continuous increasing process Λ and a positive P-F local martingale N such that Λ₀=0 and N_te^−Λ_t≤1, we construct a probability measure Q^Z and a random time τ such that Q|_F∞=P|_F∞ and Q[τ>t|F_t]=Z_t. The probability Q^Z is linked with the well-known Cox model by an explicit density function. Various properties exist, which characterize Q^Z from others. Let G=(G_t)_t≥0 with G_t=F_t∨σ({τ≤s}:s≤t). We establish the (H^′)-property between the filtrations F and G, and we provide the enlargement of filtration formula.     

Analysis of customer lifetime value and marketing expenditure decisions through a Markovian-based model

The general aim of this study is to provide a guide to the future marketing decisions of a firm, using a model to predict customer lifetime values. The proposed framework aims to eliminate the limitations and drawbacks of the majority of models encountered in the literature through a simple and industry-specific model with easily measurable and objective indicators. In addition, this model predicts the potential value of the current customers rather than measuring the current value, which has generally been used in the majority of previous studies. This study contributes to the literature by helping to make future marketing decisions via Markov decision processes for a company that offers several types of products. Another contribution is that the states for Markov decision processes are also generated using the predicted customer lifetime values where the prediction is realized by a regression-based model. Finally, a real world application of the proposed model is provided in the banking sector to show the empirical validity of the model. Therefore, we believe that the proposed framework and the developed model can guide both practitioners and researchers.     

A model to study the relationship between forecast error,inventory investment,and customer service level in a multiperiod,probabilistic environment

With the advent of Just-In-Time manufacturing strategies, reduction of inventory costs have once again become the focus of all attention. However, efforts to reduce inventory while continuing to live with poor forecasts and unduly high service level requirements, is likely to be futile.This work uses a dynamic programming approach to establish trade-off curves, tying in forecast error, customer service level, and inventory investment. This work applies to realistic, dynamic settings wherein there is uncertainty associated with demand, and lead time could be fixed or probabilistic. This work further assumes form-free probability distributions and thus avoids errors introduced into the analysis from estimating parameters of the distribution from limited data.     

A hybrid fuzzy-optimization approach to customer grouping-based logistics distribution operations

This paper presents an integrated fuzzy-optimization customer grouping based logistics distribution methodology for quickly responding to a variety of customer demands. The proposed methodology involves three main mechanisms: (1) pre-route customer classification using fuzzy clustering techniques, (2) determination of customer group-based delivery service priority and (3) en-route goods delivery using multi-objective optimization programming methods. In the process of pre-route customer classification, the proposed method groups customers’ orders primarily based on the multiple attributes of customer demands, rather than by static geographic attributes, which are mainly considered in classical vehicle routing algorithms. Numerical studies including a real-world application are conducted to illustrate the applicability of the proposed method and its potential advantages over existing operational strategies. Using the proposed method, it is shown that the overall performance of a logistics distribution system can be improved by more than 20%, according to the numerical results from the case studied.     

A mixed integer linear programming model to support e-fulfillment strategies in warehouse-based supermarket chains

Central European Journal of Operations Research - The use of the online channel has greatly increased the logistics costs of supermarket chains. Even the difficulty of managing order picking and...