مدیریت ریسک طول عمر فردی پس از بازنشستگی تحت ریسک سیستماتیک مرگ و میر

This paper analyzes an individual’s post-retirement longevity risk management strategy allowing for systematic longevity risk, recent product innovations, and product loadings. A complete-markets discrete state model and multi-period simulations of portfolio strategies are used to assess individual longevity insurance product portfolios with different levels of systematic and idiosyncratic longevity risk. Portfolios include: fixed life annuities, deferred annuities, inflation-indexed annuities, phased withdrawals and recently proposed group self-annuitization (GSA) plans. GSA plans are found to replace even inflation-indexed annuity products when there are loadings on guaranteed life annuity products. With a bequest motive and loadings, coinsurance portfolio strategies with phased withdrawals and GSA’s dominate portfolios with life annuities or deferred annuities.

Most developed countries use subsidies and tax incentives to increase mandatory and voluntary retirement savings so individuals are less reliant on public pay-as-you-go pension systems. Payout phases of pension systems are organized very differently across countries with individuals increasingly responsible for post-retirement financial decisions (Rocha et al., 2010 and Rocha and Vittas, 2010). Although most private retirement savings are in defined contribution plans, limited attention has been paid to the decumulation of these funds. Individuals face a complex problem of post-retirement financial planning. They have to take into account investment risk, inflation risk, product loadings and guarantees and both systematic and unsystematic (idiosyncratic) longevity risk.1 Recent product innovations in the form of group self-annuitization (GSA) plans provide new opportunities to manage longevity risk. The importance and complexity of the post-retirement financial planning problem has been well recognized in the literature. There is a longstanding literature on optimal annuitization dating back to Yaari (1965). Recent studies consider individuals’ retirement portfolio choice with longevity insurance products such as life annuities and deferred annuities (e.g., Horneff et al., 2010b, Post, 2012, Purcal and Piggott, 2008, Schulze and Post, 2010 and Stevens, 2010), variable annuities (e.g., Doyle and Piggott, 2003, Milevsky and Kyrychenko, 2008 and Horneff et al., 2010a), variable deferred annuities (Kartashov et al., 2011), or group self-annuitization plans (e.g., Piggott et al., 2005, Valdez et al., 2006 and Stamos, 2008). Only a few of these papers distinguish between idiosyncratic and systematic longevity risk, model inflation risk or assess new product innovations including group self-annuitization. This paper makes two contributions. First, we develop a complete markets framework to study the optimal management of systematic and idiosyncratic longevity risk, in a structure which distinguishes between idiosyncratic and systematic risk. This affirms the Yaari result in the special case of zero loadings and no bequest motive, but shows how products other than a standard annuity come into play when these assumptions are relaxed. Second, we adapt this framework to allow numerical simulation analysis, to show how portfolios of retirement products can improve welfare in this environment when product loadings and a bequest motive are introduced. New insights into the impact of systematic longevity risk, loadings for guarantee products and the potential impact of insurer insolvency risk on an individual’s optimal product portfolio are provided. Multi-period simulation allows us to assess a broader range of realistic individual portfolio strategies. The multi-period model incorporates stochastic mortality and inflation risk. Investment risk is not included in order to focus on products that manage longevity risk rather than investment risk.2 Individual welfare is compared for different product portfolios motivated by the optimal insurance literature (see,e.g., Borch, 1960, Arrow, 1971, Arrow, 1973 and Raviv, 1979). Coinsurance for longevity risk is represented in the portfolio by self-insurance, referred to as phased withdrawals, and deductibles by deferred annuities. Applying concepts from the optimal insurance literature is complicated by the effect of a bequest motive in the longevity case. We address this by including cases with and without bequests to highlight the impact on individual longevity risk management. The paper consists of two main parts. Section 2 presents a two-period discrete state model to derive the optimal longevity insurance strategy for an individual facing both idiosyncratic and systematic longevity risk subject to a budget constraint. The individual has access to a complete market of financial and insurance products that allows the individual to attain optimal consumption in current and future states. The products required to complete the market are a risk-free bond, a life annuity, a longevity bond, and a GSA plan. Products are priced using a contingent claims approach. Frictional costs and insolvency risk are introduced. An example is used to highlight the main insights of the model. In Section 3 the two-period model is extended into a more comprehensive multi-period expected utility framework. Simulation analysis is used to compare a range of longevity insurance strategies developed from the insights of the two-period model and optimal insurance concepts. The longevity insurance strategies include portfolios of life annuities, deferred annuities, inflation-indexed annuities, GSA’s, and phased withdrawals. Expected utilities and certainty equivalent consumption are used for welfare comparisons. The market price for insurer annuity products is determined as the actuarially fair insurance premium plus loadings typically observed in annuity markets. In practice loadings cover costs of guarantees, adverse selection and costs of capital from regulatory capital requirements. The stochastic evolution of mortality rates is based on a multivariate mortality model in Wills and Sherris (2010) designed to study the pricing and risk management of longevity risk. The market model developed by Ngai and Sherris (2011) is used to generate future stochastic inflation and economic scenarios. This model simulates gross domestic product, interest rates, stock prices, and inflation. The results of the study show that for individuals with no bequest motive and with no product loadings, annuitization strategies including small GSA holdings are optimal under systematic longevity risk. Inflation indexed annuities dominate, and because life annuities insure both systematic and idiosyncratic longevity risk, GSA’s have a limited role. With loadings on guaranteed life annuity products, GSA plans, which are mutual and non-guaranteed, become significantly more attractive for individuals in managing their post-retirement longevity risk, replacing even annuitization products with inflation guarantees. For individuals with a bequest motive, coinsurance strategies in the form of self annuitization (phased withdrawals) dominate. Holdings of GSA plans increase significantly where there are loadings on guarantee products typical of these products.

The paper has assessed individual post-retirement longevity risk strategies for an individual facing both idiosyncratic and systematic longevity risk, inflation risk, and allows for bequests and product loadings. The individual holds a portfolio of financial and insurance products to optimize retirement consumption needs. Products include life annuities as well as recent innovations such as GSA’s. A theoretical framework based on state-contingent consumption and complete markets was presented and insight into the optimal strategy for an individual provided using a two-period model. The impact of a bequest motive was also assessed allowing for utility in the death state. The model showed that, with a bequest, strategies that include a risk free bond were optimal and that loadings on products that change their relative pricing have a significant effect on the optimal product portfolio required to manage both systematic and idiosyncratic longevity risk. Multi-period simulation was used to assess and compare a broader range of retirement strategies with realistic simulations of economic variables and stochastic mortality with systematic and idiosyncratic risk. The product portfolios included traditional products such fixed life annuities, deferred annuities, inflation-indexed annuities, as well as group self-annuitization (GSA) plans, and phased withdrawals. The analysis included product loadings and expected insolvency risk for annuity providers. Individuals with no bequest motive, and with no product loadings, prefer annuitization strategies with small holdings of GSA plans under systematic longevity risk. With loadings on guaranteed life annuity products, GSA plans which are mutual and non-guaranteed, increase significantly in the preferred portfolios, replacing even annuitization products with inflation guarantees. For individuals with a bequest motive, portfolio strategies including self annuitization and GSA’s dominate full annuitization. Deferred annuity portfolios based on simple but reasonable retirement income portfolios are not preferred. They must be constructed carefully to include an appropriate level of consumption in the deferral period if they are to provide optimal consumption outcomes for individuals. Importantly, under realistic assumptions, recent product innovations that manage systematic longevity risk (GSA plans) play a significant role in preferred portfolios. Product loadings for guarantee products can undermine the insurance welfare benefits of traditional annuity products.