Journal of the American Statistical Association: William S. Cleveland, Marylyn E. McGill and Robert McGill, “The shape parameter for a two variable graph” 83 (1988) 289–300

Life annuities and pension products usually involve a number of guarantees, such as minimum accumulation rates, minimum annual payments or a minimum total payout. Packaging different types of guarantees is the feature of so-called variable annuities. Basically, these products are unit-linked investment policies providing a post-retirement income. The guarantees, commonly referred to as GMxBs (namely, Guaranteed Minimum Benefits of type ‘x’), include minimum benefits both in the case of death and survival. In this paper we propose a unifying framework for the valuation of variable annuities under quite general model assumptions. We compute and compare contract values and fair fee rates under ‘static’ and ‘mixed’ valuation approaches, via ordinary and least squares Monte Carlo methods, respectively.

Variable annuities: A unifying valuation approach

Life annuities and pension products usually involve a number of guarantees, such as minimum accumulation rates, minimum annual payments or a minimum total payout. Packaging different types of guarantees is the feature of so-called variable annuities. Basically, these products are unit-linked investment policies providing a post-retirement income. The guarantees, commonly referred to as GMxBs (namely, Guaranteed Minimum Benefits of type 'x'), include minimum benefits both in case of death and survival. In this paper we propose a unifying framework for the valuation of variable annuities under quite general model assumptions. We compute and compare contract values and fair fee rates under 'static' and 'mixed' valuation approaches, via ordinary and least squares Monte Carlo methods, respectively.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1810522_code437995.pdf?abstractid=1810522&type=2

Variable Annuities: A Unifying Valuation Approach

In this article, an extensive study of the no-arbitrage fee for Guaranteed Minimum Withdrawal Benefit (GMWB) variable annuity riders is carried out. The value of the GMWB guarantee increases substantially when taking into account the separation of mutual fund fees and the fees earmarked for hedging the guarantee. In addition, the fee is also adversely affected if the underlying risky asset follows a jump diffusion process. We also explore the effects of various modelling assumptions on the optimal withdrawal strategy of the contract holder, as well as the impact on the guarantee value of sub-optimal withdrawal behaviour. Our general conclusions are that only if several unrealistic modelling assumptions are made is it possible to obtain GMWB fees in the same range as is normally charged. In all other cases, it would appear that typical fees are not enough to cover the cost of hedging these guarantees.

/pdf/the-effect-of-modelling-parameters-on-the-value-of-gmwb-54bepo7juz.pdf

The effect of modelling parameters on the value of GMWB guarantees

In this paper, we outline an impulse stochastic control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB) assuming the policyholder is allowed to withdraw funds continuously. We develop a numerical scheme for solving the Hamilton–Jacobi–Bellman (HJB) variational inequality corresponding to the impulse control problem. We prove the convergence of our scheme to the viscosity solution of the continuous withdrawal problem, provided a strong comparison result holds. The scheme can be easily generalized to price discrete withdrawal contracts. Numerical experiments are conducted, which show a region where the optimal control appears to be non-unique.

/pdf/a-numerical-scheme-for-the-impulse-control-formulation-for-2ul44fxhwe.pdf

A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB)

Variable Annuities with embedded guarantees are very popular in the US market. There exists a great variety of products with both, guaranteed minimum death benefits (GMDB) and guaranteed minimum living benefits (GMLB). Although several approaches for pricing some of the corresponding guarantees have been proposed in the academic literature, there is no general framework in which the existing variety of such guarantees can be priced consistently. The present paper fills this gap by introducing a model, which permits a consistent and extensive analysis of all types of guarantees currently offered within Variable Annuity contracts. Besides a valuation assuming that the policyholder follows a given strategy with respect to surrender and withdrawals, we are able to price the contract under optimal policyholder behavior. Using both, MonteCarlo methods and a generalization of a finite mesh discretization approach, we find that some guarantees are overpriced, whereas others, e.g. guaranteed annuities within guaranteed minimum income benefits (GMIB), are offered significantly below their risk-neutral value.

/pdf/a-universal-pricing-framework-for-guaranteed-minimum-5duf9195ac.pdf

A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities

Newly introduced government-subsidized pension products in Germany are required to contain a promise by the seller to provide a “money-back guarantee” at the end of the term. The client is also given the right to stop paying premiums at any time (paid-up option). In this case, the amount of all premiums paid must also be guaranteed by the seller at maturity, no matter when the client stopped paying the premiums. Previous analyses of guarantees in such government-subsidized pension products have ignored this additional option. Within a generalized Black/Scholes framework, we analyze the value of the paid-up option for different products, market scenarios, and client behavior. Our results indicate that the paid-up option significantly increases the value of the money-back guarantee. Furthermore, we find that reducing volatility by shifting the client's assets from stocks to bonds as maturity approaches is a suitable means of reducing the risk arising from the “pure” money-back guarantee but much less effective in reducing the risk arising from the paid-up option.

https://link.springer.com/content/pdf/10.1007%2Fs10713-006-9467-9.pdf

Analysis of embedded options in individual pension schemes in Germany

Increasing life expectancy and thus decreasing mortality rates constitute a global trend that can be observed in almost all countries worldwide. Estimating the current rate at which mortality rates decrease and modeling the future rate of decrease is important for e.g. demographers and actuaries. This task is commonly referred to as mortality trend modeling. In many applications however one needs to carefully distinguish between two different mortality trends: The actual (but unobservable) mortality trend (AMT) prevailing at a certain point in time and the estimated mortality trend (EMT) that an observer would estimate given the (observable) realized mortality up to that point in time. Since the AMT is not observable, an actuary or demographer might misestimate the AMT at any point in time. In particular, he would typically not be able to distinguish between a recent change in the actual trend and a “normal” random fluctuation around the previous long term trend. Depending on the question at hand, the AMT or the EMT or both need to be considered and modeled in analyses. The paper provides a clear definition of and distinction between the actual mortality trend and the estimated mortality trend, discusses their connection, and explains which of the two is relevant for which kind of question. Moreover, a numerically efficient combined model for both trends is specified and calibrated to mortality data. The model component for the actual mortality trend builds on recent findings that mortality appears to evolve log-linear over time with random changes in slope. The model component for the estimated mortality trend is specified such that, given the assumed dynamics for the actual mortality trend, the estimated mortality trend matches the actual trend as close as possible. This provides valuable information on how best estimate mortality assumptions should be derived from the available data in general. Finally, we apply the combined model in practical examples and illustrate the importance of distinguishing between AMT and EMT. We show that, if the AMT is wrongfully assumed observable, the hedge effectiveness of a longevity hedge or the SCR for longevity risk will typically be misestimated significantly.

It takes two: Why mortality trend modeling is more than modeling one mortality trend

We use data from a large US life expectancy provider to test for asymmetric information in the secondary life insurance—or life settlement—market. We compare the average difference between realized lifetimes and estimated life expectancies for a sub-sample of settled policies relative to the entire sample. We find a significant positive difference indicating private information on mortality prospects. Using non-parametric estimates for the excess mortality and survival regressions, we show that the informational advantage is temporary and wears off over five to six years. We argue this is in line with adverse selection on an individual’s condition, which has important economic consequences for the life settlement market and beyond. JEL classification: D12; G22; J10

https://business.uc.edu/content/dam/business/centers/insurance-center/Documents/BauerRussZhu_AdvSelSecondMark.pdf

Adverse Selection in Secondary Insurance Markets: Evidence from the Life Settlement Market

The quality of life expectancy estimates is one key consideration for an investor in life settlements. The predominant metric for assessing this quality is the so-called A-to-E ratio, which relies on a comparison of the actual to the predicted number of deaths. In this article, we explain key issues with this metric: In the short run, it is subject to estimation uncertainty for small and moderately sized portfolios; and, more critically, in the long run, it converges to 100% even if the underwriting is systematically biased. As an alternative, we propose and discuss a set of new metrics based on the difference in (temporary) life expectancies. We examine the underwriting quality of a leading U.S. life expectancy provider based on this new methodology.

/pdf/evaluating-life-expectancy-evaluations-2vj7dk8kne.pdf

Evaluating Life Expectancy Evaluations.

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