Emilia Di Lorenzo

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.

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

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

Approximate solutions of severity of ruins

ZusammenfassungDurch Anwendung numerischer Algorithmen, die auf Approximationen durch kubische Splinefunktionen basieren, bestimmen die Autorinnen die Ruinhöhe. Für die Berechnungen wird “Mathematica” verwendet. Auch für die Ruinwahrscheinlichkeit werden Resultate angegeben.SummaryLet G (u, y) be the severity of ruin, i.e. the probability that, starting with the initial surplus u, ruin occurs and the deficit at the time of ruin is less than y (cf. Gerber, Goovaerts and Kaas, 1987; Dickson, 1989). The Authors determine approximate solutions for the severity of ruin using a numerical algorithm based on cubic spline approximation (cf. Deligoenuel and Bilgen, 1984; Kremer, 1989). The algorithm is performed using Mathematica. Approximations for ruin probability are also obtained.

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.

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.

Empirical Evidences on Predictive Accuracy of Survival Models

The paper focuses on a stochastic proportional hazard model depicting the evolution of the force of mortality; in particular the real data are plotted against a specific survival model by means of the stochastic process that describes their ratio. The predictive accuracy of the survival model is investigated, since, by means of the calibrated “ratio process”, its forecasting skills are assessed. A statistical analysis is developed in order to test the capacity the assumed survival model has to follow the real behavior of the observed data.

Internal risk control by solvency measures

The paper deals with the solvency analysis through internal models in the case of a portfolio of life annuities, focusing on the interplay between stochastic interest rate dynamics and the survival evolution in time. This specific aspect is investigated basing the survival death rates on Poisson Lee Carter model approached according to the Iterative Procedure and two simulated approaches on the Poisson Lee Carter: the Standard Procedure and the Stratified Sampling Procedure. The financial aspect, particularly notable in portfolios with long duration and multiplicity of payments as in the considered case, is tackled assuming different stochastic hypotheses on the interest rates evolution. Aim of the paper is to deepen the reaction of solvency measures as the surplus index and the ruin probability to the specific financial and demographic scenario. The indexes are studied in different loading factor assumptions and several numerical applications illustrate the model setup.

A Dynamic Solvency Approach for Life Insurance

The paper investigates risk management processes in life insurance, in a perspective consistent with the framework of Solvency II. The paper starts with the breakdown of the business dynamics. This analysis provides for a complete depiction of risk and value driver within life business. The corresponding map is then put into the solvability context, in order to formally identify the equilibrium conditions. Considerations about the technical equilibrium of an insurance portfolio and the financial regulation lead to a dynamic system of solvency assessment. The formal model is applied to a life annuity cohort in a stochastic context in order to exemplify the potential of the model, especially referred to the need to frame solvency assessment in a dynamic perspective.

A stochastic model for the force of interest

ZusammenfassungDiese Arbeit vermittelt Charakterisierungen für ein stochastisches Modell für die Zinsintensität, das von einem Ornstein-Uhlenbeck Prozess erzeugt ist, insbesondere bei Vorhandensein von Barrieren.Weitere Charakterisierungen für den Kapitalisierungsprozess werden bereitgestellt.SummaryThe paper provides characterizations for a stochastic model for the force of interest, generated by an Ornstein-Uhlenbeck process, particularly in presence of barriers. Further characterization for the capitalization process are given.