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.     

Longevity risk management for life and variable annuities: The effectiveness of static hedging using longevity bonds and derivatives

For many years, the longevity risk of individuals has been underestimated, as survival probabilities have improved across the developed world. The uncertainty and volatility of future longevity has posed significant risk issues for both individuals and product providers of annuities and pensions. This paper investigates the effectiveness of static hedging strategies for longevity risk management using longevity bonds and derivatives (q-forwards) for the retail products: life annuity, deferred life annuity, indexed life annuity, and variable annuity with guaranteed lifetime benefits. Improved market and mortality models are developed for the underlying risks in annuities. The market model is a regime-switching vector error correction model for GDP, inflation, interest rates, and share prices. The mortality model is a discrete-time logit model for mortality rates with age dependence. Models were estimated using Australian data. The basis risk between annuitant portfolios and population mortality was based on UK experience. Results show that static hedging using q-forwards or longevity bonds reduces the longevity risk substantially for life annuities, but significantly less for deferred annuities. For inflation-indexed annuities, static hedging of longevity is less effective because of the inflation risk. Variable annuities provide limited longevity protection compared to life annuities and indexed annuities, and as a result longevity risk hedging adds little value for these products.     

Risk aggregation and stochastic claims reserving in disability insurance

We consider a large, homogeneous portfolio of life or disability annuity policies. The policies are assumed to be independent conditional on an external stochastic process representing the economic–demographic environment. Using a conditional law of large numbers, we establish the connection between claims reserving and risk aggregation for large portfolios. Further, we derive a partial differential equation for moments of present values. Moreover, we show how statistical multi-factor intensity models can be approximated by one-factor models, which allows for solving the PDEs very efficiently. Finally, we give a numerical example where moments of present values of disability annuities are computed using finite-difference methods and Monte Carlo simulations.     

Modelling lifetime dependence for older ages using a multivariate Pareto distribution

The main driver of longevity risk is uncertainty in old-age mortality, especially surrounding potential dependence structures. We investigate a multivariate Pareto distribution that allows for the exploration of a variety of applications, from portfolios of standard annuities to joint-life annuity products for couples. Given the anticipated continued increase of supercentenarians, the heavy-tailed nature of the Pareto distribution is appropriate for this application. In past work, it has been shown that even a little dependence between lives can lead to much higher uncertainty. Therefore, the ability to assess and incorporate the appropriate dependence structure, whilst allowing for extreme observations, significantly improves the pricing and risk management of life-benefit products.     

Longevity risk in portfolios of pension annuities

We analyze the importance of longevity risk for the solvency of portfolios of pension annuities. We distinguish two types of mortality risk. Micro-longevity risk quantifies the risk related to uncertainty in the time of death if survival probabilities are known with certainty, while macro-longevity risk is due to uncertain future survival probabilities. We use a generalized two-factor Lee-Carter mortality model to produce forecasts of future mortality rates, and to assess the relative importance of micro- and macro-longevity risk for funding ratio uncertainty. The results show that if financial market risk is fully hedged so that uncertainty in future lifetime is the only source of uncertainty, pension funds are exposed to a substantial amount of risk. Systematic and non-systematic deviations from expected survival imply that, depending on the size of the portfolio, buffers that reduce the probability of underfunding to 2.5% at a 5-year horizon have to be of the order of magnitude of 7% to 8% of the initial value of the liabilities.     

A Markov Process Modeling and Analysis of Indifference Pricing of Insurance Contracts for Home Reversion Plan for a Pair of Insureds

The intent of the home reversion plan for retired homeowners in United Kingdom is to help house-rich but cash-poor seniors by the way of releasing their home equity into cash so as to meet living expenses and continue to enjoy the right to live in their home until their death. We present multiple finite state Markov models to price the continuous annuities of insurance policies relevant to home reversion plan for a pair of insureds (meaning husband and wife). Our modeling assumes that the home value follows a geometric Brownian motion. By applying the principle of equivalent utility, we derive the partial differential equation system that the indifferent annuities satisfy under the exponential utility function, and find their explicit representations. Furthermore, we employ an explicit finite difference scheme to calculate the numerical solution and discuss the impacts of the risk aversion of insurer, the volatility of home value, and the interest rate on the annuities of home reversion plan for a couple. We observe that these results reflect our intuition.     

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.     

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。     

Comonotonic bounds on the survival probabilities in the Lee–Carter model for mortality projection

In the Lee–Carter framework, future survival probabilities are random variables with an intricate distribution function. In large homogeneous portfolios of life annuities, value-at-risk or conditional tail expectation of the total yearly payout of the company are approximately equal to the corresponding quantities involving random survival probabilities. This paper aims to derive some bounds in the increasing convex (or stop-loss) sense on these random survival probabilities. These bounds are obtained with the help of comonotonic upper and lower bounds on sums of correlated random variables.     

Using fuzzy random variables in life annuities pricing

This paper develops life annuity pricing with stochastic representation of mortality and fuzzy quantification of interest rates. We show that modelling the present value of annuities with fuzzy random variables allows quantifying their expected price and risk resulting from the uncertainty sources considered. So, we firstly describe fuzzy random variables and define some associated measures: the mathematical expectation, the variance, distribution function and quantiles. Secondly, we show several ways to estimate the discount rates to price annuities. Subsequently, the present value of life annuities is modelled with fuzzy random variables. We finally show how an actuary can quantify the price and the risk of a portfolio of annuities when their present value is given by means of fuzzy random variables.     

Mean‐variance hedging with basis risk

Basis risk arises in a number of financial and insurance risk management problems when the hedging assets do not perfectly match the underlying asset in a hedging program. Notable examples in insurance include the hedging for longevity risks, weather index–based insurance products, variable annuities, etc. In the presence of basis risk, a perfect hedging is impossible, and in this paper, we adopt a mean‐variance criterion to strike a balance between the expected hedging error and its variability. Under a time‐dependent diffusion model setup, explicit optimal solutions are derived for the hedging target being either a European option or a forward contract. The solutions are obtained by a delicate application of the linear quadratic control theory, the method of backward stochastic differential equation, and Malliavin calculus. A numerical example is presented to illustrate our theoretical results and their interesting implications.     

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.     

Regulatory capital decisions in the Context of consumer loan portfolios

A topic of interest in recent literature is regulatory capital requirements for consumer loan portfolios. Banks are required to hold regulatory capital for unexpected losses, while expected losses are to be covered by either provisions or future income. In this paper, we show the set of efficient operating points in the market share and profit space for a portfolio manager operating under Basel II capital requirement and under capital constraints are a union of single-cutoff-score and double-cutoff-score operating points. For a portfolio manager to increase market-share beyond the maximum allowable under a single-cutoff score policy (eg, with binding capital constraints) requires granting loans to higher than optimal risk applicants. We show this result in greater portfolio risk but without an increase in regulatory capital requirement amount. The increase in forecasted losses is assumed to be absorbed by provisions or future margin income. Given portfolio managers take on higher risk under the same regulatory capital amount, our findings call for greater focus on provision amounts and future margin income under the supervisory review pillar of Basel II. This research raises the issue of whether the design of the regulatory formula for consumer loan portfolios is flawed.     

Enhanced annuities and the impact of individual underwriting on an insurer’s profit situation

We analyze the effect of enhanced annuities on an insurer engaging in individual underwriting. We use a frailty model for heterogeneity of the insured population and model individual underwriting by a random variable that positively correlates with the corresponding frailty factor. For a given annuity portfolio, we analyze the effect of the quality of the underwriting on the insurer’s profit/loss situation and the impact of adverse selection effects.     

A PDE approach to risk measures of derivatives

This paper proposes a partial differential equation (PDE) approach to calculate coherent risk measures for portfolios of derivatives under the Black-Scholes economy. It enables us to define the risk measures in a dynamic way and to deal with American options in a relatively effective way. Our risk measure is based on the representation form of coherent risk measures. Through the use of some earlier results the PDE satisfied by the risk measures are derived. The PDE resembles the standard Black-Scholes type PDE which can be solved using standard techniques from the mathematical finance literature. Indeed, these results reveal that the PDE approach can provide practitioners with a more applicable and flexible way to implement coherent risk measures for derivatives in the context of the Black-Scholes model.     

Analytic bounds and approximations for annuities and Asian options

Even in case of the Brownian motion as most natural rate of return model it appears too difficult to obtain analytic expressions for most risk measures of constant continuous annuities. In literature the so-called comonotonic approximations have been proposed but these still require the evaluation of integrals. In this paper we show that these integrals can sometimes be computed, and we obtain explicit approximations for some popular risk measures for annuities.Next, we show how these results can be used to obtain fully analytic expressions for lower and upper bounds for the price of a continuously sampled European-style Asian option with fixed exercise price. These analytic lower bound prices are as sharp as those from [Rogers, L.C.G., Shi, Z., 1995. The value of an Asian option. J. Appl. Probab. 32, 1077–1088], if not sharper, but in contrast do not require any longer the evaluation of a two-dimensional or a one-dimensional integral.     

Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model

This paper focuses on the constant elasticity of variance (CEV) model for studying the optimal investment strategy before and after retirement in a defined contribution pension plan where benefits are paid under the form of annuities; annuities are supposed to be guaranteed during a certain fixed period of time. Using Legendre transform, dual theory and variable change technique, we derive the explicit solutions for the power and exponential utility functions in two different periods (before and after retirement). Each solution contains a modified factor which reflects an investor’s decision to hedge the volatility risk. In order to investigate the influence of the modified factor on the optimal strategy, we analyze the property of the modified factor. The results show that the dynamic behavior of the modified factor for the power utility mainly depends on the time and the investor’s risk aversion coefficient, whereas it only depends on the time in the exponential case.     

Semi-static hedging of variable annuities

This paper focuses on hedging financial risk in variable annuities with guarantees. We show that insurers should incorporate the specificity of the periodic payment of variable annuities fees to best hedge embedded guarantees and should focus on hedging the net liability. We develop a new hedging strategy based on semi-static hedging techniques, which takes into account the periodically collected fees, and confirm that it is more effective than delta-hedging with same rebalancing dates, as well as traditional semi-static hedging strategies that do not consider the specificity of the payments of fees in their optimization. It is also verified that short-selling or using put options as hedging instruments allows more effective hedging.     

Optimal Portfolio Diversification: Empirical Bayes versus Classical Approach

This study extends the research on international portfolio diversification with estimation risk. The model suggested by Jorion is used to derive the predictive distribution of future returns. The estimator is then compared with the classical one for both with and without short sales. It is found that the improvement for various optimal portfolios is greater for the case of short sales than the case of no short sales.     

Optimal portfolio policies under bounded expected loss and partial information

In a market with partial information we consider the optimal selection of portfolios for utility maximizing investors under joint budget and shortfall risk constraints. The shortfall risk is measured in terms of expected loss. Stock returns satisfy a stochastic differential equation. Under general conditions on the corresponding drift process we provide the optimal trading strategy using Malliavin calculus. We give extensive numerical results in the case that the drift is modeled as a continuous-time Markov chain with finitely many states. To deal with the problem of time-discretization when applying the results to market data, we propose a method to detect and correct possible tracking errors.