Maximum Market Price of Longevity Risk under Solvency Regimes: The Case of Solvency II.

Longevity risk constitutes an important risk factor for life insurance companies, and it can be managed through longevity-linked securities. The market of longevity-linked securities is at present far from being complete and does not allow finding a unique pricing measure. We propose a method to estimate the maximum market price of longevity risk depending on the risk margin implicit within the calculation of the technical provisions as defined by Solvency II. The maximum price of longevity risk is determined for a survivor forward (S-forward), an agreement between two counterparties to exchange at maturity a fixed survival-dependent payment for a payment depending on the realized survival of a given cohort of individuals. The maximum prices determined for the S-forwards can be used to price other longevity-linked securities, such as q-forwards. The Cairns–Blake–Dowd model is used to represent the evolution of mortality over time that combined with the information on the risk margin, enables us to calculate upper limits for the risk-adjusted survival probabilities, the market price of longevity risk and the S-forward prices. Numerical results can be extended for the pricing of other longevity-linked securities

Maximum Market Price of Longevity Risk under Solvency Regimes: The Case of Solvency II.

Longevity risk constitutes an important risk factor for life insurance companies, and it can be managed through longevity-linked securities. The market of longevity-linked securities is at present far from being complete and does not allow finding a unique pricing measure. We propose a method to estimate the maximum market price of longevity risk depending on the risk margin implicit within the calculation of the technical provisions as defined by Solvency II. The maximum price of longevity risk is determined for a survivor forward (S-forward), an agreement between two counterparties to exchange at maturity a fixed survival-dependent payment for a payment depending on the realized survival of a given cohort of individuals. The maximum prices determined for the S-forwards can be used to price other longevity-linked securities, such as q-forwards. The Cairns–Blake–Dowd model is used to represent the evolution of mortality over time that combined with the information on the risk margin, enables us to calculate upper limits for the risk-adjusted survival probabilities, the market price of longevity risk and the S-forward prices. Numerical results can be extended for the pricing of other longevity-linked securities

Modeling and Management of Longevity Risk

In this article we review the state of play in the use of stochastic models for the measurement and management of longevity risk. A focus of the discussion concerns how robust these models are relative to a variety of inputs: something that is particularly important in formulating a risk management strategy. On the modeling front much still needs to be done on robust multipopulation mortality models, and on the risk management front we need to develop a better understanding of what the objectives are of pension plans that need to be optimized. We propose a variety of ways forward on both counts

Bayesian stochastic mortality modelling under serially correlated local effects

The vast majority of stochastic mortality models in the academic literature are intended to explain the dynamics underpinning the process by a combination of age, period and cohort e ects. In principle, the more such e ects are included in a stochastic mortality model, the better is the in-sample t to the data. Estimates of those parameters are most usually obtained under some distributional assumption about the occurrence of deaths, which leads to the optimisation of a relevant objective function. The present Thesis develops an alternative framework where the local mortality effect is appreciated, by employing a parsimonious multivariate process for modelling the latent residual e ects of a simple stochastic mortality model as dependent rather than conditionally independent variables. Under the suggested extension the cells of the examined data-set are supplied with a serial dependence structure by relating the residual terms through a simple vector autoregressive model. The method is applicable for any of the popular mortality modelling structures in academia and industry, and is accommodated herein for the Lee-Carter and Cairns-Blake-Dowd models. The additional residuals model is used to compensate for factors of a mortality model that might mostly be a ected by local e ects within given populations. By using those two modelling bases, the importance of the number of factors for a stochastic mortality model is emphasised through the properties of the prescribed residuals model. The resultant hierarchical models are set under the Bayesian paradigm, and samples from the joint posterior distribution of the latent states and parameters are obtained by developing Markov chain Monte Carlo algorithms. Along with the imposed short-term dynamics, we also examine the impact of the joint estimation in the long-term factors of the original models. The Bayesian solution aids in recognising the di erent levels of uncertainty for the two naturally distinct type of dynamics across di erent populations. The forecasted rates, mortality improvements, and other relevant mortality dependent metrics under the developed models are compared to those produced by their benchmarks and other standard stochastic mortality models in the literature

De-risking strategy: Longevity spread buy-in

The paper proposes a demographic de-risking strategy for a pension provider, to deal with the future uncertainty in longevity over a long time horizon. The innovative idea of a longevity spread buy-in is presented. The formulae for calculating the buy-in premium are proposed in the case of pension plans. The proposal directly impacts the pension provider’s risk management systems and hence can be an important part of the overall approach to risk management. The numerical results, developed under specified stochastic hypotheses for the dynamics of the underlying financial and demographic processes, show how the proposal of the paper can be practically implemented

An Analysis of the Australian Mortality

This thesis evaluates and compares the goodness-of-fit of six selected stochastic mortality models, based on Australian mortality data. These models are applied to both sexes, across three different age-group scenarios, and four lookback windows and five look forward windows. Four different criteria were applied in the model selection and evaluation and results indicate that the Lee-Carter model is more efficient. The forecast is for a period of 50-years (from 2012 to 2061) and results reveal that mortality rates are improving with age stratification

Longevity Risk and Natural Hedge Potential in Portfolios Of Life Insurance Products: The Effect of Investment Risk

Payments of life insurance products depend on the uncertain future evolution of survival probabilities. This uncertainty is referred to as longevity risk. Existing literature shows that the effect of longevity risk on single life annuities can be substantial, and that there exists a (natural) hedge potential from combining single life annuities with death benefits or from investing in survivor swaps. The effect of financial risk on these hedge effects is typically ignored. The aim of this paper is to quantify longevity risk in portfolios of mortality-linked assets and liabilities, taking into account the effect of financial risk. We find that investment risk significantly affects the impact of longevity risk in life insurance products. It also significantly affects the hedge potential that arises from combining life insurance products, or from investing in longevity-linked assets. For example, our results suggest that ignoring the effect of financial risk can lead to severe overestimation of the natural hedge potential from death benefits, and underestimation of the hedge effects of survivor swaps.Life insurance;life annuities;death benefits;survivor swaps;risk management;financial risk;longevity risk;insolvency risk;capital adequacy

Essays on Lifetime Uncertainty: Models, Applications, and Economic Implications

My doctoral thesis “Essays on Lifetime Uncertainty: Models, Applications, and Economic Implications” addresses economic and mathematical aspects pertaining to uncertainties in human lifetimes. More precisely, I commence my research related to life insurance markets in a methodological direction by considering the question of how to forecast aggregate human mortality when risks in the resulting projections is important. I then rely on the developed method to study relevant applied actuarial problems. In a second strand of research, I consider the uncertainty in individual lifetimes and its influence on secondary life insurance market transactions. Longevity risk is becoming increasingly crucial to recognize, model, and monitor for life insurers, pension plans, annuity providers, as well as governments and individuals. One key aspect to managing this risk is correctly forecasting future mortality improvements, and this topic has attracted much attention from academics as well as from practitioners. However, in the existing literature, little attention has been paid to accurately modeling the uncertainties associated with the obtained forecasts, albeit having appropriate estimates for the risk in mortality projections, i.e. identifying the transiency of different random sources affecting the projections, is important for many applications. My first essay “Coherent Modeling of the Risk in Mortality Projections: A Semi-Parametric Approach” deals with stochastically forecasting mortality. In contrast to previous approaches, I present the first data-driven method that focuses attention on uncertainties in mortality projections rather than uncertainties in realized mortality rates. Specifically, I analyze time series of mortality forecasts generated from arbitrary but fixed forecasting methodologies and historic mortality data sets. Building on the financial literature on term structure modeling, I adopt a semi-parametric representation that encompasses all models with transitions parameterized by a Normal distributed random vector to identify and estimate suitable specifications. I find that one to two random factors appear sufficient to capture most of the variation within all of our data sets. Moreover, I observe similar systematic shapes for their volatility components, despite stemming from different forecasting methods and/or different mortality data sets. I further propose and estimate a model variant that guarantees a non-negative process of the spot force of mortality. Hence, the resulting forward mortality factor models present parsimonious and tractable alternatives to the popular methods in situations where the appraisal of risks within medium or long-term mortality projections plays a dominant role. Relying on a simple version of the derived forward mortality factor models, I take a closer look at their applications in the actuarial context in the second essay “Applications of Forward Mortality Factor Models in Life Insurance Practice. In the first application, I derive the Economic Capital for a stylized UK life insurance company offering traditional product lines. My numerical results illustrate that (systematic) mortality risk plays an important role for a life insurer\u27s solvency. In the second application, I discuss the valuation of different common mortality-contingent embedded options within life insurance contracts. Specifically, I present a closed-form valuation formula for Guaranteed Annuity Options within traditional endowment policies, and I demonstrate how to derive the fair option fee for a Guaranteed Minimum Income Benefit within a Variable Annuity Contract based on Monte Carlo simulations. Overall my results exhibit the advantages of forward mortality factor models in terms of their simplicity and compatibility with classical life contingencies theory. The second major part of my doctoral thesis concerns the so-called life settlement market, i.e. the secondary market for life insurance policies. Evolving from so-called “viatical settlements” popular in the late 1980s that targeted severely ill life insurance policyholders, life settlements generally involve senior insureds with below average life expectancies. Within such a transaction, both the liability of future contingent premiums and the benefits of a life insurance contract are transferred from the policyholder to a life settlement company, which may further securitize a bundle of these contracts in the capital market. One interesting and puzzling observation is that although life settlements are advertised as a high-return investment with a low “Beta”, the actual market systematically underperformed relative to expectations. While the common explanation in the literature for this gap between anticipated and realized returns falls on the allegedly meager quality of the underlying life expectancy estimates, my third essay “Coherent Pricing of Life Settlements under Asymmetric Information” proposes a different viewpoint: The discrepancy may be explained by adverse selection. Specifically, by assuming information with respect to policyholders’ health states is asymmetric, my model shows that a discrepancy naturally arises in a competitive market when the decision to settle is taken into account for pricing the life settlement transaction, since the life settlement company needs to shift its pricing schedule in order to balance expected profits. I derive practically applicable pricing formulas that account for the policyholder’s decision to settle, and my numerical results reconfirm that---depending on the parameter choices---the impact of asymmetric information on pricing may be considerable. Hence, my results reveal a new angle on the financial analysis of life settlements due to asymmetric information. Hence, all in all, my thesis includes two distinct research strands that both analyze certain economic risks associated with the uncertainty of individuals’ lifetimes---the first at the aggregate level and the second at the individual level. My work contributes to the literature by providing both new insights about how to incorporate lifetime uncertainty into economic models, and new insights about what repercussions---that are in part rather unexpected---this risk factor may have