Downside Risk Management of a Defined Benefit Plan Considering Longevity Basis Risk

To control downside risk of a defined benefit pension plan arising from unexpected mortality improvements and severe market turbulence, this article proposes an optimization model by imposing two conditional value at risk constraints to control tail risks of pension funding status and total pension costs. With this setup, we further examine two longevity risk hedging strategies subject to basis risk. While the existing literature suggests that the excess-risk hedging strategy is more attractive than the ground-up hedging strategy as the latter is more capital intensive and expensive, our numerical examples show that the excess-risk hedging strategy is much more vulnerable to longevity basis risk, which limits its applications for pension longevity risk management. Hence, our findings provide important insight on the effect of basis risk on longevity hedging strategies.     

Managing longevity risk under adverse selection: the influence of policyholder age and contract duration

In recent years, the rising life expectancy in almost all industrialized countries has led to an increasing demand by life insurers for possibilities to hedge longevity risk. Two of the most prominent alternative risk management instruments in this regard are the transfer of longevity risk to the capital market, e.g. through the purchase of mortality contingent bonds, and natural hedging, i.e. hedging longevity risk through portfolio composition. In this paper, we study the effectiveness of these risk management instruments under adverse selection, which here refers to the difference between annuitant mortality and the mortality of the population as a whole. Special emphasis is thereby placed on analyzing the impact of policyholders’ age at contract inception and contract duration on the effectiveness of risk management.     

Application of Relational Models in Mortality Immunization

The prediction of future mortality rates by any existing mortality models is hardly exact, which causes an exposure to mortality (longevity) risk for life insurers (annuity providers). Since a change in mortality rates has opposite impacts on the surpluses of life insurance and annuity, hedging strategies of mortality and longevity risks can be implemented by creating an insurance portfolio of both life insurance and annuity products. In this article, we apply relational models to capture the mortality movements by assuming that the realized mortality sequence is a proportional change and/or a constant shift of the expected one, and the size of the changes varies in the length of the sequences. Then we create a variety of non-size-free matching strategies to determine the weights of life insurance and annuity products in an insurance portfolio for mortality immunization, where the weights depend on the sizes of the proportional and/or constant changes. Comparing the hedging performances of four non-size-free matching strategies with corresponding size-free ones proposed by Lin and Tsai, we demonstrate with simulation illustrations that the non-size-free matching strategies can hedge against mortality and longevity risks more effectively than the size-free ones.     

Cohort and value-based multi-country longevity risk management

ABSTRACT

Multi-country risk management of longevity risk provides new opportunities to hedge mortality and interest rate risks in guaranteed lifetime income streams. This requires consideration of both interest rate and mortality risks in multiple countries. For this purpose, we develop value-based longevity indexes for multiple cohorts in two different countries that take into account the major sources of risks impacting life insurance portfolios, mortality and interest rates. To construct the indexes we propose a cohort-based affine model for multi-country mortality and use an arbitrage-free multi-country Nelson–Siegel model for the dynamics of interest rates. Index-based longevity hedging strategies have the advantages of efficiency, liquidity and lower cost but introduce basis risk. Graphical risk metrics are a way to effectively capture the relationship between an insurer's portfolio and hedging strategies. We illustrate the effectiveness of using a value-based index for longevity risk management between two countries using graphical basis risk metrics. To show the impact of both interest rate and mortality risk we use Australia and the UK as domestic and foreign countries, and, to show the impact of mortality only, we use the male populations of the Netherlands and France with common interest rates and basis risk arising only from differences in mortality risks.     

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.     

Statistical Inference for Lee-Carter Mortality Model and Corresponding Forecasts

Although the Lee-Carter model has become a benchmark in modeling mortality rates, forecasting mortality risk, and hedging longevity risk, some serious issues exist on its inference and interpretation in the actuarial science literature. After pointing out these pitfalls, this article proposes a modified Lee-Carter model, provides a rigorous statistical inference, and derives the asymptotic distributions of the proposed estimators and unit root test when the mortality index is nearly integrated and errors in the model satisfy some mixing conditions. After a unit root hypothesis is not rejected, future mortality forecasts can be obtained via the proposed inference. An application of the proposed unit root test to U.S. mortality rates rejects the unit root hypothesis for the female and combined mortality rates but does not reject the unit root hypothesis for the male mortality rates.     

Measuring Basis Risk in Longevity Hedges

Abstract

In examining basis risk in index longevity hedges, it is important not to ignore the dependence between the population underlying the hedging instrument and the population being hedged. We consider four extensions to the Lee-Carter model that incorporate such dependence: Both populations are jointly driven by the same single time-varying index, the two populations are cointegrated, the populations depend on a common age factor, and there is an augmented common factor model in which a population-specific time-varying index is added to the common factor model with the property that it will tend toward a certain constant level over time. Using data from the female populations of Canada and the United States, we show the augmented common factor model is preferred in terms of both goodness-of-fit and ex post forecasting performance. This model is then used to quantify the basis risk in a longevity hedge of 65-year old Canadian females structured using a portfolio of q-forward contracts predicated on U.S. female population mortality. The hedge effectiveness is estimated at 56% on the basis of longevity value-at-risk and 81.61% on the basis of longevity risk reduction.     

Partial Hedging for Equity-Linked Products Using Risk-Minimizing Strategies

In this article, we consider the pricing and hedging of equity-indexed annuities (EIAs) using local risk-minimizing strategies as well as evaluating the capital requirement for these products. Since these products involve mortality as well as financial risks, we integrate mortality risk and propose partial hedging strategies that protect the insurer based on risk measures. The framework we present makes use of sequential local risk-minimizing strategies to take into account all intermediate requirements. To demonstrate the flexibility of this framework we present numerical examples featuring point-to-point EIAs with a two-state regime-switching equity model.     

基于双因素Wang转换方法的长寿风险债券定价研究

在比较国外经典债券设计的基础上,基于离散型死亡率模型假设,设计一种可调整上触碰点的触发型长寿债券,运用带永久跳跃的APC模型和双因素Wang转换定价方法对长寿债券进行定价,实证结果表明:在不同的参数组合下的风险溢价均处在一个合理的范围,由于模型参数多、可用死亡率数据年限短,风险溢价的结果对无风险利率等参数敏感性较高.     

Three Approaches to Risk Management—and How and Why Swedish Companies Use Them

Companies can manage risk by using derivatives or through operational hedging. But there is a third possibility: to leave their operating cash flows unhedged while ensuring that the firm has access to external finance in adverse states of the world. This article reports the findings of a recent survey of over 800 Swedish companies that aims to shed light on the relative importance of these three risk management methods, as well as how they interact in corporate risk management programs. The results show that risk management practices aimed at ensuring access to external finance are the main method used by the largest number of companies, followed by operational hedging methods and financial hedging with derivatives. Large companies hedge using both operational methods and derivatives, whereas small firms are less likely to use derivatives but nevertheless attach great importance to the other two ways of managing risk. Even among the largest companies, operational hedging tends to deemed more important than hedging with derivatives—a finding that, although perhaps a surprise to financial professionals, underscores the authors’ finding that operational and derivative‐based hedges function as complements rather than substitutes. Indeed, the authors report that the most financially sophisticated companies tend to use all three of these common forms of risk management.     

Creating Value at the Intersection of Sourcing,Hedging and Trading

Higher commodity prices, along with higher currency and commodity price volatility, have combined with challenging economic circumstances to make for difficult economics within many industries today. These factors can introduce risk to both top‐line revenue and the cost structure, and wreak havoc on net cash flow and profitability. To the extent that high prices and increasing price volatility continue to be the rule in many global commodity categories, the authors suggest that both sourcing and hedging will soon be (if they are not already) near the top of the strategic agenda for many companies. Many companies now design their hedging programs—and in some cases their sourcing—to achieve the goals of reducing cash flow volatility and optimizing value (as opposed to the more conventional aim of minimizing sourced or manufactured unit cost). And a growing number of corporate managements have expressed interest in an even more systematic approach to risk management. Rising pressure for growth and profitability has led companies with large commodities exposures—both those that are naturally long and those with a natural short—to explore a more strategic role for commodity hedging and trading, as well as the use of innovative risk‐shifting mechanisms for inbound and outbound material flows. This article shows how companies can design their commodity risk management programs to make the greatest use of the expertise and capabilities of four different corporate groups: Purchasing, Treasury, Selling, and Marketing. To that end, the authors presents a five‐step program for creating a company‐wide strategic risk management program:

• ? The first step involves making active design choices about what risks to “own” and what risks to limit based on the company's strategy, core competencies, and relative competitive advantages in owning that risk.
• ? The second step is to establish relevant risk guidelines based on capacity to own risks and, to a lesser extent, risk appetite, with specific hedging targets and benchmarks. This involves defining objectives, priorities, and constraints (for example, protecting liquidity or increasing debt capacity by reducing cash flow volatility).
• ? The third step is to identify and characterize a complete inventory of all exposures—source, size and drivers—and those exposures to be managed. This involves defining, measuring and analyzing all exposures (e.g., commodities, FX, interest rates), with special attention to aggregating, netting, natural offsets, and correlations.
• ? The fourth step involves comparing the suitability of various hedging tools and determining how to incorporate these tools into a systematic program that will achieve stated goals, views, and risk preferences for each exposure.
• ? The fifth and last step is establishing an appropriate risk management operating model, which involves considerations of organizational architecture, management processes, decision rights, information flows, and governance.

     

Longevity/Mortality Risk Modeling and Securities Pricing

Securitizing longevity/mortality risk can transfer longevity/mortality risk to capital markets. Modeling and forecasting mortality rate is key to pricing mortality‐linked securities. Catastrophic mortality and longevity jumps occur in historical data and have an important impact on security pricing. This article introduces a stochastic diffusion model with a double‐exponential jump diffusion process that captures both asymmetric rate jumps up and down and also cohort effect in mortality trends. The model exhibits calibration advantages and mathematical tractability while better fitting the data. The model provides a closed‐form pricing solution for J.P. Morgan’s q‐forward contract usable as a building block for hedging.     

Modeling and Pricing Longevity Derivatives Using Stochastic Mortality Rates and the Esscher Transform

The Lee-Carter mortality model provides a structure for stochastically modeling mortality rates incorporating both time (year) and age mortality dynamics. Their model is constructed by modeling the mortality rate as a function of both an age and a year effect. Recently the MBMM model (Mitchell et al. 2013) showed the Lee Carter model can be improved by fitting with the growth rates of mortality rates over time and age rather than the mortality rates themselves. The MBMM modification of the Lee-Carter model performs better than the original and many of the subsequent variants. In order to model the mortality rate under the martingale measure and to apply it for pricing the longevity derivatives, we adapt the MBMM structure and introduce a Lévy stochastic process with a normal inverse Gaussian (NIG) distribution in our model. The model has two advantages in addition to better fit: first, it can mimic the jumps in the mortality rates since the NIG distribution is fat-tailed with high kurtosis, and, second, this mortality model lends itself to pricing of longevity derivatives based on the assumed mortality model. Using the Esscher transformation we show how to find a related martingale measure, allowing martingale pricing for mortality/longevity risk–related derivatives. Finally, we apply our model to pricing a q-forward longevity derivative utilizing the structure proposed by Life and Longevity Markets Association.     

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.     

A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration

In this article, we consider the evolution of the post‐age‐60 mortality curve in the United Kingdom and its impact on the pricing of the risk associated with aggregate mortality improvements over time: so‐called longevity risk. We introduce a two‐factor stochastic model for the development of this curve through time. The first factor affects mortality‐rate dynamics at all ages in the same way, whereas the second factor affects mortality‐rate dynamics at higher ages much more than at lower ages. The article then examines the pricing of longevity bonds with different terms to maturity referenced to different cohorts. We find that longevity risk over relatively short time horizons is very low, but at horizons in excess of ten years it begins to pick up very rapidly. A key component of the article is the proposal and development of a method for calculating the market risk‐adjusted price of a longevity bond. The proposed adjustment includes not just an allowance for the underlying stochastic mortality, but also makes an allowance for parameter risk. We utilize the pricing information contained in the November 2004 European Investment Bank longevity bond to make inferences about the likely market prices of the risks in the model. Based on these, we investigate how future issues might be priced to ensure an absence of arbitrage between bonds with different characteristics.     

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.     

Human Survival at Older Ages and the Implications for Longevity Bond Pricing

Abstract

Governments are concerned about the future of pension plans, for which increasing longevity is judged to be an important risk to their future viability. We focus on human survival at age 65, the starting age point for many pension products. Using a simple model, we link basic measures of life expectancy to the shape of the human survival function and consider its various forms. The model is then used as the basis for investigating actual survival in England and Wales. We find that life expectancy is increasing at a faster rate than at any time in history, with no evidence of this trend slowing or any upper age limit. With interest growing in the use of longevity bonds as a way to transfer longevity risks from pension providers to the capital markets, we seek to understand how longevity drift affects pension liabilities based on mortality rates at the point of annuitization, versus what actually happens as a cohort ages. The main findings are that longevity bonds are an effective hedge against longevity risk; however, it is not only the oldest old that are driving risk, but also more 65-year-olds reaching less extreme ages such as 80. In addition, we find that the possibility of future inflation and interest rates could be as an important a risk to annuities as longevity itself.     

Asset Pricing with Conditioning Information: A New Test

This paper presents a new test of conditional versions of the Sharpe-Lintner CAPM, the Jagannathan and Wang (1996) extension of the CAPM, and the Fama and French (1993) three-factor model. The test is based on a general nonparametric methodology that avoids functional form misspecification of betas, risk premia, and the stochastic discount factor. Our results provide a novel view of empirical performance of these models. In particular, we find that a nonparametric version of the Fama and French model performs well, even when challenged by momentum portfolios.     

Klugman,S.A., Panjer,H.H., and Willmot,G.E., 1998, Los flwfodels: From Data to Decisions

Abstract

Basis risk is an important consideration when hedging longevity risk with instruments based on longevity indices, since the longevity experience of the hedged exposure may differ from that of the index. As a result, any decision to execute an index-based hedge requires a framework for (1) developing an informed understanding of the basis risk, (2) appropriately calibrating the hedging instrument, and (3) evaluating hedge effectiveness. We describe such a framework and apply it to a U.K. case study, which compares the population of assured lives from the Continuous Mortality Investigation with the England and Wales national population. The framework is founded on an analysis of historical experience data, together with an appreciation of the contextual relationship between the two related populations in social, economic, and demographic terms. Despite the different demographic profiles, the case study provides evidence of stable long-term relationships between the mortality experiences of the two populations. This suggests the important result that high levels of hedge effectiveness should be achievable with appropriately calibrated, static, index-based longevity hedges. Indeed, this is borne out in detailed calculations of hedge effectiveness for a hypothetical pension portfolio where the basis risk is based on the case study. A robustness check involving populations from the United States yields similar results.     

Canonical Valuation of Mortality‐Linked Securities

A fundamental question in the study of mortality‐linked securities is how to place a value on them. This is still an open question, partly because there is a lack of liquidly traded longevity indexes or securities from which we can infer the market price of risk. This article develops a framework for pricing mortality‐linked securities on the basis of canonical valuation. This framework is largely nonparametric, helping us avoid parameter and model risk, which may be significant in other pricing methods. The framework is then applied to a mortality‐linked security, and the results are compared against those derived from other methods.