Optimal Petrov--Galerkin Methods through Approximate Symmetrization

^*Present address: Department of Mathematics, Imperial College, London SW7 2BZ. A technique of approximate symmetrization is used to derivea test space from a given trial space for a Petrov—Galerkinmethod. This is applied to one-dimensional diffusion—convectionproblems to give approximations which are near optimal in anenergy norm. Rigorous and precise error bounds are derived todemonstrate the uniformly good behaviour and near optimalityof the procedure over all values of the mesh Péclet number.     

A multicriteria approach for rating the credit risk of financial institutions

Within the new bank regulatory context, the assessment of the credit risk of financial institutions is an important issue for supervising authorities and investors. This study explores the possibility of a developing risk assessment model for financial institutions using a multicriteria classification method. The analysis is based on publicly available financial data for UK firms. The results indicate that the proposed multicriteria methodology provides promising results compared to well known statistical methods.     

The Computation of all the Derivatives of a B-spline Basis

Splines are currently much used in the field of interpolationto functions and their derivatives. In this context for a givenargument two relationships between derivatives of B-spline basesof consecutive orders are derived. Using these relationshipsit is shown there are (K—1)!((K—m1)!m!) schemesfor the evaluation of the mth derivative of a B-spline basisof order k. Analyses of error growth in terms of a matrix notationare carried out in order to see which of the schemes is themost numerically stable, for uniform or highly non-uniform knotsets. The computation of the B-spline basis of order ^K and its(K—1)th derivative are shown to have small a priori relativeerror bounds.     

Multilevel flexible specification of the production function in health economics

^** Email: grassetti{at}stat.unipd.it^*** Email: e.gori{at}dss.uniud.it^**** Email: simona.minotti{at}unicatt.it Previous studies on hospitals' efficiency often refer to quiterestrictive functional forms for the technology (Aigner et al.,1977, J. Econom., 6, 21–37). In this paper, referringto a study about some hospitals in Lombardy, we formulate convenientcorrectives to a statistical model based on the translogarithmicfunction—the most widely used flexible functional form(Christensen et al., 1973, Rev. Econ. Stat., 55, 28–45).More specifically, in order to take into consideration the hierarchicalstructure of the data (as in Gori et al., 2002, Stat. Appl.,14, 247–275), we propose a multilevel model, ignoringfor the moment the one-side error specification, typical ofstochastic frontier analysis (Aigner et al., 1977, J. Econom.,6, 21–37). Given this simplification, however, we areeasily able to take into account some typical econometric problemsas, e.g. heteroscedasticity. The estimated production functioncan be used to identify the technical inefficiency of hospitals(as already seen in previous works), but also to draw some economicconsiderations about scale elasticity, scale efficiency andoptimal resource allocation of the productive units. We willshow, in fact, that for the translogarithmic specification itis possible to obtain the elasticity of the output (regardingan input) at hospital level as a weighted sum of elasticitiesat ward level. Analogous results can be achieved for scale elasticity,which measures how output changes in response to simultaneousinputs variation. In addition, referring to scale efficiencyand to optimal resource allocation, we will consider the resultsof Ray (1998, J. Prod. Anal., 11, 183–194) to our context.The interpretation of the results is surely an interesting administrativeinstrument for decision makers in order to analyse the productiveconditions of each hospital and its single wards and also todecide the preferable interventions.     

An Embedding Method for the Cauchy--Riemann Equations

^*Present address: Department of Applied Mathematics, University College of Wales, Aberystwyth SY23 3 BZ. A technique is proposed in which the Cauchy—Riemann equationsare embedded in a second-order system of equations. The solutionof the original Cauchy—Riemann equations is then retrievedif appropriate additional boundary conditions are imposed onthe embedded system. This system is also shown to result froma least-squares formulation of the first-order equations.     

Reimbursement systems and regional inpatient allocation: a non-linear optimisation model

^** Email: marion.rauner{at}univie.ac.at^*** Email: georg.schneider{at}univie.ac.at^**** Email: kurt.heidenberger{at}univie.ac.at This study presents a non-linear optimisation model for investigatingthe optimal allocation of both budgets and inpatients with differenttreatments among hospitals within a geographic region such asVienna. The objective function maximises the overall qualityof treatment provided by regional hospitals. We compare theeffects of two different reimbursement systems—fixed versusvariable budgets—on optimal allocation strategies. Thecombination of modelling ideas from hospital location-allocationmodels and economic models to solve such a problem is new accordingto the literature. We found that fixed budgets outperformedvariable budgets as fewer Euros had to be invested for an incrementalunit of quality of care provided in most of the policy scenariosanalysed. Regional demand and supply patterns for differenttreatments affect the decision makers' choice of the most suitablereimbursement system. In our illustrative example, two hospitalsappeared inefficient regardless of the reimbursement system.Vienna policy makers are currently considering restructuringthese hospitals. They plan to merge one with nearby hospitalsand transform the other into a nursing home.     

Applicability of generic linear scoring mosels in the U.S. credit-union environment

Although credit-scoring models represent a widely used managerialaid for large financial intermediaries, the vast majority ofU.S. credit unions—relatively small cooperatively ownedretail intermediaries, constrained by sample and funding limitations—haveyet to adopt such techniques. Lovie & Lovie (1986) havetheorized that the flat-maximum effect or curve of insensitivityassociated with linear scoring models could be advantageousin areas of applied prediction such as credit scoring. In thiscontext, we reported the relative predictive power of genericcredit-scoring models versus customized models in an earlierpaper (Overstreet et al. 1992). Unfortunately, these findingswere not readily adaptable to the credit-union industry dueto a dated sample with incomplete credit-bureau information.Consequently, from 1988 to 1991, we gathered a refined databasefrom which to further develop and field-test generic scoringmodels in the credit-union environment. The results reportedherein not only confirm, but amplify, the relative predictivepower of such models found earlier. Relative costs and benefitsof generic versus customized models are modelled for a representativecredit union. Future research directions are set forth in theconclusions.     

Convergence of Methods for the Numerical Solution of the Korteweg--de Vries Equation

We prove that a family of methods for the numerical solutionof the Korteweg—de Vries equation is convergent. Thisfamily includes as particular cases some known finite differenceand finite element schemes. It is also found that the stabilityproperties of the methods vary significantly with the treatmentof the non-linear term. On leave from Shanghai University of Science and Technology,Shanghai, China.     

A bi‐level neural‐based fuzzy classification approach for credit scoring problems

The credit scoring is a risk evaluation task considered as a critical decision for financial institutions in order to avoid wrong decision that may result in huge amount of losses. Classification models are one of the most widely used groups of data mining approaches that greatly help decision makers and managers to reduce their credit risk of granting credits to customers instead of intuitive experience or portfolio management. Accuracy is one of the most important criteria in order to choose a credit‐scoring model; and hence, the researches directed at improving upon the effectiveness of credit scoring models have never been stopped. In this article, a hybrid binary classification model, namely FMLP, is proposed for credit scoring, based on the basic concepts of fuzzy logic and artificial neural networks (ANNs). In the proposed model, instead of crisp weights and biases, used in traditional multilayer perceptrons (MLPs), fuzzy numbers are used in order to better model of the uncertainties and complexities in financial data sets. Empirical results of three well‐known benchmark credit data sets indicate that hybrid proposed model outperforms its component and also other those classification models such as support vector machines (SVMs), K‐nearest neighbor (KNN), quadratic discriminant analysis (QDA), and linear discriminant analysis (LDA). Therefore, it can be concluded that the proposed model can be an appropriate alternative tool for financial binary classification problems, especially in high uncertainty conditions. © 2013 Wiley Periodicals, Inc. Complexity 18: 46–57, 2013     

Credit risk optimization using factor models

We study portfolio credit risk management using factor models, with a focus on optimal portfolio selection based on the tradeoff of expected return and credit risk. We begin with a discussion of factor models and their known analytic properties, paying particular attention to the asymptotic limit of a large, finely grained portfolio. We recall prior results on the convergence of risk measures in this “large portfolio approximation” which are important for credit risk optimization. We then show how the results on the large portfolio approximation can be used to reduce significantly the computational effort required for credit risk optimization. For example, when determining the fraction of capital to be assigned to particular ratings classes, it is sufficient to solve the optimization problem for the large portfolio approximation, rather than for the actual portfolio. This dramatically reduces the dimensionality of the problem, and the amount of computation required for its solution. Numerical results illustrating the application of this principle are also presented. JEL Classification G11     

Extending the intensity model with joint defaults to incorporate the lasting effects from common credit events

This paper studies how the lasting effects of common credit events influence default probability distribution and the prices of multiname credit derivatives. Based on a joint defaults model where common credit events are used to generate simultaneous defaults, we extend the model to allow for their impacts to last for a longer while. The default intensity of each entity is heightened significantly while the impact still has an influence, until some time later when this effect fades away. Incorporating these lasting effects helps to generate higher default correlation, which is more consistent with today's highly correlated financial markets. The proposed model can be either formulated as a Markov chain or implemented by Monte Carlo simulation in order to calculate the default probability distributions and multiname derivatives prices. Our numerical results demonstrate the strong influences from the lasting effects and provide a justification of their incorporation.     

Scheduling fixtures for New Zealand Cricket

^** Email: m.wright{at}lancaster.ac.uk This paper describes the problem faced every year by New ZealandCricket in scheduling the principal inter-regional fixtures.This is a combinatorial optimization problem with few constraintsbut many objectives, which are described in detail. A techniqueknown as Subcost-Guided Simulated Annealing is used to solvethis problem, producing one or more schedules of high quality.One particular feature of the problem requires great care—thedetermination of adequate neighbourhoods for such a tightlyconstrained problem, where most simple changes lead to infeasibility.The approach adopted is to use a complex and unorthodox definitionof a perturbation, each one leading to several possible neighbouringsolutions which are generated by means of a tree search procedure.The system will be used in practice for the 2003–2004cricket season and beyond.     

On the Application of Extrapolation, Deferred Correction and Defect Correction to Discrete-time Galerkin Methods for Parabolic Problems

Conditions which ensure the applicability of extrapolation-in-timeof the backward-difference Galerkin and Crank—NicolsonGalerkin methods for linear parabolic initial-boundary problemsare discussed. Deferred correction and defect correction proceduresfor these discrete-time Galerkin methods are formulated andtheir theoretical properties summarized. These procedures yieldapproximations which are up to fourth-order correct in time,and whose computation requires less work per time step thanthat required by extrapolation. The results of numerical experiments,which demonstrate the expected rates of convergence with respectto the time discretization of the various techniques, are presented. ^*The work of this author was supported in part by the NationalScience Foundation under grant MCS-8102295     

The Caratheodory--Fejer Extension of a Finite Geometric Series

It is shown that the Caratheodory—Fejer extension of afinite geometric series can be given explicitly up to a simplepolynomial equation in an auxiliary variable. This result allowsus to analyse the Caratheodory-Fejer approximation method inthe case where the quotients of successive Maclaurin coefficientsof the given function tend to a limit. ^*Research carried out while this author was at ETH Zurich partiallysupported by a Royal Society European Visiting Fellowship.     

Application of Spline Functions to Systems of Volterra Integral Equations of the First and Second Kinds

Continuous approximations to the solution of systems of Volterraintegral equations of the first and second kinds are soughtby methods using spline functions of degree m, deficiency-(k—1),i.e. in C^m—k, and a fixed quadrature rule of degree p-1,p m-1. The resulting method is called an (m, k)-method. Thestability behaviour of the (m, 1)- and the (m, m)-method isstudied for arbitrarily finite m. Also studied is the stabilityof the (m, m-1)-method for second-kind systems. Convergenceresults and asymptotic formulae for the discretization errorare obtained.     

Credit risk assessment using a multicriteria hierarchical discrimination approach: A comparative analysis

Corporate credit risk assessment decisions involve two major issues: the determination of the probability of default and the estimation of potential future benefits and losses for credit granting. The former issue is addressed by classifying the firms seeking credit into homogeneous groups representing different levels of credit risk. Classification/discrimination procedures commonly employed for such purposes include statistical and econometric techniques. This paper explores the performance of the M.H.DIS method (Multi-group Hierarchical DIScrimination), an alternative approach that originates from multicriteria decision aid (MCDA). The method is used to develop a credit risk assessment model using a large sample of firms derived from the loan portfolio of a leading Greek commercial bank. A total of 1411 firms are considered in both training and holdout samples using financial information through the period 1994–1997. A comparison with discriminant analysis (DA), logit analysis (LA) and probit analysis (PA) is also conducted to investigate the relative performance of the M.H.DIS method as opposed to traditional tools used for credit risk assessment.     

Enhanced decision support in credit scoring using Bayesian binary quantile regression

Fierce competition as well as the recent financial crisis in financial and banking industries made credit scoring gain importance. An accurate estimation of credit risk helps organizations to decide whether or not to grant credit to potential customers. Many classification methods have been suggested to handle this problem in the literature. This paper proposes a model for evaluating credit risk based on binary quantile regression, using Bayesian estimation. This paper points out the distinct advantages of the latter approach: that is (i) the method provides accurate predictions of which customers may default in the future, (ii) the approach provides detailed insight into the effects of the explanatory variables on the probability of default, and (iii) the methodology is ideally suited to build a segmentation scheme of the customers in terms of risk of default and the corresponding uncertainty about the prediction. An often studied dataset from a German bank is used to show the applicability of the method proposed. The results demonstrate that the methodology can be an important tool for credit companies that want to take the credit risk of their customer fully into account.     

Cam design by hyperbolic spline functions of fourth order

E-mail: kohaupt{at}tfh-berlin.de In this paper, cam design using splines is discussed. The splinesare of 4th order so as to guarantee the continuity of the valve-liftcurve up to the third derivative in order to obtain valves withbetter dynamic behaviour. First, ordinary polynomial splinesare investigated, and their limited potential in cam designis illustrated. Then, the reason for introducing the new hyperbolicsplines is explained, and their potential for cam design isshown. Finally, specific cam designs based on the new splinesare compared with cams developed by Kurz. It turns out thatspline cams are an interesting alternative to Kurz's designsince—in some cases—they show a better dynamic behaviourwhen compared with Kurz's cams. The spline cam has been employed,e.g. in the M103 engine of Mercedes-Benz.     

Shape-preserving interpolation by G1 and G2 PH quintic splines

The interpolation of a planar sequence of points p₀, ..., p_Nby shape-preserving G¹ or G² PH quintic splines with specifiedend conditions is considered. The shape-preservation propertyis secured by adjusting ‘tension’ parameters thatarise upon relaxing parametric continuity to geometric continuity.In the G² case, the PH spline construction is based on applyingNewton–Raphson iterations to a global system of equations,commencing with a suitable initialization strategy—thisgeneralizes the construction described previously in NumericalAlgorithms 27, 35–60 (2001). As a simpler and cheaperalternative, a shape-preserving G¹ PH quintic spline schemeis also introduced. Although the order of continuity is lower,this has the advantage of allowing construction through purelylocal equations.     

A fraud-alert model for credit cards during the authorization process

^*Correspondence regarding this paper should be addressed to the first author In this paper, we set forth an expert-system model to help alertbanks and other financial institutions to fraudulent usage ofconsumer credit during the authorization process. The paperfirst addresses the model-building process, then briefly describesprototype development, and finally presents results from analysisof real data from a Canadian bank.