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.     

Pricing and hedging of variable annuities with state-dependent fees

We investigate the problem of pricing and hedging variable annuity contracts for which the fee deducted from the policyholder’s account depends on the account value. It is believed that state-dependent fees are beneficial to policyholders and insurers since they reduce policyholders’ incentives to lapse the policies and match the costs incurred by policyholders with the pay-offs received from embedded guarantees. We consider an incomplete financial market which consists of two risky assets modelled with a two-dimensional Lévy process. One of the assets is a security which can be traded by the insurer, and the second asset is a security which is the underlying fund for the variable annuity contract. In our model we derive an equation from which the fee for the guaranteed benefit can be calculated and we characterize a strategy which allows the insurer to hedge the benefit. To solve the pricing and hedging problem in an incomplete financial market we apply a quadratic objective.     

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.     

Partial splitting of longevity and financial risks: The longevity nominal choosing swaptions

The previous attempts to launch liquid and standardized longevity derivatives in the market failed because banks do not seem to be ready to take longevity risk. Therefore, instead of trying to transfer longevity risk to investors, it could be interesting for financial institutions to propose interest rate hedges adapted to longevity portfolios, in the spirit of liability driven investments. In this paper, we introduce a new structured financial product: the so-called Longevity Nominal Chooser Swaption. Thanks to such a contract, insurers could keep pure longevity risk and transfer to financial markets a great part of interest rate risk underlying annuity portfolios.We use a population dynamics longevity model and a classical two-factor interest rate model to price this product. Numerical results show that the option offered to the insurer (in terms of choice of nominal) is not too expensive in many real-world cases. We also discuss the pros and the cons of the product and of our methodology.     

Accounting and actuarial smoothing of retirement payouts in participating life annuities

Life insurers use accounting and actuarial techniques to smooth reporting of firm assets and liabilities, seeking to transfer surpluses in good years to cover benefit payouts in bad years. Yet these techniques have been criticized as they make it difficult to assess insurers’ true financial status. We develop stylized and realistically-calibrated models of a participating life annuity, an insurance product that pays retirees guaranteed lifelong benefits along with variable non-guaranteed surplus. Our goal is to illustrate how accounting and actuarial techniques for this type of financial contract shape policyholder wellbeing, along with insurer profitability and stability. Smoothing adds value to both the annuitant and the insurer, so curtailing smoothing could undermine the market for long-term retirement payout products.     

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.     

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.     

Systematic mortality risk: An analysis of guaranteed lifetime withdrawal benefits in variable annuities

Guaranteed lifetime withdrawal benefits (GLWB) embedded in variable annuities have become an increasingly popular type of life annuity designed to cover systematic mortality risk while providing protection to policyholders from downside investment risk. This paper provides an extensive study of how different sets of financial and demographic parameters affect the fair guaranteed fee charged for a GLWB as well as the profit and loss distribution, using tractable equity and stochastic mortality models in a continuous time framework. We demonstrate the significance of parameter risk, model risk, as well as the systematic mortality risk component underlying the guarantee. We quantify how different levels of equity exposure chosen by the policyholder affect the exposure of the guarantee providers to systematic mortality risk. Finally, the effectiveness of a static hedge of systematic mortality risk is examined allowing for different levels of equity exposure.     

A dynamic equivalence principle for systematic longevity risk management

This paper addresses systematic longevity risk in long-term insurance business. We analyze the consequences of working under unknown survival probabilities on the efficiency of the Law of Large Numbers and point out the need for appropriate and feasible risk management techniques. We propose a setting for risk sharing schemes between the insurer and policyholders via a dynamic equivalence principle. We focus on a pure endowment contract and derive conditions for a viable risk sharing scheme which enhances the solvency situation of the insurer while being more favorably priced for the policyholders.     

Valuing variable annuities with guaranteed minimum lifetime withdrawal benefits

Variable annuities with guaranteed minimum lifetime withdrawal benefits (VA/GLWB) offer retirees longevity protection, exposure to equity markets, and access to flexible withdrawals in emergencies. We model how risk-averse retirees optimally withdraw from the products, balancing returns and the embedded longevity protection. Assuming reasonable individual preferences, the resulting cash flow generates a Money’s Worth Ratio of around 0.9 for our stylized VA/GLWB in the post-crisis product, considerably lower than what was offered prior to 2008. Sensitivity analyses with respect to portfolio choice, mortality, fees, and guaranteed withdrawal rates show that MWRs range from 0.80 to 1.0, with the portfolio choice making the biggest difference. For most parameter choices, the utility value of the VA/GLWB exceeds that of a similar mutual fund, but it is less than for a fixed annuity. Interestingly, VA/GLWB withdrawals in early retirement can optimally exceed contract maximum withdrawals, despite the fact that this reduces future withdrawal guarantees.     

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.     

Analytical valuation and hedging of variable annuity guaranteed lifetime withdrawal benefits

Variable annuity is a retirement planning product that allows policyholders to invest their premiums in equity funds. In addition to the participation in equity investments, the majority of variable annuity products in today’s market offer various types of investment guarantees, protecting policyholders from the downside risk of their investments. One of the most popular investment guarantees is known as the guaranteed lifetime withdrawal benefit (GLWB). In current market practice, the development of hedging portfolios for such a product relies heavily on Monte Carlo simulations, as there were no known closed-form formulas available in the existing actuarial literature. In this paper, we show that such analytical solutions can in fact be determined for the risk-neutral valuation and delta-hedging of the plain-vanilla GLWB. As we demonstrate by numerical examples, this approach drastically reduces run time as compared to Monte Carlo simulations. The paper also presents a novel technique of fitting exponential sums to a mortality density function, which is numerically more efficient and accurate than the existing methods in the literature.     

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.     

Valuation perspectives and decompositions for variable annuities with GMWB riders

The guaranteed minimum withdrawal benefit (GMWB) rider, as an add on to a variable annuity (VA), guarantees the return of premiums in the form of periodic withdrawals while allowing policyholders to participate fully in any market gains. GMWB riders represent an embedded option on the account value with a fee structure that is different from typical financial derivatives. We consider fair pricing of the GMWB rider from a financial economic perspective. Particular focus is placed on the distinct perspectives of the insurer and policyholder and the unifying relationship. We extend a decomposition of the VA contract into components that reflect term-certain payments and embedded derivatives to the case where the policyholder has the option to surrender, or lapse, the contract early.     

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.     

On the management of life insurance company risk by strategic choice of product mix,investment strategy and surplus appropriation schemes

The aim of this paper is to analyze the impact of management’s strategic choice of asset and liability composition in life insurance on shortfall risk and the shareholders’ fair risk charge. In contrast to previous work, we focus on the effectiveness of management decisions regarding the product mix and the riskiness of the asset side under different surplus appropriation schemes. We propose a model setting that comprises temporary life annuities and endowment insurance contracts. Our numerical results show that the effectiveness of management decisions in regard to risk reduction strongly depends on the surplus appropriation scheme offered to the customer and their impact on guaranteed benefit payments, which thus presents an important control variable for the insurer.     

Life Anuities with Stochastic Survival Probabilities: A Review

An insurance company selling life annuities has to use projected life tables to describe the survival of policyholders. Such life tables are generated by stochastic processes governing the future path of mortality. To fix the ideas, the standard Lee-Carter model for mortality projection is adopted here. In that context, the paper purposes to examine the consequences of working with random survival probabilities. Various stochastic inequalities are derived, showing that the risk borne by the annuity provider is increased compared to the classical independent case. Moreover, the type of dependence existing between the insured life times is carefully examined. The paper also deals with the computation of ruin probabilities and large portfolio approximations.      

Modern tontine with bequest: Innovation in pooled annuity products

We introduce a new pension product that offers retirees the opportunity for a lifelong income and a bequest for their estate. Based on a tontine mechanism, the product divides pension savings between a tontine account and a bequest account. The tontine account is given up to a tontine pool upon death while the bequest account value is paid to the retiree’s estate. The values of these two accounts are continuously re-balanced to the same proportion, which is the key feature of our new product.Our main research question about the new product is what proportion of pension savings should a retiree allocate to the tontine account. Under a power utility function, we show that more risk averse retirees allocate a fairly stable proportion of their pension savings to the tontine account, regardless of the strength of their bequest motive. The proportion declines as the retiree becomes less risk averse for a while. However, for the least risk averse retirees, a high proportion of their pension savings is optimally allocated to the tontine account. This surprising result is explained by the least risk averse retirees seeking the potentially high value of the bequest account at very old ages.     

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.