Efficient algorithms for basket default swap pricing with multivariate Archimedean copulas

We introduce a new importance sampling method for pricing basket default swaps employing exchangeable Archimedean copulas and nested Gumbel copulas. We establish more realistic dependence structures than existing copula models for credit risks in the underlying portfolio, and propose an appropriate density for importance sampling by analyzing multivariate Archimedean copulas. To justify efficiency and accuracy of the proposed algorithms, we present numerical examples and compare them with the crude Monte Carlo simulation, and finally show that our proposed estimators produce considerably smaller variances.     

Modelling dynamic portfolio risk using risk drivers of elliptical processes

The situation of a limited availability of historical data is frequently encountered in portfolio risk estimation, especially in credit risk estimation. This makes it difficult, for example, to find statistically significant temporal structures in the data on the single asset level. By contrast, there is often a broader availability of cross-sectional data, i.e. a large number of assets in the portfolio. This paper proposes a stochastic dynamic model which takes this situation into account. The modelling framework is based on multivariate elliptical processes which model portfolio risk via sub-portfolio specific volatility indices called portfolio risk drivers. The dynamics of the risk drivers are modelled by multiplicative error models (MEMs)-as introduced by Engle [Engle, R.F., 2002. New frontiers for ARCH models. J. Appl. Econom. 17, 425-446]-or by traditional ARMA models. The model is calibrated to Moody’s KMV Credit Monitor asset returns (also known as firm-value returns) given on a monthly basis for 756 listed European companies at 115 time points from 1996 to 2005. This database is used by financial institutions to assess the credit quality of firms. The proposed risk drivers capture the volatility structure of asset returns in different industry sectors. A characteristic cyclical as well as a seasonal temporal structure of the risk drivers is found across all industry sectors. In addition, each risk driver exhibits idiosyncratic developments. We also identify correlations between the risk drivers and selected macroeconomic variables. These findings may improve the estimation of risk measures such as the (portfolio) Value at Risk. The proposed methods are general and can be applied to any series of multivariate asset or equity returns in finance and insurance.     

Analytic solutions for infinite horizon stochastic optimal control problems via finite horizon approximation: A practical guide

We show how infinite horizon stochastic optimal control problems can be solved via studying their finite horizon approximations. This often leads to analytical solutions for the infinite horizon problem by studying phase diagrams, even in cases where the complexity of the finite horizon case does not permit analytic solutions. Our approach can be applied to many problems in dynamic economics.     

Random orthogonal matrix simulation

This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. We introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices L matrices. They may be deterministic, parametric or data specific in nature. The target moments determine the L matrix then infinitely many random samples with the same exact moments may be generated by multiplying the L matrix by arbitrary random orthogonal matrices. This methodology is thus termed “ROM simulation”. Considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But no parametric assumptions are required (unless parametric L matrices are used) so there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.     

Estimating the term structure of mortality

In modeling and forecasting mortality the Lee-Carter approach is the benchmark methodology. In many empirical applications the Lee-Carter approach results in a model that describes the log central death rates by means of linear trends. However, due to the volatility in (past) mortality data, the estimation of these trends, and, thus, the forecasts based on them, might be rather sensitive to the sample period employed. We allow for time-varying trends, depending on a few underlying factors, to make the estimates of the future trends less sensitive to the sampling period. We formulate our model in a state-space framework, and use the Kalman filtering technique to estimate it. We illustrate our model using Dutch mortality data.     

A comparison of the Lee-Carter model and AR-ARCH model for forecasting mortality rates

With the decline in the mortality level of populations, national social security systems and insurance companies of most developed countries are reconsidering their mortality tables taking into account the longevity risk. The Lee and Carter model is the first discrete-time stochastic model to consider the increased life expectancy trends in mortality rates and is still broadly used today. In this paper, we propose an alternative to the Lee-Carter model: an AR(1)-ARCH(1) model. More specifically, we compare the performance of these two models with respect to forecasting age-specific mortality in Italy. We fit the two models, with Gaussian and t-student innovations, for the matrix of Italian death rates from 1960 to 2003. We compare the forecast ability of the two approaches in out-of-sample analysis for the period 2004-2006 and find that the AR(1)-ARCH(1) model with t-student innovations provides the best fit among the models studied in this paper.     

Cyclic orders: Equivalence and duality

Cyclic orders of graphs and their equivalence have been promoted by Bessy and Thomassé’s recent proof of Gallai’s conjecture. We explore this notion further: we prove that two cyclic orders are equivalent if and only if the winding number of every circuit is the same in the two. The proof is short and provides a good characterization and a polynomial algorithm for deciding whether two orders are equivalent. We then derive short proofs of Gallai’s conjecture and a theorem “polar to” the main result of Bessy and Thomassé, using the duality theorem of linear programming, total unimodularity, and the new result on the equivalence of cyclic orders.     

Effective graph resistance

This paper studies an interesting graph measure that we call the effective graph resistance. The notion of effective graph resistance is derived from the field of electric circuit analysis where it is defined as the accumulated effective resistance between all pairs of vertices. The objective of the paper is twofold. First, we survey known formulae of the effective graph resistance and derive other representations as well. The derivation of new expressions is based on the analysis of the associated random walk on the graph and applies tools from Markov chain theory. This approach results in a new method to approximate the effective graph resistance. A second objective of this paper concerns the optimisation of the effective graph resistance for graphs with given number of vertices and diameter, and for optimal edge addition. A set of analytical results is described, as well as results obtained by exhaustive search. One of the foremost applications of the effective graph resistance we have in mind, is the analysis of robustness-related problems. However, with our discussion of this informative graph measure we hope to open up a wealth of possibilities of applying the effective graph resistance to all kinds of networks problems.     

Twistor fibrations over Hermitian symmetric spaces and harmonic maps

Given a twistor space over a Hermitian symmetric space of compact type we construct a map onto a twistor space over another inner symmetric space of compact type. This map is holomorphic and preserves the superhorizontal distributions. We describe an application to harmonic maps.     

Optimal reinsurance under VaR and CTE risk measures

Let X denote the loss initially assumed by an insurer. In a reinsurance design, the insurer cedes part of its loss, say f(X), to a reinsurer, and thus the insurer retains a loss I_f(X)=X−f(X). In return, the insurer is obligated to compensate the reinsurer for undertaking the risk by paying the reinsurance premium. Hence, the sum of the retained loss and the reinsurance premium can be interpreted as the total cost of managing the risk in the presence of reinsurance. Based on a technique used in [Müller, A., Stoyan, D., 2002. Comparison Methods for Stochastic Models and Risks. In: Willey Series in Probability and Statistics] and motivated by [Cai J., Tan K.S., 2007. Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measure. Astin Bull. 37 (1), 93–112] on using the value-at-risk (VaR) and the conditional tail expectation (CTE) of an insurer’s total cost as the criteria for determining the optimal reinsurance, this paper derives the optimal ceded loss functions in a class of increasing convex ceded loss functions. The results indicate that depending on the risk measure’s level of confidence and the safety loading for the reinsurance premium, the optimal reinsurance can be in the forms of stop-loss, quota-share, or change-loss.     

Optimal reinsurance with positively dependent risks

In the individual risk model, one is often concerned about positively dependent risks. Several notions of positive dependence have been proposed to describe such dependent risks. In this paper, we assume that the risks in the individual risk model are positively dependent through the stochastic ordering (PDS). The PDS risks include independent, comonotonic, conditionally stochastically increasing (CI) risks, and other interesting dependent risks. By proving the convolution preservation of the convex order for PDS random vectors, we show that in individualized reinsurance treaties, to minimize certain risk measures of the retained loss of an insurer, the excess-of-loss treaty is the optimal reinsurance form for an insurer with PDS dependent risks among a general class of individualized reinsurance contracts. This extends the study in Denuit and Vermandele (1998) on individualized reinsurance treaties to dependent risks. We also derive the explicit expressions for the retentions in the optimal excess-of-loss treaty in a two-line insurance business model.     

Edgeworth expansion for an estimator of the adjustment coefficient

We establish an Edgeworth expansion for an estimator of the adjustment coefficient R, directly related to the geometric-type estimator for general exponential tail coefficients, proposed in [Brito, M., Freitas, A.C.M., 2003. Limiting behaviour of a geometric-type estimator for tail indices. Insurance Math. Econom. 33, 211-226].Using the first term of the expansion, we construct improved confidence bounds for R. The accuracy of the approximation is illustrated using an example from insurance (cf. [Schultze, J., Steinebach, J., 1996. On least squares estimates of an exponential tail coefficient. Statist. Dec. 14, 353-372]).     

On the invariant properties of notions of positive dependence and copulas under increasing transformations

Notions of positive dependence and copulas play important roles in modeling dependent risks. The invariant properties of notions of positive dependence and copulas under increasing transformations are often used in the studies of economics, finance, insurance and many other fields. In this paper, we examine the notions of the conditionally increasing (CI), the conditionally increasing in sequence (CIS), the positive dependence through the stochastic ordering (PDS), and the positive dependence through the upper orthant ordering (PDUO). We first use counterexamples to show that the statements in Theorem 3.10.19 of Müller and Stoyan (2002) about the invariant properties of CIS and CI under increasing transformations are not true. We then prove that the invariant properties of CIS and CI hold under strictly increasing transformations. Furthermore, we give rigorous proofs for the invariant properties of PDS and PDUO under increasing transformations. These invariant properties enable us to show that a continuous random vector is PDS (PDUO) if and only of its copula is PDS (PDUO). In addition, using the properties of generalized left-continuous and right-continuous inverse functions, we give a rigorous proof for the invariant property of copulas under increasing transformations on the components of any random vector. This result generalizes Proposition 4.7.4 of Denuit et al. (2005) and Proposition 5.6. of McNeil et al. (2005).     

A new approach to the credibility formula

The usual credibility formula holds whenever, (i) claim size distribution is a member of the exponential family of distributions, (ii) prior distribution conjugates with claim size distribution, and (iii) square error loss has been considered. As long as, one of these conditions is violent, the usual credibility formula no longer holds. This article, using the mean square error minimization technique, develops a simple and practical approach to the credibility theory. Namely, we approximate the Bayes estimator with respect to a general loss function and general prior distribution by a convex combination of the observation mean and mean of prior, say, approximate credibility formula. Adjustment of the approximate credibility for several situations and its form for several important losses are given.     

On the link between credibility and frequency premium

This paper questions the equidistribution assumption for the random effects in a frequency risk model. Two models are presented, which use parametric and nonparametric links between the variance of the random effect and frequency risk. They are estimated on a Spanish automobile insurance portfolio, for which a decreasing link is obtained. Conclusions are drawn for credibility and bonus-malus coefficients.     

Extensions by Means of Expansions and Selections

We provide proper mapping-characterizations of some embedding-like properties weaker than -embedding. For instance, we show that a subset A of a space X is -embedded in X if and only if for every continuous map g: A → Y into a Banach space Y of weight w(Y) ⩽ λ, there exists a continuous set-valued mapping φ of X into the nonempty compact subsets of Y such that g is a selection for φ∣A (i.e., g(x) ∈ φ(x) for every x ∈ A). On the other hand, we show that a subset A is C*-embedded in X if and only if for every continuous set-valued mapping φ of X into the non-empty compact subsets of a Banach space Y, every continuous selection g: A → Y for φ∣A is continuously extendable to the whole of X. Combining both results we get the well-known mapping-characterization of -embedding which makes more transparent the relation ‘’. Other weak components of -embedding are described in terms of expansions and selections, possible applications are demonstrated as well.     

The feasibility pump

In this paper we consider the NP-hard problem of finding a feasible solution (if any exists) for a generic MIP problem of the form min{c^Tx:Ax≥b,x_j integer ∀j ∈ }. Trivially, a feasible solution can be defined as a point x* ∈ P:={x:Ax≥b} that is equal to its rounding , where the rounded point is defined by := x*_j if j ∈ and := x*_j otherwise, and [·] represents scalar rounding to the nearest integer. Replacing “equal” with “as close as possible” relative to a suitable distance function Δ(x*, ), suggests the following Feasibility Pump (FP) heuristic for finding a feasible solution of a given MIP.We start from any x* ∈ P, and define its rounding . At each FP iteration we look for a point x* ∈ P that is as close as possible to the current by solving the problem min {Δ(x, ): x ∈ P}. Assuming Δ(x, ) is chosen appropriately, this is an easily solvable LP problem. If Δ(x*, )=0, then x* is a feasible MIP solution and we are done. Otherwise, we replace by the rounding of x*, and repeat.We report computational results on a set of 83 difficult 0-1 MIPs, using the commercial software ILOG-Cplex 8.1 as a benchmark. The outcome is that FP, in spite of its simple foundation, proves competitive with ILOG-Cplex both in terms of speed and quality of the first solution delivered. Interestingly, ILOG-Cplex could not find any feasible solution at the root node for 19 problems in our test-bed, whereas FP was unsuccessful in just 3 cases.     

Some results on the multivariate truncated normal distribution

This note formalizes some analytical results on the n-dimensional multivariate truncated normal distribution where truncation is one-sided and at an arbitrary point. Results on linear transformations, marginal and conditional distributions, and independence are provided. Also, results on log-concavity, A-unimodality and the MTP₂ property are derived.     

Heterogeneous information lags and evolutionary stability

We investigate the effect of information lags in discrete time evolutionary game dynamics on symmetric games. At the end of each period, some players obtain information about the distribution of strategies among the entire population. They update their strategies according to this information. In contrast to the previous literature (e.g., Tao and Wang (1997)) where large delays lead to instability, we show that the relationship between information lags and the stability of equilibria is not “monotonic.” In anti-coordination games under smoothed best-response dynamics, a small probability of delay can stabilize the equilibrium, while a large probability can destabilize it.     

Specification and estimation of multiple output technologies: A primal approach

This paper addresses specification and estimation of multiple-outputs and multiple-inputs production technology in the presence of technical inefficiency. The primary focus is on the primal formulations. Several competing specifications such as production function, input (output) distance function, input requirement function are considered. We show that all these specifications come from the same transformation function and are algebraically identical. We also show that: (i) unless the transformation function is separable (i.e., outputs are separable from inputs), the input (output) ratios in the input (output) distance function can not be treated as exogenous (uncorrelated with technical inefficiency) resulting inconsistent estimates of the input (output) distance function parameters. (ii) Even if input (output) ratios are exogenous, estimation of the input (output) distance function will result in inconsistent parameter estimates if outputs (inputs) are endogenous. We address endogeneity and instrumental variable issues in details in the context of flexible (translog) functional forms. Estimation of several specifications using both single and system approaches are discussed using Norwegian dairy farming data.