Correlation expansions for CDO pricing

This paper develops numerical approximations for pricing collateralized debt obligations (CDOs) and other portfolio credit derivatives in the multifactor Normal Copula model. A key aspect of pricing portfolio credit derivatives is capturing dependence between the defaults of the elements of the portfolio. But, compared with an independent-obligor model, pricing in a model with correlated defaults is more challenging. Our approach strikes a balance by reducing the problem of pricing in a model with correlated defaults to calculations involving only independent defaults. We develop approximations based on power series expansions in a parameter that scales the underlying correlations. These expansions express a CDO tranche price in a multifactor model as a series of prices in independent-obligor models, which are easy to compute. The approach builds on a classical approximation for multivariate Gaussian probabilities; we introduce an alternative representation that greatly reduces the number of terms required to evaluate the coefficients in the expansion. We also apply this method to the underlying problem of computing joint probabilities of multivariate normal random variables for which the correlation matrix has a factor structure.     

Heterogeneity effects on the management of retirement fund

This article studies the importance of plan members’ heterogeneity to the management of defined benefit (DB) pension fund. We propose a new multi-member model of DB pension fund that allows for heterogeneity in plan members’ retirement ages, salary growths and other characteristics. We first solve analytically for optimal management strategy and show that the sponsor’s supplementary contribution and the fund’s allocation in risky assets are determined by the cross-product between the fund’s expected retirement liabilities and some heterogeneity-adjusted discount factors. We then demonstrate that the presence of heterogeneity can have a significant influence on the optimal management strategy and that a management decision made while ignoring heterogeneity will be suboptimal. The knowledge of desirable and undesirable effects of heterogeneity that we uncover in this article also provides implications to the grouping of fund members. Introducing a presence of young member whose salary is positively correlated with the risky asset and avoid stacking members with negative correlations will all help the management.