Multiobjective optimization of credit capital allocation in financial institutions

The evolution of international regulation leads to new capital requirements imposed on globally active companies. Financial services firms allocate capital to business lines in order to withstand the materializing credit losses and to measure the performance of various business lines. In this study, we introduce a methodology for optimal credit capital allocation based on operations research approach. In particular, we focus on the efficient allocation of capital to business lines characterized by credit risk losses and cost of capital. We compare different allocation methods and provide a rationale behind using the OR approach. Finally, we formulate a multiobjective optimization model to capital allocation problem and apply it to a real-world case of two financial conglomerates.     

An optimization approach to adaptive multi-dimensional capital management

Firms should keep capital to offer sufficient protection against the risks they are facing. In the insurance context methods have been developed to determine the minimum capital level required, but less so in the context of firms with multiple business lines including allocation. The individual capital reserve of each line can be represented by means of classical models, such as the conventional Cramér–Lundberg model, but the challenge lies in soundly modelling the correlations between the business lines. We propose a simple yet versatile approach that allows for dependence by introducing a common environmental factor. We present a novel Bayesian approach to calibrate the latent environmental state distribution based on observations concerning the claim processes. The calibration approach is adjusted for an environmental factor that changes over time. The convergence of the calibration procedure towards the true environmental state is deduced. We then point out how to determine the optimal initial capital of the different business lines under specific constraints on the ruin probability of subsets of business lines. Upon combining the above findings, we have developed an easy-to-implement approach to capital risk management in a multi-dimensional insurance risk model.     

Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation

In the renewal risk model, we study the asymptotic behavior of the expected time-integrated negative part of the process. This risk measure has been introduced by Loisel (2005) [1]. Both heavy-tailed and light-tailed claim amount distributions are investigated. The time horizon may be finite or infinite. We apply the results to an optimal allocation problem with two lines of business of an insurance company. The asymptotic behavior of the two optimal initial reserves is computed.     

The effect of the nature of the liabilities on the solvency and maturity payouts of a UK life office fund: a stochastic evaluation

Several studies have shown that a simulation model of participating life insurance business must reflect explicitly the major decisions which are left to the discretion of management. In the UK these decisions include asset allocation and bonus distribution. It is also common in the UK to use premium bases which do not explicitly account for the expectation that a terminal bonus will be paid, while using a uniform system of reversionary bonus for business of all terms. Here, we show that this results in short-term business being at much greater risk of insolvency than long-term business, under a variety of investment strategies. Our main purpose is to study the effect of small changes to the parameters of the investment model used on the outcomes described above. For this purpose we compare results using two versions of the Wilkie model (Wilkie, A.D., 1986. A stochastic investment model for actuarial use. Transactions of the Faculty of Actuaries 39, 341–403, Wilkie, A.D., 1995. More on a stochastic asset model for actuarial use. British Actuarial Journal 1, 1–168.) We show that small changes in the variances of the economic series modelled have a marked impact on solvency, while changes in maturity values are mostly the result of a small change in the rate of real dividend growth.     

Transfer pricing and decentralized dynamic lot-sizing in multistage,multiproduct production processes

This paper presents the use of a transfer pricing system to coordinate business units in a Wagner–Whitin type model for a decentralized lot-sizing problem in a dynamic multistage, multiproduct environment. The paper includes two major proofs: (1) a transfer pricing system enabling optimal decentralized lot-sizing is characterized. The transfer pricing system can be interpreted as a generalization of the reciprocal method or step-down allocation method in cost location accounting; and (2) based on a Wagner–Whitin type theorem and the Kakutani fixed point theorem, it is shown that such a transfer pricing system exists.     

Managing variances in manufacturing system design

The goal of this paper is to investigate how uncertainties in demand and production should be incorporated into manufacturing system design problems. We examine two problems in manufacturing system design: the resource allocation problem and the product grouping problem. In the resource allocation problem, we consider the issue of how to cope with uncertainties when we utilize two types of resources: actual processing capacity and stored capacity (inventory). A closed form solution of the optimal allocation scheme for each type of capacity is developed, and its performance is compared to that of the conventional scheme where capacity allocation and inventory control decisions are made sequentially. In the product grouping problem, we consider the issue of how we design production lines when each line is dedicated to a certain set of products. We formulate a mathematical program in which we simultaneously determine the number of production lines and the composition of each line. Two heuristics are developed for the problem.     

Optimal scoring cutoff policies and efficient frontiers

Multiple business objectives are increasingly important in determining account acquisition and management policies in scored retail credit and loan portfolios. These business objectives include profit and market share, as well as the more traditional management of risk. We formulate a mathematical model that addresses the problem of how acquisition decisions should be made with multiple, conflicting objectives when one, or more than one, scorecard is available to the portfolio manager. We show that iso-contours for expected profit, volume and loss are straight lines in the receiver operating characteristic (ROC) space and develop results that establish equivalence between ROC dominance, maximum expected profit, and efficient-frontier dominance in the space of multiple business measures. For two non-dominating scorecards, we derive the efficient frontiers in the profit-volume space and provide guidelines for choosing optimal policies based on the decision maker's trade-offs between objectives.     

Group forecasting accuracy in hotels

Yield management helps hotels more profitably manage the capacity of their rooms. Hotels tend to have two types of business: transient and group. Yield management research and systems have been designed for transient business in which the group forecast is taken as a given. In this research, forecast data from approximately 90 hotels of a large North American hotel chain were used to determine the accuracy of group forecasts and to identify factors associated with accurate forecasts. Forecasts showed a positive bias and had a mean absolute percentage error (MAPE) of 40% at two months before arrival; 30% at one month before arrival; and 10–15% on the day of arrival. Larger hotels, hotels with a higher dependence on group business, and hotels that updated their forecasts frequently during the month before arrival had more accurate forecasts.     

Modelling a multi-objective allocation problem in a government sponsored entrepreneur development programme

Typically the vendor allocation problem for a commercial company requires the buyer to allocate business between the selected vendor suppliers to meet its needs at minimum cost, taking account of its quality and delivery performance requirements. However in a government sponsored entrepreneur development programme, the allocation of business cannot only be made dependent on criteria that reflect the current ability of the vendor nor solely on criteria fulfilling the current needs of the buyer. It must also consider how it may aid the vendors to improve their business performance in the future and also on how it affects the development of that indigenous industry sector. Furthermore, conditions change from one year to the next so the problem is dynamic over time.This paper describes how a combination of Goal Programming model (GP) with pre-emptive priority ranking of goal constraints with a linear programming model facilitates allocation of businesses to entrepreneurs. In particular, the subject used in this paper is a Malaysia Government sponsored entrepreneur development programme for furniture maker, which administered by a commercial company. An innovative aspect of the work is the use of GP as a modelling rather than a solution technique. It is used to convert the conflicting multi-objectives that were expressed only in general policy terms to operational terms on which business could be allocated. The paper shows how it can be used to try to understand, and then formally model how managers use their judgement and experience in a complex multi-criteria situation. The paper goes on to show that using the formal model leads to more consistency in decision-making and an improvement in the achievement of the objectives. This is important as several different managers are having to make independent decisions on subsets of the vendors.     

A simulated annealing approach for buffer allocation in reliable production lines

We describe a simulated annealing approach for solving the buffer allocation problem in reliable production lines. The problem entails the determination of near optimal buffer allocation plans in large production lines with the objective of maximizing their average throughput. The latter is calculated utilizing a decomposition method. The allocation plan is calculated subject to a given amount of total buffer slots in a computationally efficient way.     

商业银行激励费用分配的数学模型

研究商业银行激励费用分配问题，建立了激励费用分配的数学模型，并对模型进行了分析和求解。目前对激励费用分配问题尚缺少科学的系统研究，本文给出了处理该问题的一种理论依据，其方法和结论也可用于其它类似的问题。     

Allocation of flexible and indivisible resources with decision postponement and demand learning

We consider a firm that uses two perishable resources to satisfy two demand types. Resources are flexible such that each resource can be used to satisfy either demand type. Resources are also indivisible such that the entire resource must be allocated to the same demand type. This type of resource flexibility can be found in different applications such as movie theater complexes, cruise lines, and airlines. In our model, customers arrive according to independent Poisson processes, but the arrival rates are uncertain. Thus, the manager can learn about customer arrival rates from earlier demand figures and potentially increase the sales by postponing the resource allocation decision. We consider two settings, and derive the optimal resource allocation policy for one setting and develop a heuristic policy for the other. Our analysis provides managerial insights into the effectiveness of different resource allocation mechanisms for flexible and indivisible resources.     

Symmetrically multilateral-bargained allocations in multi-sided assignment markets

We extend the notion of symmetrically pairwise-bargained (SPB) allocations (Rochford, J Econ Theory, 34:262–281, 1984) to balanced assignment games with more than two sides. A symmetrically multilateral-bargained (SMB) allocation is a core allocation such that any agent’s payoff remains invariant after a negotiation process between all agents based on what they could receive—and use as a threat—in their preferred alternative matching to any given optimal matching. We prove that, for balanced multi-sided assignment games, the set of SMB is always nonempty and that, unlike the two-sided case, it does not coincide in general with the kernel (Davis and Maschler, Naval Res Logist Q 12:223–259, 1965). We also give an answer to an open question formulated by Rochford by introducing a kernel-based set whose intersection with the core coincides with the set of SMB.     

Multi-year non-life insurance risk of dependent lines of business in the multivariate additive loss reserving model

We derive analytical estimators of non-life insurance risk in multi-year view for the multivariate additive loss reserving model. Thereby we jointly assess reserve and premium risks of multiple years for portfolios of possibly dependent lines of business in one integrated approach. By extending existing formulae for the univariate additive model to the multivariate case, risk estimators for the aggregated portfolio now include the inherent dependencies among all lines of business. The resulting risk evaluation over one-year and general multi-year horizons is fundamental to regulatory reporting (e.g. the ORSA process in Solvency II) and risk-based business planning of non-life insurers with multiple lines of business. A case study illustrates the fruitful application of our formulae and reproduces previous findings for the special case of ultimo view.     

The Determination of the Optimum Joint Replenishment Policy for a Group of Discount-Connected Stock Lines

This paper investigates the situation where a joint replenishment order for a group of stock lines from a common supplier qualifies for a quantity discount provided the total order equals a specified number of units. It is assumed that, when any line in the group reaches its reorder point, an order of this size is made for the whole group and it is the problem of how to determine the optimum order "mix" which is considered here. In the case of a group consisting of two lines only, an analytical solution is derived which, assuming Poisson sales-rates, gives the individual stock levels for which the expected number of sales between successive replenishments is maximized for a given total stock. However, for groups of more than two lines, it was necessary to resort to numerical methods and a computer program is described which uses a quasi dynamic-programming approach to determine the optimum set of individual stocks for any given set of sales-rates and total stock. Based on the results from this program, a "ready reckoner" has been constructed which permits an easy look-up to obtain the optimum allocation of any given total stock between the lines of a group of any size and sales-rates. Finally, the possible extension of this approach to other multi-item inventory problems is discussed.     

TVaR-based capital allocation with copulas

Because of regulation projects from control organisations such as the European solvency II reform and recent economic events, insurance companies need to consolidate their capital reserve with coherent amounts allocated to the whole company and to each line of business. The present study considers an insurance portfolio consisting of several lines of risk which are linked by a copula and aims to evaluate not only the capital allocation for the overall portfolio but also the contribution of each risk over their aggregation. We use the tail value at risk (TVaR) as risk measure. The handy form of the FGM copula permits an exact expression for the TVaR of the sum of the risks and for the TVaR-based allocations when claim amounts are exponentially distributed and distributed as a mixture of exponentials. We first examine the bivariate model and then the multivariate case. We also show how to approximate the TVaR of the aggregate risk and the contribution of each risk when using any copula.     

Proposing a method for fixed cost allocation using DEA based on the efficiency invariance and common set of weights principles

One of the applications of data envelopment analysis is fixed costs allocation among homogenous decision making units. In this paper, we first prove that Beasley’s method (Eur J Oper Res 147(1):198–216, 2003), whose infeasibility has been claimed by Amirteimoori and Kordrostami (Appl Math Comput 171(1):136–151, 2005), always has a feasible solution and the efficiency invariance principle does not necessarily satisfy in Amirteimoori and Kordrostami’s method (Appl Math Comput 171(1):136–151, 2005). Hence, we present two equitable methods for fixed cost allocation based on the efficiency invariance and common set of weights principles such that, if possible, they help meet these two principles. In the first method, the costs are allocated to DMU in such a way that the efficiency score of DMUs does not change, and simultaneously this allocation has the minimum distance from the allocation that has been obtained with a common set of weights. However, in the second method, the costs are allocated in such a way that input and output of all units have a common set of weights and it has the minimum distance from the allocation that satisfies the efficiency invariance principle. Moreover, both methods, consider the satisfaction of each unit of the allocated cost. Finally, the proposed method is illustrated by two real world examples.     

A constrained nonlinear 0–1 program for data allocation

This paper analyzes the problem of allocating copies of relations from a global database to the sites of a geographically distributed communication network. The objective of the allocation is to minimize the total cost due to transmissions generated by queries from the various sites, including queries that access multiple relations. This allocation problem is modeled as a constrained nonlinear 0–1 subproblems generated during subgradient optimization are solved as optimization. Some of the unconstrained quadratic 0–1 subproblems generated during subgradient optimization are solved as maximum flow problems, while the others require implicit enumeration, depending on the nature of the objective function coefficients of the subproblems. Our solution approach is tested extensively on data allocation problems with as many as 100 sites and 20 relations. On a set of randomly generated test problems our approach was close to two orders of magnitude faster than the general purpose integer programming code OSL.     

TVaR-based capital allocation with copulas

Because of regulation projects from control organisations such as the European solvency II reform and recent economic events, insurance companies need to consolidate their capital reserve with coherent amounts allocated to the whole company and to each line of business. The present study considers an insurance portfolio consisting of several lines of risk which are linked by a copula and aims to evaluate not only the capital allocation for the overall portfolio but also the contribution of each risk over their aggregation. We use the tail value at risk (TVaR) as risk measure. The handy form of the FGM copula permits an exact expression for the TVaR of the sum of the risks and for the TVaR-based allocations when claim amounts are exponentially distributed and distributed as a mixture of exponentials. We first examine the bivariate model and then the multivariate case. We also show how to approximate the TVaR of the aggregate risk and the contribution of each risk when using any copula.     

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This note deals with an insurance company with multiple lines of business. In the context of heavy-tailed heterogeneous claim amounts with the 1st upper-orthant tail dependence, based on the so-called k-out-of-n ruin set, we can exhibit the Radon measure mu and derive the asymptotic ruin probability for some of all lines business to ruin in a finite time. One numerical example is also presented to illustrate our main results.