Marilena Sibillo

The Poisson Log-Bilinear Lee-Carter Model: Applications Of efficient bootstrap methods to annuity analyses

Abstract Life insurance companies deal with two fundamental types of risks when issuing annuity contracts: financial risk and demographic risk. Recent work on the latter has focused on modeling the trend in mortality as a stochastic process. A popular method for modeling death rates is the Lee-Carter model. This methodology has become widely used, and various extensions and modifications have been proposed to obtain a broader interpretation and to capture the main features of the dynamics of mortality rates. In order to improve the measurement of uncertainty in survival probability estimates, in particular for older ages, the paper proposes an extension based on simulation procedures and on the bootstrap methodology. It aims to obtain more reliable and accurate mortality projections, based on the idea of obtaining an acceptable accuracy of the estimate by means of variance reducing techniques. In this way the forecasting procedure becomes more efficient. The longevity question constitutes a critical element in the solvency appraisal of pension annuities. The demographic models used for the cash flow distributions in a portfolio impact on the mathematical reserve and surplus calculations and affect the risk management choices for a pension plan. The paper extends the investigation of the impact of survival uncertainty for life annuity portfolios and for a guaranteed annuity option in the case where interest rates are stochastic. In a framework in which insurance companies need to use internal models for risk management purposes and for determining their solvency capital requirement, the authors consider the surplus value, calculated as the ratio between the market value of the projected assets to that of the liabilities, as a meaningful measure of the company’s financial position, expressing the degree to which the liabilities are covered by the assets.

Mathematical and statistical methods in insurance and finance

Least Squares Predictors for Threshold Models: Properties and Forecast Evaluation.- Estimating Portfolio Conditional Returns Distribution Through Style Analysis Models.- A Full Monte Carlo Approach to the Valuation of the Surrender Option Embedded in Life Insurance Contracts.- Spatial Aggregation in Scenario Tree Reduction.- Scaling Laws in Stock Markets. An Analysis of Prices and Volumes.- Bounds for Concave Distortion Risk Measures for Sums of Risks.- Characterization of Convex Premium Principles.- FFT, Extreme Value Theory and Simulation to Model Non-Life Insurance Claims Dependences.- Dynamics of Financial Time Series in an Inhomogeneous Aggregation Framework.- A Liability Adequacy Test for Mathematical Provision.- Iterated Function Systems, Iterated Multifunction Systems, and Applications.- Remarks on Insured Loan Valuations.- Exploring the Copula Approach for the Analysis of Financial Durations.- Analysis of Economic Fluctuations: A Contribution from Chaos Theory.- Generalized Influence Functions and Robustness Analysis.- Neural Networks for Bandwidth Selection in Non-Parametric Derivative Estimation.- Comparing Mortality Trends via Lee-Carter Method in the Framework of Multidimensional Data Analysis.- Decision Making in Financial Markets Through Multivariate Ordering Procedure.- A Biometric Risks Analysis in Long Term Care Insurance.- Clustering Financial Data for Mutual Fund Management.- Modeling Ultra-High-Frequency Data: The S&P 500 Index Future.- Simulating a Generalized Gaussian Noise with Shape Parameter 1/2.- Further Remarks on Risk Profiles for Life Insurance Participating Policies.- Classifying Italian Pension Funds via GARCH Distance.- The Analysis of Extreme Events - Some Forecasting Approaches.

Solvency analysis and demographic risk measures

Purpose - The demographic risk is the risk due to the uncertainty in the demographic scenario assumptions by which life insurance products are designed and valued. The uncertainty lies both in the accidental (insurance risk) and systematic (longevity risk) deviations of the number of deaths from the value anticipated for it. This last component gives rise to the risk due to the randomness in the choice of the survival model for valuations (model risk or projection risk). If the insurance risk component can be assumed negligible for well-diversified portfolios, as in the case of pension annuities, longevity risk is crucial in the actuarial valuations. The question is particularly decisive in contexts in which the longevity phenomenon of the population is strong and pension annuity portfolios constitute a meaningful slice of the financial market – both typical elements of Western economies. The paper aims to focus on the solvency appraisal for a portfolio of life annuities, deepening the impact of the demographic risk according to suitable risk indexes apt to describe its evolution in time. Design/methodology/approach - The financial quantity proposed for representing the economic wealth of the life insurance company is the stochastic surplus, and the paper analyses the impact on it of different demographic assumptions by means of risk indicators as the projection risk index, the quantile surplus valuation and the ruin probability. By means of the proposed models, the longevity risk is mainly taken into account in a stochastic scenario for the financial risk component, in order to consider their interactions, too. In order to furnish practical details significant in the portfolio risk management, several numerical applications clarify the practical meaning of the models in the solvency context. Findings - This paper studies the impact on the portfolio surplus of the systematic demographic risk, taking into account their interaction with the financial risk sources. In this order of ideas, the internal risk profile of a life annuity portfolio is deeply investigated by means of suitable risk indexes: in a solvency analysis perspective, some possible scenarios for the evolution of death rates (generated by different survival models) are considered and this paper evaluates the impact on the portfolio surplus caused by different choices of the demographic model. The first index is deduced by a variance decomposition formula, the other ones involve the conditional quantile calculus and the ruin probability. Such indexes constitute benchmarks, whose conjoined use provides useful information to the meeting of the solvency requirements. Originality/value - With respect to the recent actuarial literature, in which the most important contribution on the surplus analysis has been given by Lisenko

Measuring demographic uncertainty via actuarial indexes

Aim of the paper is the analysis of the behaviour of risk filters connected to the demographic risk drivers for a portfolio of life annuities. The model, easily suitable to the rase of pension annuities, involves the evolution in time of the mortality rates, taking into account the randomness of the financial scenario. Within this context, the uncertainty in the choice of the deiriograpllic scenario is measured and the analysis is also supported by the VaR sensitivity to this risk source.

A stochastic model for financial evaluation: applications to actuarial contracts

This paper presents a model for the force of interest which is based on the consideration of a real force of interest deviations from its estimated value; the resulting stochastic process for financial evaluation is characterized by its expected value and autocovariance functions. Then applications of the results to actuarial contracts are proposed. In particular, the cases of temporary life annuity and n‐year term life insurance are considered and their expected values and variances are illustrated. Copyright

Using Interest Rate Models to Improve Mortality Forecast

Future evolution of mortality poses important challenges for life insurance, pension funds, public policy and fiscal planning. Indeed, when fair values, premium rates and risk reserves are computed, sound and accurate models to forecast stochastic longevity are needed. In this paper, we propose a methodological approach in order to improve the predictive accuracy of the existing survival models. The central idea is to model the ratio between the observed death rates and the corresponding fitted values obtained as outputs of a survival model we select, by means of the Cox-Ingersoll-Ross (CIR) model. For our numerical application, we choose to apply the CIR correction to the Cairns-Blake-Dowd (or M5) model. Using the Italian females mortality data and implementing the backtesting procedure, over both a static time horizon and fixed-length windows rolling one-year ahead through time, we empirically test the performance of the CBD model in forecasting death rates both for itself (CBD) and corrected by the CIR process (mCBD). On the basis of average measures of forecasting errors and information criteria we demonstrate that the mCBD model is a parsimonious model providing better results in terms of predictive accuracy than the CBD model.

A financial analysis of surplus dynamics for deferred life schemes

The paper investigates the financial dynamics of the surplus evolution in the case of deferred life schemes, in order to evaluate both the distributable earnings and the expected worst occurence for the portfolio surplus. The evaluation is based on a compact formulation of the insurance surplus defined as the difference between accrued assets and present value of relevant liabilities. The dynamic analysis is performed by means of Monte Carlo simulations in order to provide a year-by-year valuation. The analysis is applied to a deferred life scheme exemplar, considering that the selected contract constitutes the basis for many life insurance policies and pension plans. The evaluation is put into an asset and liability management decision-making context, where the relationships between profits and risks are compared in order to evaluate the main features of the whole portfolio.

Life Annuity Portfolios: Risk-Adjusted Valuations and Suggestions on the Product Attractiveness

Solvency assessing is a compelling issue for the insurance industry, also in light of the current international risk-based regulations. Internal models have to take into account risk/profit indicators, in order to provide flexible tools aimed at valuing solvency. We focus on a variable annuity with an embedded option involving a participation level which depends on the period financial result. We realize a performance evaluation by means of a suitable indicator, which properly captures both financial and demographic risk drivers. In fact, in the case of life annuity business, assessing solvency has to be framed within a wide time horizon, where specific financial and demographic risks are realized. In this order of ideas, solvency indicators have to capture the amount of capital to cope with the impact of those risk sources over the considered period. The analysis is carried out in accordance with a management perspective, apt to measure the business performance, which requires a correct risk control; in particular we present a study of the dynamics of the profit realized per unit of the total financial value of the contract. On the other hand, the consumer profitability is also measured by means of an utility-equivalent fixed life annuity. Ac-cording to the insureds point of view, we measure their perception of the contract profitability within the expected utility approach.

A Liability Adequacy Test for Mathematical Provision

This paper deals with the application of the Value at Risk of the mathematical provision within a fair valuation context. Through the VaR calculation, the estimate of an appropriate contingency reserve is connected to the predicted worst case additional cost, at a specific confidence level, projected over a fixed accounting period. The numerical complexity is approached by means of a simulation methodology, particularly suitable also in the case of a large number of risk factors.

What If Two Different Interest Rates Datasets Allow for Describing the Same Financial Product

The chance to choose among more than one dataset for representing and describing the movements in the financial market of the same financial entity has noteworthy effects on the practical quantifications. The case we consider in the paper concerns two datasets, different and deemed to be equivalent between them, referred to risk free interest rates. In light of the volatility term structure discrepancies between the two databases and of some closed formulas for stochastically describing the behavior of the financial valuation discrepancies by means of the Vasicek interest rate process, we show two relevant practical evidences. The application concerns the pricing of two derivative cases. The aim is to quantify how much the use of one dataset rather than the other impacts on the final result.