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.     

An index for longevity risk transfer

This paper discusses the choice of an appropriate longevity index to track the improvements in mortality in industrialized countries. Period life expectancies computed from national life tables turn out to be efficient in this context. A detailed analysis of the predictive distribution of this longevity index is performed in the Lee–Carter model where the period life expectancy is just a functional of the underlying time index.     

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.     

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.     

Entropy, longevity and the cost of annuities

This paper presents an extension of the application of the concept of entropy to annuity costs. Keyfitz (1985) introduced the concept of entropy, and analysed this in the context of continuous changes in life expectancy. He showed that a higher level of entropy indicates that the life expectancy has a greater propensity to respond to a change in the force of mortality than a lower level of entropy. In other words, a high level of entropy means that further reductions in mortality rates would have an impact on measures like life expectancy. In this paper, we apply this to the cost of annuities and show how it allows the sensitivity of the cost of a life annuity contract to changes in longevity to be summarized in a single figure index.     

Reverse mortgage pricing and risk analysis allowing for idiosyncratic house price risk and longevity risk

Reverse mortgages provide an alternative source of funding for retirement income and health care costs. The two main risks that reverse mortgage providers face are house price risk and longevity risk. Recent real estate literature has shown that the idiosyncratic component of house price risk is large. We analyse the combined impact of house price risk and longevity risk on the pricing and risk profile of reverse mortgage loans in a stochastic multi-period model. The model incorporates a new hybrid hedonic–repeat-sales pricing model for houses with specific characteristics, as well as a stochastic mortality model for mortality improvements along the cohort direction (the Wills–Sherris model). Our results show that pricing based on an aggregate house price index does not accurately assess the risks underwritten by reverse mortgage lenders, and that failing to take into account cohort trends in mortality improvements substantially underestimates the longevity risk involved in reverse mortgage loans.     

Redistribution of longevity risk: The effect of heterogeneous mortality beliefs

Existing literature regarding the natural hedge potential that arises from combining different longevity-linked liabilities typically does not address the question how changes in the liability mix can be obtained. We consider firms who aim to exploit the benefits of natural hedge potential by redistributing their risks, and characterize the risk redistributions that will arise when the parties bargain for a redistribution of risk that weakly benefits them all. We analyze the effects of heterogeneity in the beliefs regarding the probability distribution of future mortality rates on the properties of these risk redistributions, and provide a numerical illustration for a case where an insurer with a portfolio of term assurance contracts and a pension fund with a portfolio of life annuities redistribute their risks.     

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.     

Swiss coherent mortality model as a basis for developing longevity de-risking solutions for Swiss pension funds: A practical approach

Pension funds in Switzerland are exposed to longevity risk possibly to a greater extent than in many other developed economies. The ground for this is a dearth of financial products to combat longevity risk, with a lack of buy-in and very limited variety of buy-out solutions available. The solutions that do exist frequently come at a very high price and many pension funds are in deficit on a buy-out basis. From our point of view creating an approach for evaluating the longevity risk faced by each pension fund and integrating it into dynamic risk budgeting strategies will help Swiss pension funds better understand the mechanism behind different longevity de-risking solutions and decide on the most suitable as well as affordable solution for them. To develop capital market solutions for longevity hedging strategies it is crucial that both hedgers (pension funds) as well as solution providers are able to quantify the longevity risk in the framework of a holistic risk management and to develop an adequate pricing approach.In this publication we present our stochastic coherent mortality model developed for Swiss pension funds based on the reference population of fifteen countries and discuss the robustness of the forecasts relative to the sample period used to fit the model, biological reasonableness of the forecasts and other modelling parameters as well as possible impact on results. The model has taken into account past single population modelling techniques and allows flexible age effect to capture the spread behaviour introduced by the target population. The augmented terms for the spread function are chosen based on their forecast accuracy and a coherent behaviour is expected in the long term. The idea behind is fairly simple and yields a design with both transparency and robustness. The model usage is not limited to Switzerland.     

A note on yardstick regulation of fishery monopolies

This note considers a two-period model of a fishery exploited by two firms selling their harvests in separate local markets. Assuming the harvesting cost is a private information to the firms, there is an agency that can regulate the market. I propose a contract in the realm of yardstick competition framework that allows the regulator to solve the information asymmetry problem and achieve cooperation.     

Two error-correcting pooling designs from symplectic spaces over a finite field

In this paper, we construct two classes of -disjunct matrix with subspaces in a symplectic space and prove that the ratio efficiency t/n of two constructions are smaller than that of D’yachkov et al. (2005) [2].     

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.     

Demand for longevity securities under relative performance concerns: Stochastic differential games with cointegration

This paper investigates the impact of relative performance concerns on the longevity risk transfer market. When an insurer concerns about the relative performance in a two-insurer economy, she maximizes the expected utility of her terminal wealth benchmarked against her competitor’s. The problem formulation for a general utility, a general interest rate process and cointegrated mortality rates uses a nonzero sum stochastic differential game approach. Explicit solution of the Nash equilibrium is derived for constant relative risk adverse insurers under the Vasicek-type stochastic interest and mortality rates. Existence and uniqueness of the Nash equilibrium are established for the CIR-type models, which rule out negative interest and mortality rates. While previous studies based on the single-agent approaches have shown a high investment demand in longevity bonds, the launch of it was unsuccessful in reality. Ours supplements that the demand is much lower subject to the relative performance concerns.     

Optimal consumption and portfolio choice for pooled annuity funds

This paper presents the optimal continuous time dynamic consumption and portfolio choice for pooled annuity funds. A pooled annuity fund constitutes an alternative way to protect against mortality risk compared to purchasing a life annuity. The crucial difference between the pooled annuity fund and purchase of a life annuity offered by an insurance company is that participants of a pooled annuity fund still have to bear some mortality risk while insured annuitants bear no mortality risk at all. The population of the pool is modelled by employing a Poisson process with time-dependent hazard-rate. It follows that the pool member’s optimization problem has to account for the stochastic investment horizon and for jumps in wealth which occur if another pool member dies. In case the number of pool members goes to infinity analytical solutions are provided. For finite pool sizes the solution of the optimization problem is reduced to the numerical solution of a set of ODEs. A simulation and welfare analysis show that pooled annuity funds insure very effectively against longevity risk even if their pool size is rather small. Only very risk averse investors or those without access to small pools are more inclined to pay a risk premium to access private life annuity markets in order to lay off mortality risk completely. As even families constitute such small pools the model provides theoretical justification for the low empirical annuity demand.     

Stochastic portfolio specific mortality and the quantification of mortality basis risk

In the last decade a vast literature on stochastic mortality models has been developed. However, these models are often not directly applicable to insurance portfolios because:

(a) For insurers and pension funds it is more relevant to model mortality rates measured in insured amounts instead of measured in the number of policies.

(b) Often there is not enough insurance portfolio specific mortality data available to fit such stochastic mortality models reliably.

Therefore, in this paper a stochastic model is proposed for portfolio specific mortality experience. Combining this stochastic process with a stochastic country population mortality process leads to stochastic portfolio specific mortality rates, measured in insured amounts. The proposed stochastic process is applied to two insurance portfolios, and the impact on the Value at Risk for longevity risk is quantified. Furthermore, the model can be used to quantify the basis risk that remains when hedging portfolio specific mortality risk with instruments of which the payoff depends on population mortality rates.     

Step by step. The benefits of stage-based R&D licensing contracts

We examine how a licensor can optimally design licensing contracts for multi-phase R&D projects when he does not know the licensee’s project valuation, leading to adverse selection, and cannot enforce the licensee’s effort level, resulting in moral hazard. We focus on the effect of the phased nature typical of such projects, and compare single-phase and multi-phase contracts. We determine the optimal values for the upfront payment, milestone payments and royalties, and the optimal timing for outlicensing. Including multiple milestones and accompanying payments can be an effective way of discriminating between licensees holding different valuations, without having to manipulate the royalty rate, which induces licensees to invest less, resulting in lower project values and socially suboptimal solutions. Interestingly, we also find that multiple milestone payments are beneficial even when the licensor is risk-averse, contrary to standard contract theory results, which recommend that only an upfront payment should be used. In terms of licensing timing, we show that the optimal time depends on the licensor’s risk aversion, the characteristics of the licensee and the project value.     

A new defined benefit pension risk measurement methodology

Defined benefit pension plan sponsors have taken on greater risks for sponsoring these plans in the last several years. Due to ever increasing concerns of longevity risk and the weak economic environment, sponsors are eager to understand their pension-related risks to facilitate optimal enterprise decision-making. Borrowing an analytical framework from the life insurance and annuity industry where the amount of risk is framed in terms of the total assets required to remain solvent over a one-year period with a high level of confidence, i.e., the economic capital approach, this paper develops a benchmark risk measure for pension sponsors by obtaining a total asset requirement for sustaining the pension plan. The difference between the total asset requirement and the actual trust assets thus provides a measure of sponsor assets at risk due to plan sponsorship. Two factor-based approaches are proposed for this calculation. The first approach develops a set of pension-specific factors as if the pension plan were a group annuity. The second approach directly simulates the risk drivers of the pension plan and develops a framework for obtaining factors and calculating the pension risk given a desired confidence level. Our approach is very easy to implement and monitor in practice.     

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.