Developing Mortality Improvement Formulas

Abstract

Longevity improvements have contributed to widespread underfunding of pension plans and losses in insured annuity portfolios. Insurers might reasonably expect some upside from the effect of lower mortality on their life business. Although mortality improvement scales, such as the Society of Actuaries Scale AA, are widely employed in pension and annuity valuation, the derivation of these scales appears heuristic, leading to problems in deriving meaningful measures of uncertainty. We explore the evidence on mortality trends for the Canadian life insurance companies, data, using stochastic models. We use the more credible population data to benchmark the insured lives data. Finally, we derive a practical, model-based formula for actuaries to incorporate mortality improvement and the associated uncertainty into their calculations.     

To Hedge or Not to Hedge: Managing Demographic Risk in Life Insurance Companies

Demographic risk, i.e., the risk that life tables change in a nondeterministic way, is a serious threat to the financial stability of an insurance company having underwritten life insurance and annuity business. The inverse influence of changes in mortality laws on the market value of life insurance and annuity liabilities creates natural hedging opportunities. Within a realistically calibrated shareholder value (SHV) maximization framework, we analyze the implications of demographic risk on the optimal risk management mix (equity capital, asset allocation, and product policy) for a limited liability insurance company operating in a market with insolvency‐averse insurance buyers. Our results show that the utilization of natural hedging is optimal only if equity is scarce. Otherwise, hedging can even destroy SHV. A sensitivity analysis shows that a misspecification of demographic risk has severe consequences for both the insurer and the insured. This result highlights the importance of further research in the field of demographic risk.     

Enterprise Risk and Return Management for Financial Institutions

It is important to include both risk and return in finding the optimal balance of assets and liabilities for financial institutions. Insurance companies, with their range of sophisticated assets and liabilities, are perhaps the best example of the value of such an approach. Examples in the paper refer to the insurance industry, but parallels to other types of financial institutions are easily drawn. The analysis in the paper shows that a comprehensive approach to risk and return produces some interesting conclusions with respect to asset allocation, active versus passive asset management, and the mix and pricing of liabilities.     

Market Value Margin für die Fair Value-Bilanzierung versicherungstechnischer Verpflichtungen : Ableitung auf Basis des CAPM

The International Accounting Standards Board (IASB) started a project on Insurance Accounting to apply the principles of fair value to insurance businesses. The so called ?asset and liability approach“ would focus on the balance sheet, with both assets and liabilities being reported at fair values, while income and expenses are defined in terms of changes in the values of those assets and liabilities. Indeed, there are no deep and liquid markets for insurance liabilities. Thus, the fair value has to be calculated as a theoretical value, using assumptions concerning future events, risk provisions and discount rates. Both in theory and in practice a generally accepted modelling of fair value is missing. Particularly with regard to the deviation of a Market Value Margin (MVM), which reflects the premium that a marketplace participant would demand for bearing the uncertainty inherent in the cash flows, there is a need for adequate modelling. Transforming the CAPM for determining risk loads in insurance will mean measuring the correlation between insurance companies’ returns from underwriting and market returns on its shareholders’ equity. The criticism on an underwriting beta focuses on (a) the basic assumptions of the CAPM, (b) the absence of active markets for insurance liabilities and (c) the unreliability of estimating underwriting betas.     

Individual Annuity Demand Under Aggregate Mortality Risk

Aggregate mortality risk—the risk that the mortality trend in a population changes in a nondeterministic way—and its implications for corporate decisions has recently been the subject of lively scientific discussion. We show that aggregate mortality risk is also a key determinant for individual annuitization decisions. Aggregate mortality risk appears to be a risk very difficult to transfer for individuals. Whether its existence leads to a higher or lower annuity demand depends on objective factors (e.g., insurers’ vulnerability to aggregate mortality changes). Subjective factors (i.e., individuals’ preferences) determine only the intensity of the annuity demand reaction to aggregate mortality risk. Our results are of significant importance not only for financial planning approaches of individual annuity buyers but also for strategic decisions in insurance companies and for solvency regulators. Furthermore, consideration of aggregate mortality risk may alleviate, but also intensify, the annuity puzzle.     

Lifetime asset allocation with idiosyncratic and systematic mortality risks

This paper considers a lifetime asset allocation problem with both idiosyncratic and systematic mortality risks. The novelty of the paper is to integrate stochastic mortality, stochastic interest rate and stochastic income into a unified framework. An investor, who is a wage earner receiving a stochastic income, can invest in a financial market, consume part of his wealth and purchase life insurance or annuity so as to maximize the expected utility from consumption, terminal wealth and bequest. The problem is solved via the dynamic programming principle and the Hamilton–Jacobi–Bellman equation. Analytical solutions to the problem are derived, and numerical examples are provided to illustrate our results. It is shown that idiosyncratic mortality risk has significant impacts on the investor’s investment, consumption, life insurance/annuity purchase and bequest decisions regardless of the length of the decision-making horizon. The systematic mortality risk is largely alleviated by trading the longevity bond. However, its impacts on consumption, purchase of life insurance/annuity and bequest as well as the value function are still pronounced, when the decision-making horizon is sufficiently long.     

财产保险公司资产负债管理多维量化评估体系研究

良好的资产负债管理是保险业可持续发展的基石,也是支持保险业在日益复杂的风险环境中保持稳健发展、防范系统性风险的重要保障。近年来,随着我国金融市场发展,业务产品创新加快,保险业在资产端与负债端的业务结构和风险特征出现了新情况、新变化。特别是部分保险公司缺乏有效的治理结构,采取激进经营、激进投资的策略,导致业务快进快出、风险敞口过大以及流动性问题,对保险公司资产负债匹配管理、风险控制提出了挑战。本文介绍了财产保险公司资产负债多维度量化评估规则设计原理、主要评估模型和评估方法,针对财产保险公司的负债特性提出的沉淀资金匹配,在成本收益匹配中有机地将资产投资收益与承保业务综合成本进行匹配,在现金流匹配模式中打破了僵化的匹配模式,解决了长期困扰财产保险公司的资产负债期限不匹配的问题,对财产保险公司资产负债管理具有重要意义。     

Discretion in accounting for pensions under IAS 19: using the ‘magic telescope’?

We use a panel data set of UK-listed companies over the period 2005–2009 to analyse the actuarial assumptions used to value pension plan liabilities under IAS 19. The valuation process requires companies to make assumptions about financial and demographic variables, notably discount rate, price inflation, salary inflation and mortality/life expectancy of plan members/beneficiaries. We use regression analysis to analyse the relationships between these key assumptions (except mortality, where disclosures are limited) and company-specific factors such as the pension plan funding position and duration of pension liabilities. We find evidence of selective ‘management’ of the three assumptions investigated, although the nature of this appears to differ from the findings of US authors. We conclude that IAS 19 does not prevent the use of managerial discretion, particularly by companies whose pension plan funding positions are weak, thereby reducing the representational faithfulness of the reported pension figures. We also highlight that the degree of discretion used reflects the extent to which IAS 19 defines how the assumptions are to be determined. We therefore suggest that companies should be encouraged to justify more explicitly their choice of assumptions.     

On systematic mortality risk and risk-minimization with survivor swaps

A new market for so-called mortality derivatives is now appearing with survivor swaps (also called mortality swaps), longevity bonds and other specialized solutions. The development of these new financial instruments is triggered by the increased focus on the systematic mortality risk inherent in life insurance contracts, and their main focus is thus to allow the life insurance companies to hedge their systematic mortality risk. At the same time, this new class of financial contract is interesting from an investor's point of view, since it increases the possibility for an investor to diversify the investment portfolio. The systematic mortality risk stems from the uncertainty related to the future development of the mortality intensities. Mathematically, this uncertainty is described by modeling the underlying mortality intensities via stochastic processes. We consider two different portfolios of insured lives, where the underlying mortality intensities are correlated, and study the combined financial and mortality risk inherent in a portfolio of general life insurance contracts. In order to hedge this risk, we allow for investments in survivor swaps and derive risk-minimizing strategies in markets where such contracts are available. The strategies are evaluated numerically.     

寿险公司负债价值计量研究

有效的资产负债管理以寿险公司资产和负债价值的正确计量为基础。寿险负债的长期性和多种嵌入期权的特性使寿险公司资产负债管理更具复杂性。寿险负债价值计量的精算方法注重于它的死亡率风险测定的技术秘诀和期限的长期性,但该方法使期权的效应无从体现,也使在利率和被保险人行为的静态假设不成立的前提下,负债价值被低估的情况发生。因此,必须辅之于财务方法来测量期权在寿险合约中的价值。本文对寿险合约中期权价值的测量及其对ALM的作用也做了探讨。     

Accounting of the German Statutory Pension Scheme: Balance Sheet,Cross‐Sectional Internal Rate of Return and Implicit Tax Rate

We present a framework for accounting of the German statutory pension scheme and calculate a balance sheet for the period 2005–14. Estimating a funding ratio of about 90 per cent, we present some policy recommendations in order to restore balancing of assets and liabilities. Extending and applying the methodology proposed by Settergren and Mikula (2005), we additionally estimate the aggregate cross‐sectional internal rate of return of the German pension scheme over this period. We are able to show that these internal rates of return are mainly financed by increasing contributions and by increasing unfunded liabilities. Additionally, our analysis reveals that from an expenditure perspective, the major part of the internal rate of return results from changing longevity rather than other changes. We also estimate the implicit tax rates from a cross‐sectional perspective and find that they can mainly be interpreted as an ‘implicit wealth tax’ on pension wealth. Finally, we analyse the impact of demographic change on the balance sheet employing a population projection. While the pure demographic effect leads to a decreasing funding ratio during population ageing, the automatic balancing mechanisms of the German pension scheme lead to a significant overfunding in the long run from the accounting perspective adopted in the Swedish pension system.     

Multivariate Tweedie lifetimes: the impact of dependence

Systematic longevity risk is increasingly relevant for public pension schemes and insurance companies that provide life benefits. In view of this, mortality models should incorporate dependence between lives. However, the independent lifetime assumption is still heavily relied upon in the risk management of life insurance and annuity portfolios. This paper applies a multivariate Tweedie distribution to incorporate dependence, which it induces through a common shock component. Model parameter estimation is developed based on the method of moments and generalized to allow for truncated observations. The estimation procedure is explicitly developed for various important distributions belonging to the Tweedie family, and finally assessed using simulation.     

Stochastic mortality under measure changes

We provide a self-contained analysis of a class of continuous-time stochastic mortality models that have gained popularity in the last few years. We describe some of their advantages and limitations, examining whether their features survive equivalent changes of measures. This is important when using the same model for both market-consistent valuation and risk management of life insurance liabilities. We provide a numerical example based on the calibration to the French annuity market of a risk-neutral version of the model proposed by Lee & Carter (1992).     

An Optimal Product Mix for Hedging Longevity Risk in Life Insurance Companies: The Immunization Theory Approach

This article investigates the natural hedging strategy to deal with longevity risks for life insurance companies. We propose an immunization model that incorporates a stochastic mortality dynamic to calculate the optimal life insurance–annuity product mix ratio to hedge against longevity risks. We model the dynamic of the changes in future mortality using the well‐known Lee–Carter model and discuss the model risk issue by comparing the results between the Lee–Carter and Cairns–Blake–Dowd models. On the basis of the mortality experience and insurance products in the United States, we demonstrate that the proposed model can lead to an optimal product mix and effectively reduce longevity risks for life insurance companies.     

动态死亡率下个人年金的长寿风险分析

传统的精算定价方法假定死亡率是静态的,实际上死亡率是随时间而变动的具有动态不确定性的变量。在动态死亡率的框架下定量分析长寿风险对于个人年金产品定价的影响:引入Wang转换的风险定价方法度量长寿风险的市场价格,并运用模拟分析的方法分析长寿风险对个人年金定价的影响。最后,基于分析结果,就保险公司如何管理这一风险给出建议。     

Pricing pension buy-outs under stochastic interest and mortality rates

Pension buy-out is a special financial asset issued to offload the pension liabilities holistically in exchange for an upfront premium. In this paper, we concentrate on the pricing of pension buy-outs under dependence between interest and mortality rates risks with an explicit correlation structure in a continuous time framework. Change of measure technique is invoked to simplify the valuation. We also present how to obtain the buy-out price for a hypothetical benefit pension scheme using stochastic models to govern the dynamics of interest and mortality rates. Besides employing a non-mean reverting specification of the Ornstein–Uhlenbeck process and a continuous version of Lee–Carter setting for modeling mortality rates, we prefer Vasicek and Cox–Ingersoll–Ross models for short rates. We provide numerical results under various scenarios along with the confidence intervals using Monte Carlo simulations.     

Forecasting Longevity Gains for a Population with Short Time Series Using a Structural SUTSE Model: An Application to Brazilian Annuity Plans

In this article, a multivariate structural time series model with common stochastic trends is proposed to forecast longevity gains of a population with a short time series of observed mortality rates, using the information of a related population for which longer mortality time series exist. The state space model proposed here makes use of the seemingly unrelated time series equation and applies the concepts of related series and common trends to construct a proper model to predict the future mortality rates of a population with little available information. This common trends approach works by assuming the two populations’ mortality rates are affected by common factors. Further, we show how this model can be used by insurers and pension funds to forecast mortality rates of policyholders and beneficiaries. We apply the proposed model to Brazilian annuity plans where life expectancies and their temporal evolution are predicted using the forecast longevity gains. Finally, to demonstrate how the model can be used in actuarial practice, the best estimate of the liabilities and the capital based on underwriting risk are estimated by means of Monte Carlo simulation. The idiosyncratic risk effect in the process of calculating an amount of underwriting capital is also illustrated using that simulation.     

Financing Gaps of Public Pension Schemes and Market Potential for the Private Retirement Market

Abstract

Increasing longevity, declining birth rates, and high unemployment severely threaten the financial basis of many public pension plans. These problems are most pronounced in continental Europe, where public pension plans tend to be relatively generous and are usually funded on a pay-as-you-go basis. Given the demographic development, future pension payments exceed the expected contribution payments. The resulting financing gaps can be seen as implicit public debts (net pension liabilities), which often exceed the value of GDP figures and are in many cases higher than the explicit public debt figures. If people would decide to cover these financing gaps via life insurance, life insurance premiums would triple in Germany, more than double in Italy, and double in Canada and France. The increase would be only moderate in the U.S. and particularly small in the U.K.     

Parameter risk in time-series mortality forecasts

The projection of mortality rates is an essential part of valuing liabilities in life insurance portfolios and pension schemes. An important tool for risk management and solvency purposes is a stochastic projection model. We show that ARIMA models can be better representations of mortality time-series than simple random-walk models. We also consider the issue of parameter risk in time-series models from the point of view of an insurer using them for regulatory risk reporting – formulae are given for decomposing overall risk into undiversifiable trend risk (parameter uncertainty) and diversifiable volatility. Particular attention is given to the contrasts in how academic researchers might view these models and how insurance regulators and practitioners in life offices might use them. Using a bootstrap method we find that, while certain kinds of parameter risk are negligible, others are too material to ignore. We also find that an objective model selection criterion, such as goodness of fit to past data, can result in the selection of a model with unstable parameter values. While this aspect of the model is superficially undesirable, it also leads to slightly higher capital requirements and thus makes the model of keen interest to regulators. Our conclusions have relevance to insurers using value-at-risk capital assessments in the European Union under Solvency II, but also territories using conditional tail expectations such as Australia, Canada and Switzerland.     

Do a firm's equity returns reflect the risk of its pension plan?

This paper examines the empirical question of whether systematic equity risk of US firms as measured by beta from the capital asset pricing model reflects the risk of their pension plans. There are a number of reasons to suspect that it might not. Chief among them is the opaque set of accounting rules used to report pension assets, liabilities, and expenses. Pension plan assets and liabilities are off-balance sheet and are often viewed as segregated from the rest of the firm, with its own trustees. Pension accounting rules are complicated. Furthermore, the role of the Pension Benefit Guaranty Corporation clouds the real relation between pension plan risk and firm equity risk. The empirical findings in this paper are consistent with the hypothesis that equity risk does reflect the risk of the firm's pension plan despite arcane accounting rules for pensions. This finding is consistent with informational efficiency of the capital markets. It also has implications for corporate finance practice in the determination of the cost of capital for capital budgeting. Standard procedure uses de-leveraged equity return betas to infer the cost of capital for operating assets. But the de-leveraged betas are not adjusted for the risk of the pension assets and liabilities. Failure to make this adjustment typically biases upward estimates of the discount rate for capital budgeting. The magnitude of the bias is shown here to be large for a number of well-known US companies. This bias can result in positive net present value projects being rejected.