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.     

The design of equity-indexed annuities

There is a rich variety of tailored investment products available to the retail investor in every developed economy. These contracts combine upside participation in bull markets with downside protection in bear markets. Examples include equity-linked contracts and other types of structured products. This paper analyzes these contracts from the investor’s perspective rather than the issuer’s using concepts and tools from financial economics. We analyze and critique their current design and examine their valuation from the investor’s perspective. We propose a generalization of the conventional design that has some interesting features. The generalized contract specifications are obtained by assuming that the investor wishes to maximize end of period expected utility of wealth subject to certain constraints. The first constraint is a guaranteed minimum rate of return which is a common feature of conventional contracts. The second constraint is new. It provides the investor with the opportunity to outperform a benchmark portfolio with some probability. We present the explicit form of the optimal contract assuming both constraints apply and we illustrate the nature of the solution using specific examples. The paper focusses on equity-indexed annuities as a representative type of such contracts but our approach is applicable to other types of equity-linked contracts and structured products.     

The design of equity-indexed annuities

There is a rich variety of tailored investment products available to the retail investor in every developed economy. These contracts combine upside participation in bull markets with downside protection in bear markets. Examples include equity-linked contracts and other types of structured products. This paper analyzes these contracts from the investor’s perspective rather than the issuer’s using concepts and tools from financial economics. We analyze and critique their current design and examine their valuation from the investor’s perspective. We propose a generalization of the conventional design that has some interesting features. The generalized contract specifications are obtained by assuming that the investor wishes to maximize end of period expected utility of wealth subject to certain constraints. The first constraint is a guaranteed minimum rate of return which is a common feature of conventional contracts. The second constraint is new. It provides the investor with the opportunity to outperform a benchmark portfolio with some probability. We present the explicit form of the optimal contract assuming both constraints apply and we illustrate the nature of the solution using specific examples. The paper focusses on equity-indexed annuities as a representative type of such contracts but our approach is applicable to other types of equity-linked contracts and structured products.     

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.     

Valuation of guaranteed minimum maturity benefits in variable annuities with surrender options

We present a numerical approach to the pricing of guaranteed minimum maturity benefits embedded in variable annuity contracts in the case where the guarantees can be surrendered at any time prior to maturity that improves on current approaches. Surrender charges are important in practice and are imposed as a way of discouraging early termination of variable annuity contracts. We formulate the valuation framework and focus on the surrender option as an American put option pricing problem and derive the corresponding pricing partial differential equation by using hedging arguments and Itô’s Lemma. Given the underlying stochastic evolution of the fund, we also present the associated transition density partial differential equation allowing us to develop solutions. An explicit integral expression for the pricing partial differential equation is then presented with the aid of Duhamel’s principle. Our analysis is relevant to risk management applications since we derive an expression of the delta for the sensitivity analysis of the guarantee fees with respect to changes in the underlying fund value. We provide algorithms for implementing the integral expressions for the price, the corresponding early exercise boundary and the delta of the surrender option. We quantify and assess the sensitivity of the prices, early exercise boundaries and deltas to changes in the underlying variables including an analysis of the fair insurance fees.     

Valuation of equity-indexed annuities with regime-switching jump diffusion risk and stochastic mortality risk

This paper extends the model and analysis of Lin,Tan and Yang(2009).We assume that the financial market follows a regime-switching jump-diffusion model and the mortality satisfies Lvy process.We price the point to point and annual reset EIAs by Esscher transform method under Merton’s assumption and obtain the closed form pricing formulas.Under two cases:with mortality risk and without mortality risk,the effects of the model parameters on the EIAs pricing are illustrated through numerical experiments.     

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.     

Longevity bond premiums: The extreme value approach and risk cubic pricing

The purpose of this study is to analyze the securitization of longevity risk with an emphasis on longevity risk modeling and longevity bond premium pricing. Various longevity derivatives have been proposed, and the capital market has experienced one unsuccessful attempt by the European Investment Bank (EIB) in 2004. After carefully analyzing the pros and cons of previous securitizations, we present our proposed longevity bonds, whose payoffs are structured as a series of put option spreads. We utilize a random walk model with drift to fit small variations of mortality improvements and employ extreme value theory to model rare longevity events. Our method is a new approach in longevity risk securitization, which has the advantage of both capturing mortality improvements within sample and extrapolating rare, out-of- sample longevity events. We demonstrate that the risk cubic model developed for pricing catastrophe bonds can be applied to mortality and longevity bond pricing and use the model to calculate risk premiums for longevity bonds.     

Currency options as theoretical and practical instrument in hedging the exchange risk in excess of loss reinsurance

The paper demonstrates that a ceding company can fully hedge itself against adverse movements of the exchange rate in the case of excess of loss foreign reinsurance by using the currency option markets.     

A geostatistical approach for dynamic life tables: The effect of mortality on remaining lifetime and annuities

Dynamic life tables arise as an alternative to the standard (static) life table, with the aim of incorporating the evolution of mortality over time. The parametric model introduced by Lee and Carter in 1992 for projected mortality rates in the US is one of the most outstanding and has been used a great deal since then. Different versions of the model have been developed but all of them, together with other parametric models, consider the observed mortality rates as independent observations. This is a difficult hypothesis to justify when looking at the graph of the residuals obtained with any of these methods.Methods of adjustment and prediction based on geostatistical techniques which exploit the dependence structure existing among the residuals are an alternative to classical methods. Dynamic life tables can be considered as two-way tables on a grid equally spaced in either the vertical (age) or horizontal (year) direction, and the data can be decomposed into a deterministic large-scale variation (trend) plus a stochastic small-scale variation (residuals).Our contribution consists of applying geostatistical techniques for estimating the dependence structure of the mortality data and for prediction purposes, also including the influence of the year of birth (cohort). We compare the performance of this new approach with different versions of the Lee-Carter model. Additionally, we obtain bootstrap confidence intervals for predicted q_xt resulting from applying both methodologies, and we study their influence on the predictions of e_65t and a_65t.     

Managing the exchange of information in product development

In the present paper, we develop a dynamic programming (DP) model of the product development (PD) process. We conceptualize product development as a sequence of decisions: whether to incorporate a piece of information that just arrived (i.e. became available) or wait longer. We utilize this formulation to analyze different situations that depend on the type, and nature of information that is exchanged: stationary versus dynamic information. We derive optimal decision rules to determine whether (and when) to incorporate for each case. An analysis of the model results in several important findings. First, we must not necessarily incorporate all available information that is related to the design activity. Specifically, once the information collection exceeds certain value, the design team should stop collecting further information. Second, only when past design work accumulates to a certain threshold value should the team include the latest information and perform rework. Large uncertainty of the information and large sensitivity of the design activity makes the incorporation of new information less likely. Finally, managerial implications are discussed with several numerical examples.     

Longevity risk and the Grim Reaper’s toxic tail: The survivor fan charts

This paper uses survivor fan charts to illustrate the prospective density functions of future male survival rates. The fan charts are based on a version of the Cairns–Blake–Dowd model of male mortality that provides a good fit to recent mortality data for England and Wales. They indicate that although none of us can escape the Grim Reaper, survivorship uncertainty is greatest for males aged a little over 90, confirming that there exists a ‘toxic tail’ for those institutions, such as annuity and pension providers, which are obliged to make payments to them for as long as they live. We also find that taking account of uncertainty in the parameters of the underlying mortality model leads to major increases in estimates of the widths of the fan charts.     

Strategy vs risk in margining portfolios of options

The strategy-based approach to portfolio margining has been used for margining customer accounts for more than four decades. The risk-based approach was proposed in the mid eighties for margining some inventory accounts of brokers but permitted for margining customer accounts only in 2005. This paper presents a computational experiment with the strategy-based approach and the risk-based approach with the purpose of clarifying which one yields lower margin requirements under different scenarios. There exists a widespread opinion, cf. (Reuters 2007; Longo 2007; Smith 2008), that the risk-based approach is always a winner in this competition, and therefore the strategy-based approach must be disqualified as outdated. However, the results of our experiment with portfolios of stock options show that, in many practical situations, the strategy-based approach yields substantially lower margin requirements in comparison with the risk-based approach.     

Pricing catastrophe options with counterparty credit risk in a reduced form model

In this paper, we study the price of catastrophe options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model. We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.     

The effect of environmental parameters on product recovery

Economical and environmental issues are the main driving forces for the development of closed-loop supply chains. This paper examines the impact of environmental issues on long-term behaviour of a single product supply chain with product recovery. The environmental issues examined are the firm's `green image' effect on customer demand, the take back obligation imposed by legislation, and the state campaigns for proper disposal of used products. The behaviour of the system is analyzed through a dynamic simulation model based on the principles of the system dynamics (SD) methodology. This model includes all major inventories of new, used and recovered products and the flows among them. Inventory levels and flow rates are linked through differential equations. The dynamic model provides an experimental simulation tool, which can be used to evaluate the effect of environmental issues on long-term decision making in collection and remanufacturing activities and on product demand. Numerical analysis illustrates the potential uses of the methodology.     

The effect of product variety on supply-chain performance

We present a stylized model for analyzing the effect of product variety on supply-chain performance for a supply chain with a single manufacturer and multiple retailers. The manufacturer produces multiple products on a shared resource with limited capacity and the effect of changeovers on supply-chain cost is due primarily to setup time rather than setup cost. We show that the expected replenishment lead time and the retailers' costs are concave increasing in product variety and that the increase is asymptotically linear. Thus, if setup times are significant, the effect of product variety on cost is substantially greater than that suggested by the risk-pooling literature for perfectly flexible manufacturing processes, where the cost increases proportionally to the square root of product variety. We demonstrate that disregarding the effect of product variety on lead time can lead to poor decisions and can lead companies to offer product variety that is greater than optimal. The results of our analysis enable decision-makers to quantify the effect of product variety on supply-chain performance and thus to determine the optimal product variety to offer. The results can also be used to evaluate how changes in the manufacturing process, the supply-chain structure, and the customer demand rate can improve the performance of supply chains with high product variety.     

Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk

In this paper we investigate an optimal investment strategy for a defined-contribution (DC) pension plan member who is loss averse, pays close attention to inflation and longevity risks and requires a minimum performance at retirement. The member aims to maximize the expected S-shaped utility from the terminal wealth exceeding the minimum performance by investing her wealth in a financial market consisting of an indexed bond, a stock and a risk-free asset. We derive the optimal investment strategy in closed-form using the martingale approach. Our theoretical and numerical results reveal that the wealth proportion invested in each risky asset has a V-shaped pattern in the reference point level, while it always increases in the rising lifespan; with a positive correlation between salary and inflation risks, the presence of salary decreases the member’s investment in risky assets; the minimum performance helps to hedge the longevity risk by increasing her investment in risky assets.     

Mathematical analysis of a nonlinear PDE model for European options with counterparty risk

In this work, we analyze a nonlinear partial differential equation (PDE) model for the total value adjustment on European options in the presence of a counterparty risk. We transform the nonlinear PDE into an equivalent one, involving a sectorial operator, and prove the existence and uniqueness of a solution.