Eduardo F. L. de Melo

Robust Fits for Copula Models

In this article, we obtain robust estimators for copula parameters through the minimization of weighted goodness-of-fit statistics. Different weight functions emphasize different regions on the unit square and are able to handle different locations of model violation. The resulting WMDE estimators are compared to the classical maximum likelihood estimators MLE, and to their weighted version WMLE, an estimator obtained in two steps. The weights obtained in the first step by the application of a high breakdown point scatter matrix estimator are used to identify atypical points. All estimators are compared in a comprehensive simulation study. For each ε-contaminated parametric copula family considered, we showed that there is a robust estimator improving over the MLE and able to capture the correct strength of dependence of the data, despite the contamination percentual and location, and the sample size.

A estrutura a termo de taxas de juros no Brasil: modelos, estimação e testes

Neste artigo, propomos uma metodologia para a construcao da estrutura a termo da taxa de juros livre de risco no Brasil, usando o modelo de Svensson para interpolacao e extrapolacao das curvas de juros e algoritmos geneticos, em complemento aos algoritmos tradicionais de otimizacao nao linear, para a estimacao dos parâmetros do modelo. O objetivo e contribuir para que o mercado segurador brasileiro mensure suas obrigacoes descontando seus fluxos de caixa de maneira consistente e coerente, considerando a adocao, pela Superintendencia de Seguros Privados (SU-SEP), de padroes internacionais de supervisao de solvencia e de reporte financeiro. Ao longo do artigo, apresentamos os resultados encontrados na modelagem das estruturas a termo de diferentes curvas de juros no Brasil.

LOCAL ESTIMATION OF DYNAMIC COPULA MODELS

It has been empirically verified that the strength of dependence in stock markets usually rises with volatility. In this paper we exploit this stylized fact combined with local maximum likelihood estimation of copula models to analyze the dynamic joint behavior of series of financial log returns. Explanatory variables based on the estimated GARCH volatilities are considered as potential regressors for explaining the dynamics in the copula parameters. The proposed model can assess and discriminate how much of the strength of dependence is due just to the time-varying volatility. The final local-parametric estimates may be used to compute risk measures, to simulate portfolio behavior, and so on. We illustrate our methods using two American indexes. Results indicate that volatility does affect the strength of dependence. The in-sample Value-at-Risk based on the dynamic model outperforms those based on the empirical estimates.

Forecasting Surrender Rates Using Elliptical Copulas and Financial Variables

A multistage stochastic model to forecast surrender rates for life insurance and pension plans is proposed. Surrender rates are forecasted by means of Monte Carlo simulation after a sequence of GLM, ARMA-GARCH, and copula fitting is executed. The model is illustrated by applying it to age-specific time series of surrender rates derived from pension plans with annuity payments of a Brazilian insurer. In the GLM process, the only macroeconomic variable used as an explanatory variable is the Brazilian real short-term interest rate. The advantage of such a variable is that we can take future market expectation through the current term structure of interest rates. The GLM residuals of each age/gender group are then modeled by ARMA-GARCH processes to generate i.i.d. residuals. The dependence among these residuals is then modeled by multivariate Gaussian and Students t copulas. To produce a conditional forecast on a stock market index, in our application we used the residuals of an ARMA-GARCH model fitted to the Brazilian stock market index (Ibovespa) returns, which generates one of the marginal distributions used in the dependence modeling through copulas. This strategy is adopted to explain the high and uncommon surrender rates observed during the recent economic crisis. After applying known simulation methods for elliptical copulas, we proceeded backwards to obtain the forecasted distributions of surrender rates by application, in the sequel, of ARMA-GARCH and GLM models. Additionally, our approach produced an algorithm able to simulate multivariate elliptical copulas conditioned on a marginal distribution. Using this algorithm, surrender rates can be simulated conditioned on stock index residuals (in our case, the residuals of the Ibovespa returns), which allows insurers and pension funds to simulate future surrender rates assuming a financial stress scenario with no need to predict the stock market index.

Evaluating the Technical Provisions for Traditional Brazilian Annuity Plans: Continuous-Time Stochastic Approach Based on Solvency Principles

This article presents an approach for evaluating the liabilities of traditional Brazilian annuity plans, using a continuous-time stochastic approach based on modern solvency principles. The technical provisions are obtained by means of conditional expectation, under a real-world measure and considering the peculiar characteristics of each plan and the financial guarantees and profit participations (bonus and dividend plans) embedded in the annuity plans. We assume that policyholder behavior is not optimal, but we also illustrate a calculation of provision assuming optimal policyholder behavior to show the differences between both assumptions. In this article all explicit provisions formulas are derived, and several relevant conclusions about the values of these provisions are discussed.

Pricing Participating Inflation Retirement Funds Through Option Modeling and Copulas

Abstract Pension plans and life insurances offering minimum performance guarantees are very common worldwide. In the Brazilian market, the customers of a common type of defined contribution plan have the right to receive, over their savings, the positive difference between the return of a specified investment fund, usually a fixed income fund, and the minimum guaranteed rate, commonly defined as the composition of a fixed interest rate and a floating inflation rate. This instrument can be characterized as an option to exchange one asset, the minimum guaranteed rate, for another, the return of the specified investment fund. In this paper we provide a closed formula to evaluate this liability that depends on two stochastic rates assuming bivariate normality. We also explore the use of copulas for the modeling of the dependence structure and price the options using Monte Carlo simulation to compare the effects of the copula specification in their values. An application with real data is provided. The model makes use of a one-factor Vasicek framework for the term structures of interest rate and inflation rate.