Valeria D’Amato

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.

The mortality of the Italian population: Smoothing techniques on the Lee–Carter model

Several approaches have been developed for forecasting mortality using the stochastic model. In particular, the Lee-Carter model has become widely used and there have been various extensions and modifications proposed to attain a broader interpretation and to capture the main features of the dynamics of the mortality intensity. Hyndman-Ullah show a particular version of the Lee-Carter methodology, the so-called Functional Demographic Model, which is one of the most accurate approaches as regards some mortality data, particularly for longer forecast horizons where the benefit of a damped trend forecast is greater. The paper objective is properly to single out the most suitable model between the basic Lee-Carter and the Functional Demographic Model to the Italian mortality data. A comparative assessment is made and the empirical results are presented using a range of graphical analyses.

Detecting Common Longevity Trends by a Multiple Population Approach

Recently the interest in the development of country and longevity risk models has been growing. The investigation of long-run equilibrium relationships could provide valuable information about the factors driving changes in mortality, in particular across ages and across countries. In order to investigate cross-country common longevity trends, tools to quantify, compare, and model the strength of dependence become essential. On one hand, it is necessary to take into account either the dependence for adjacent age groups or the dependence structure across time in a single population setting—a sort of intradependence structure. On the other hand, the dependence across multiple populations, which we describe as interdependence, can be explored for capturing common long-run relationships between countries. The objective of our work is to produce longevity projections by taking into account the presence of various forms of cross-sectional and temporal dependencies in the error processes of multiple populations, considering mortality data from different countries. The algorithm that we propose combines model-based predictions in the Lee-Carter (LC) framework with a bootstrap procedure for dependent data, and so both the historical parametric structure and the intragroup error correlation structure are preserved. We introduce a model which applies a sieve bootstrap to the residuals of the LC model and is able to reproduce, in the sampling, the dependence structure of the data under consideration. In the current article, the algorithm that we build is applied to a pool of populations by using ideas from panel data; we refer to this new algorithm as the Multiple Lee-Carter Panel Sieve (MLCPS). We are interested in estimating the relationship between populations of similar socioeconomic conditions. The empirical results show that the MLCPS approach works well in the presence of dependence.

Computational framework for longevity risk management

Longevity risk threatens the financial stability of private and government sponsored defined benefit pension systems as well as social security schemes, in an environment already characterized by persistent low interest rates and heightened financial uncertainty. The mortality experience of countries in the industrialized world would suggest a substantial age-time interaction, with the two dominant trends affecting different age groups at different times. From a statistical point of view, this indicates a dependence structure. It is observed that mortality improvements are similar for individuals of contiguous ages (Wills and Sherris, Integrating financial and demographic longevity risk models: an Australian model for financial applications, Discussion Paper PI-0817, 2008). Moreover, considering the dataset by single ages, the correlations between the residuals for adjacent age groups tend to be high (as noted in Denton et al., J Population Econ 18:203–227, 2005). This suggests that there is value in exploring the dependence structure, also across time, in other words the inter-period correlation. In this research, we focus on the projections of mortality rates, contravening the most commonly encountered dependence property which is the “lack of dependence” (Denuit et al., Actuarial theory for dependent risks: measures. Orders and models, Wiley, New York, 2005). By taking into account the presence of dependence across age and time which leads to systematic over-estimation or under-estimation of uncertainty in the estimates (Liu and Braun, J Probability Stat, 813583:15, 2010), the paper analyzes a tailor-made bootstrap methodology for capturing the spatial dependence in deriving confidence intervals for mortality projection rates. We propose a method which leads to a prudent measure of longevity risk, avoiding the structural incompleteness of the ordinary simulation bootstrap methodology which involves the assumption of independence.

Lee-Carter error matrix simulation: heteroschedasticity impact on actuarial valuations

Recently a number of approaches have been developed for forecasting mortality. In this paper, we consider the Lee-Carter model and we investigate in particular the hypothesis about the error structure implicitly assumed in the model specification, i.e., the errors are homoschedastic. The homoschedasticity assumption is quite unrealistic, because of the observed pattern of the mortality rates showing a different variability at old ages than younger ages. Therefore, the opportunity to analyse the robustness of estimated parameter is emerging. To this aim, we propose an experimental strategy in order to assess the robustness of the Lee-Carter model by inducing the errors to satisfy the homoschedasticity hypothesis. Moreover, we apply it to a matrix of Italian mortality rates. Finally, we highlight the results through an application to a pension annuity 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.

The Solvency Capital Requirement Management for an Insurance Company

Longevity risk plays a central role in the insurance company management since only careful assumptions about future evolution of mortality phenomenon allows the company to correctly front its future obligations. According to Solvency II longevity risk represents a sub-module of the underwriting risk module in the regulatory standard formula. In this paper we examine the adequacy of the shock’s structure suggested by the standard formula studying its impact on the solvency capital requirements and liabilities at different ages. In particular, we propose an alternative to the regulatory standard model represented by a flexible internal model. The innovative approach hinges on a stochastic volatility model and a so-called coherent risk measure as the expected shortfall. An empirical analysis is provided.

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.

“Money Purchase” Pensions: Contract Proposals and Risk Analysis

The Authors propose a personal pension product, consisting of a non-traditional profit sharing life insurance contract where the insured is allowed to share the profit of the pension’s invested funds all along the contract duration, that is from the issue time till the insured’s death. In its concrete realization, the idea comes true as a sequence of premiums characterized by a level cap, followed by the sequence of benefits characterized by a level floor. The two embedded options are inserted in the basic structure of a pension annuity. Due to the negligibility of the pooling effect in such kind of portfolios, the impact of the accidental demographic risk source is investigated.

New Challenges in Pension Industry: Proposals of Personal Pension Products

Within the current post-crisis economic environment, characterized by low growth and low interest rates, retirement and long-term saving represent a crucial challenge. Furthermore, the expansion of life expectancies modifies the demand of pension products and insurers and pension providers have to guarantee the sustainability and competitiveness of their products, in spite of the economic stagnation. Within the context of the personal pension products, in the paper we propose a new contract with profit participation, which consists in a deferred life annuity with variable benefits changing according with two dynamic financial elements: the periodic financial result of the invested fund year by year and the first order financial technical base checked at the beginning of predefined intervals all along the contract life. A numerical implementation explains the forecasted trend of the inflows and outflows connected to the contract under financial and demographic stochastic assumptions.