Optimizing the equity-bond-annuity portfolio in retirement: The impact of uncertain health expenses

This paper derives optimal equity-bond-annuity portfolios for retired households who face stochastic capital market returns, differential exposures to mortality risk and uncertain uninsured health expenses, and differential Social Security and defined benefit pension coverage. The results show that the health spending risk drives household portfolios to shift from risky equities to safer assets and enhances the demand for annuities due to their increasing-with-age superiority over bonds in hedging against life-contingent health spending and longevity risks. Households with higher income have a greater incremental demand for life annuities. The annuities in turn provide greater leverage for equity investment in the remaining asset portfolios.     

Optimal life cycle portfolio choice with variable annuities offering liquidity and investment downside protection

This paper assesses optimal life cycle consumption and portfolio allocations when households have access to Guaranteed Minimum Withdrawal Benefit (GMWB) variable annuities over their adult lifetimes. Our contribution is to evaluate demand for these products which provide access to equity investments with money-back guarantees, longevity risk hedging, and partially-refundable premiums, in a realistic world with uncertain labor and capital market income as well as mortality risk. Others have predicted that consumers will only purchase such annuities late in life, but we show that they will optimally purchase GMWBs prior to retirement, consistent with their recent rapid uptick in sales. Additionally, many individuals optimally adjust their portfolios and consumption streams along the way by taking cash withdrawals from the products. These products can substantially enhance consumption, by up to 10% for those who experience highly unfavorable experiences in the stock market.     

Hedging mortality/longevity risks of insurance portfolios for life insurer/annuity provider and financial intermediary

In this paper, we propose two risk hedge schemes in which a life insurer (an annuity provider) can transfer mortality (longevity) risk of a portfolio of life (annuity) exposures to a financial intermediary by paying the hedging premium of a mortality-linked security. The optimal units of the mortality-linked security which maximize hedge effectiveness for a life insurer (an annuity provider) can be derived as closed-form formulas under the risk hedge schemes. Numerical illustrations show that the risk hedge schemes can significantly hedge the downside risk of loss due to mortality (longevity) risk for the life insurer (annuity provider) under some stochastic mortality models. Besides, finding an optimal weight of a portfolio of life and annuity business, the financial intermediary can reduce the sensitivity to mortality rates but the model risk; a security loading may be imposed on the hedge premium for a higher probability of gain to compensate the financial intermediary for the inevitable model risk.     

The determinants of mortality heterogeneity and implications for pricing annuities

Standard annuities are offered at one price to all individuals of the same age and gender. Individual mortality heterogeneity exposes insurers to adverse selection since only relatively healthy lives are expected to purchase annuities. As a result standard annuities are priced assuming above-average longevity, making them expensive for many individuals. In contrast underwritten annuity prices reflect individual risk factors based on underwriting information, as well as age and gender. While underwriting reduces heterogeneity, mortality risk still varies within each risk class due to unobservable individual risk factors, referred to as frailty. This paper quantifies the impact of heterogeneity due to underwriting factors and frailty on annuity values. Heterogeneity is quantified by fitting Generalized Linear Mixed Models to longitudinal data for a large sample of US males. The results show that heterogeneity remains after underwriting and that frailty significantly impacts the fair value of both standard and underwritten annuities. We develop a method to adjust annuity prices to allow for frailty.     

Longevity risk,cost of capital and hedging for life insurers under Solvency II

The cost of capital is an important factor determining the premiums charged by life insurers issuing life annuities. This capital cost can be reduced by hedging longevity risk with longevity swaps, a form of reinsurance. We assess the costs of longevity risk management using indemnity based longevity swaps compared to costs of holding capital under Solvency II. We show that, using a reasonable market price of longevity risk, the market cost of hedging longevity risk for earlier ages is lower than the cost of capital required under Solvency II. Longevity swaps covering higher ages, around 90 and above, have higher market hedging costs than the saving in the cost of regulatory capital. The Solvency II capital regulations for longevity risk generates an incentive for life insurers to hold longevity tail risk on their own balance sheets, rather than transferring this to the reinsurance or the capital markets. This aspect of the Solvency II capital requirements is not well understood and raises important policy issues for the management of longevity risk.     

Modeling longevity risks using a principal component approach: A comparison with existing stochastic mortality models

This research proposes a mortality model with an age shift to project future mortality using principal component analysis (PCA). Comparisons of the proposed PCA model with the well-known models—the Lee-Carter model, the age-period-cohort model (Renshaw and Haberman, 2006), and the Cairns, Blake, and Dowd model—employ empirical studies of mortality data from six countries, two each from Asia, Europe, and North America. The mortality data come from the human mortality database and span the period 1970-2005. The proposed PCA model produces smaller prediction errors for almost all illustrated countries in its mean absolute percentage error. To demonstrate longevity risk in annuity pricing, we use the proposed PCA model to project future mortality rates and analyze the underestimated ratio of annuity price for whole life annuity and deferred whole life annuity product respectively. The effect of model risk on annuity pricing is also investigated by comparing the results from the proposed PCA model with those from the LC model. The findings can benefit actuaries in their efforts to deal with longevity risk in pricing and valuation.     

Efficient versus inefficient hedging strategies in the presence of financial and longevity (value at) risk

This paper provides a closed-form Value-at-Risk (VaR) for the net exposure of an annuity provider, taking into account both mortality and interest-rate risk, on both assets and liabilities. It builds a classical risk-return frontier and shows that hedging strategies–such as the transfer of longevity risk–may increase the overall risk while decreasing expected returns, thus resulting in inefficient outcomes. Once calibrated to the 2010 UK longevity and bond market, the model gives conditions under which hedging policies become inefficient.     

应对长寿风险的分红年金:随机精算建模与应用

我国的商业养老保险作为养老金体系的重要组成部分,在实践中的发展比较缓慢,原因之一是保险公司缺乏长寿风险管理的经验。本文将探索我国商业养老保险使用分红年金管理长寿风险的可行性。研究该分红年金在给付规则和分红来源方面的特征,并基于实际数据,构建动态随机死亡率模型和随机收益率模型,采用蒙特卡洛随机模拟方法,比较分红年金和传统年金在待遇分布、资产和损失分布、破产概率等方面的特征,得出分红年金能够在精算公平原则下有效应对长寿风险,并且在待遇给付、偿付能力和盈利能力方面具有明显优势的结论。     

The impact of the determinants of mortality on life insurance and annuities

Extended risk classification has become an important issue recently in life insurance and annuity markets. Various risk factors have been explored and identified by past research. Using those risk factors, one can construct various risk classes. This enables insurers to provide more equitable life insurance and annuity benefits for individuals in different risk classes and to manage mortality/longevity risk more efficiently. The challenge of modeling mortality using various risk factors is to reflect complicated mortality dynamics in a model while maintaining statistical significance. This paper discusses the development of a mortality model that reflects the impact of various risk factors on mortality. Longitudinal survey data from the Canadian National Population Health Survey was used to determine the significant risk factors and quantify their effect on mortality. The model is used to illustrate how the various risk factors influence actuarial present values of life insurance and annuity benefits.     

Longevity risk in pension annuities with exchange options: The effect of product design

We consider defined benefit pension plans that, at retirement age, allow the participant to choose between a single life annuity and a joint and survivor annuity. We compare two plans that differ in terms of how pension rights are accrued. In one plan, the participant accrues the right to receive a single life annuity, and can exchange that annuity for an actuarially equivalent joint and survivor annuity at retirement date. The opposite holds in the other plan. We show that both plans are affected by longevity risk in two ways. First, the participants’ choices at retirement age affect the ratio of survivor benefits over single life benefits, and, therefore, affect the natural hedge potential that arises from combining single life and survivor annuities. Second, uncertainty in the rate at which the participant will be allowed to exchange one type of annuity for the other at retirement date induces uncertainty in the level of the nominal rights for single life and survivor annuities, respectively. We compare the two plans, and show that longevity risk is substantially lower in case rights are accrued in the form of a joint and survivor annuity.     

中国人口死亡率建模比较及长寿风险度量

以我国颁布的3套保险行业经验生命表为基础，结合1995-2017年国家统计局发布的《中国统计年鉴》中的死亡率数据，首先分析了中国全年龄人口数据死亡率动静态变动特点，其次比较了LC，CBD和APC 3种模型对中国死亡率数据的拟合优劣，最后采用最优APC模型度量了不同生命表下的长寿风险.死亡率的动态变化会导致以经验生命表为依据的年金产品定价出现偏差，增加养老金管理机构的承保风险.     

Options on tontines: An innovative way of combining tontines and annuities

Increases in the life expectancy, the low interest rate environment and the tightening solvency regulation have led to the rebirth of tontines. Compared to annuities, where insurers bear all the longevity risk, policyholders bear most of the longevity risk in a tontine. Following Donnelly and Young (2017), we come up with an innovative retirement product which contains the annuity and the tontine as special cases: a tontine with a minimum guaranteed payment. The payoff of this product consists of a guaranteed payoff and a call option written on a tontine. Extending Donnelly and Young (2017), we consider the tontine design described in Milevsky and Salisbury (2015) for designing the new product and find that it is able to achieve a better risk sharing between policyholders and insurers than annuities and tontines. For the majority of risk-averse policyholders, the new product can generate a higher expected lifetime utility than annuities and tontines. For the insurer, the new product is able to reduce the (conditional) expected loss drastically compared to an annuity, while the loss probability remains fairly the same. In addition, by varying the guaranteed payments, the insurer is able to provide a variety of products to policyholders with different degrees of risk aversion and liquidity needs.     

Risk-minimization for life insurance liabilities with basis risk

In this paper we study the hedging of typical life insurance payment processes in a general setting by means of the well-known risk-minimization approach. We find the optimal risk-minimizing strategy in a financial market where we allow for investments in a hedging instrument based on a longevity index, representing the systematic mortality risk. Thereby we take into account and model the basis risk that arises due to the fact that the insurance company cannot perfectly hedge its exposure by investing in a hedging instrument that is based on the longevity index, not on the insurance portfolio itself. We also provide a detailed example within the context of unit-linked life insurance products where the dependency between the index and the insurance portfolio is described by means of an affine mean-reverting diffusion process with stochastic drift.     

Statistical emulators for pricing and hedging longevity risk products

We propose the use of statistical emulators for the purpose of analyzing mortality-linked contracts in stochastic mortality models. Such models typically require (nested) evaluation of expected values of nonlinear functionals of multi-dimensional stochastic processes. Except in the simplest cases, no closed-form expressions are available, necessitating numerical approximation. To complement various analytic approximations, we advocate the use of modern statistical tools from machine learning to generate a flexible, non-parametric surrogate for the true mappings. This method allows performance guarantees regarding approximation accuracy and removes the need for nested simulation. We illustrate our approach with case studies involving (i) a Lee–Carter model with mortality shocks; (ii) index-based static hedging with longevity basis risk; (iii) a Cairns–Blake–Dowd stochastic survival probability model; (iv) variable annuities under stochastic interest rate and mortality.     

Quantitative modeling of risk management strategies: Stochastic reserving and hedging of variable annuity guaranteed benefits

Variable annuities are enhanced life insurance products that offer policyholders participation in equity investment with minimum return guarantees. There are two well-established risk management strategies in practice for variable annuity guaranteed benefits, namely, (1) stochastic reserving based on risk measures such as value-at-risk (VaR) and conditional-tail-expectation (CTE); (2) dynamic hedging using exchange-traded derivatives. The latter is increasingly more popular than the former, due to a common perception of its low cost. While both have been extensively used in the insurance industry, scarce academic literature has been written on the comparison of the two approaches. This paper presents a quantitative framework in which two risk management strategies are mathematically formulated and where the basis for decision making can be determined analytically. Besides, the paper proposes dynamic hedging of net liabilities as a more effective and cost-saving alternative to the common practice of dynamic hedging of gross liabilities. The finding of this paper does not support the general perception that dynamic hedging is always more affordable than stochastic reserving, although in many cases it is with the CTE risk measure.     

Hedging pure endowments with mortality derivatives

In recent years, a market for mortality derivatives began developing as a way to handle systematic mortality risk, which is inherent in life insurance and annuity contracts. Systematic mortality risk is due to the uncertain development of future mortality intensities, or hazard rates. In this paper, we develop a theory for pricing pure endowments when hedging with a mortality forward is allowed. The hazard rate associated with the pure endowment and the reference hazard rate for the mortality forward are correlated and are modeled by diffusion processes. We price the pure endowment by assuming that the issuing company hedges its contract with the mortality forward and requires compensation for the unhedgeable part of the mortality risk in the form of a pre-specified instantaneous Sharpe ratio. The major result of this paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation under an equivalent martingale measure. Another important result is that hedging with the mortality forward may raise or lower the price of this pure endowment comparing to its price without hedging, as determined in Bayraktar et al. (2009). The market price of the reference mortality risk and the correlation between the two portfolios jointly determine the cost of hedging. We demonstrate our results using numerical examples.     

Delta–Gamma hedging of mortality and interest rate risk

One of the major concerns of life insurers and pension funds is the increasing longevity of their beneficiaries. This paper studies the hedging problem of annuity cash flows when mortality and interest rates are stochastic. We first propose a Delta–Gamma hedging technique for mortality risk. The risk factor against which to hedge is the difference between the actual mortality intensity in the future and its “forecast” today, the forward intensity. We specialize the hedging technique first to the case in which mortality intensities are affine, then to Ornstein–Uhlenbeck and Feller processes, providing actuarial justifications for this selection. We show that, without imposing no arbitrage, we can get equivalent probability measures under which the HJM condition for no arbitrage is satisfied. Last, we extend our results to the presence of both interest rate and mortality risk. We provide a UK calibrated example of Delta–Gamma hedging of both mortality and interest rate risk.     

Multi-population mortality models: A factor copula approach

Modeling mortality co-movements for multiple populations have significant implications for mortality/longevity risk management. A few two-population mortality models have been proposed to date. They are typically based on the assumption that the forecasted mortality experiences of two or more related populations converge in the long run. This assumption might be justified by the long-term mortality co-integration and thus be applicable to longevity risk modeling. However, it seems too strong to model the short-term mortality dependence. In this paper, we propose a two-stage procedure based on the time series analysis and a factor copula approach to model mortality dependence for multiple populations. In the first stage, we filter the mortality dynamics of each population using an ARMA–GARCH process with heavy-tailed innovations. In the second stage, we model the residual risk using a one-factor copula model that is widely applicable to high dimension data and very flexible in terms of model specification. We then illustrate how to use our mortality model and the maximum entropy approach for mortality risk pricing and hedging. Our model generates par spreads that are very close to the actual spreads of the Vita III mortality bond. We also propose a longevity trend bond and demonstrate how to use this bond to hedge residual longevity risk of an insurer with both annuity and life books of business.     

Affordable and adequate annuities with stable payouts: Fantasy or reality?

This paper introduces a class of unit-linked annuities that extends existing annuities by allowing portfolio shocks to be gradually absorbed into the annuity payouts. Consequently, our new class enables insurers to offer an affordable and adequate annuity with a stable payout stream. We show how to price and adequately hedge the annuity payouts in a general financial environment. In particular, our model accounts for various stylized facts of stock returns such as asymmetry and heavy-tailedness. Furthermore, the generality of our framework makes it possible to explore the impact of a parameter misspecification on the annuity price and the hedging performance.     

Optimal portfolio‐consumption choice under stochastic inflation with nominal and indexed bonds

This research solves the intertemporal portfolio choice problems with and without interim consumption under stochastic inflation. We assume a one‐factor nominal interest rate and a one‐factor expected inflation rate, implying a two‐factor real interest rate in the economy. In contrast to other related research which adopts the one‐factor real interest rate model, the inflation‐indexed bond is not a redundant asset class even in a complete market. The infinitely risk‐averse investor would prefer to invest all her wealth in inflation‐indexed bonds maturing at the investment horizon. We also show that, with the two‐factor real interest rate model, the consumption‐wealth ratio is not determined by the real interest rate alone. The investor's consumption–wealth ratio is also affected by the nominal interest rate and expected inflation rate levels. The capital market is calibrated to U.S. stocks, bonds, and inflation data. The optimal weights show that aggressive investors hold more nominal bonds in order to earn the inflation risk premiums, while conservative investors concentrate on indexed bonds to hedge against the inflation risk. Copyright © 2011 John Wiley & Sons, Ltd.