Annuity Pricing Research Articles

Uniqueness of equilibrium with survival probability heterogeneity and endogenous annuity price

When annuitants' survival probabilities are heterogeneous, the equilibrium annuity price is affected by their annuitization choices, which further depend on the annuity price. Given this mutual dependence, it is generally difficult to establish uniqueness of the equilibrium. Based on similar expressions appearing in several annuity and insurance models, we obtain two results in an annuity model with heterogeneity in survival probability only. First, the equilibrium annuity price is always unique if the annuitization function is multiplicatively separable in survival probability and annuity price. Second, the equilibrium is unique for more general annuitization functions, provided that a sufficient condition on the distribution of survival probabilities holds. Many distributions, including the uniform, normal and gamma distributions, satisfy this condition.

Deferred annuities with gender-neutral pricing: Benefitting most women without adversely affecting too many men

Many countries emphasize gender equality and ban gender-based annuity pricing, leading to more heterogeneous health characteristics and market inefficiency. Governments may respond by offering deferred annuities when annuitants' health characteristics are more similar at an earlier age. The two combined policies benefit most female annuitants without adversely affecting too many male annuitants, contrasting to the well-known result that imposing gender-neutral pricing benefits all women but adversely affects all men, when only immediate annuities are available. Within each gender group, these two policy interventions benefit annuitants with average health, but may adversely affect those on either end.

Stochastic mortality model with respect to mixed fractional Poisson process: Calibration and empirical analysis of long-range dependence in actuarial valuation

Recently, many studies have adopted the fractional stochastic mortality process in characterising the long-range dependence (LRD) feature of mortality dynamics, while there are still fewer appropriate non-Gaussian fractional models to describe it. We propose a stochastic mortality process driven by a mixture of Brownian motion and modified fractional Poisson process to capture the LRD of mortality rates. The survival probability under this new stochastic mortality model keeps flexibility and consistency with existing affine-form mortality models, which makes the model convenient in evaluating mortality-linked products under the market-consistent method. The formula of survival probability also considers the historical information from survival data, which enables the model to capture historical health records of lives. The LRD feature is reflected by our proposed model in the empirical analysis, which includes the calibration and prediction of survival curves based on recent generation data in Japan and the UK. Finally, the consequent empirical analysis of annuity pricing illustrates the difference of whether this feature is involved in actuarial valuation.

Integrated Financial Strategies: Sinking Funds and Annuity Pricing in a Dynamic Market Environment

The study on "Integrated Financial Strategies: Sinking Funds and Annuity Pricing in a Dynamic Market Environment" seeks to investigate how sinking funds, annuity pricing, and dynamic market environments can be integrated within the context of integrated financial strategies. The study focuses on the following areas: (1) the application of actuarial tables and probability distributions to ascertain the proper values for annuities payments and sinking fund balances; (2) the role of the actuary science in sinking funding scheduling; (3) the effect of market factors, such as interest rates, inflation, and economic conditions, on the value and value of sinking investments; (4) the analysis of the relationship between market dynamics, including interest rates and time periods, and the value of a sinking asset, including the integration of the two financial instruments. The analysis revealed that changes in the interest rate and the number of periods directly influence the future value of the sinking assets. The integration of these financial instruments can help to optimize financial decision-making, enhance risk management, increase investment returns, improve financial planning, and lead to greater integration with other financial instruments, ultimately leading to better financial outcomes for both financial professionals and investors

Forecasting Age- and Sex-Specific Survival Functions: Application to Annuity Pricing

We introduce the function principal component regression (FPCR) forecasting method to model and forecast age-specific survival functions observed over time. The age distribution of survival functions is an example of constrained data whose values lie within a unit interval. Because of the constraint, such data do not reside in a linear vector space. A natural way to deal with such a constraint is through an invertible logit transformation that maps constrained onto unconstrained data in a linear space. With a time series of unconstrained data, we apply a functional time-series forecasting method to produce point and interval forecasts. The forecasts are then converted back to the original scale via the inverse logit transformation. Using the age- and sex-specific survival functions for Australia, we investigate the point and interval forecast accuracies for various horizons. We conclude that the functional principal component regression (FPCR) provides better forecast accuracy than the Lee–Carter (LC) method. Therefore, we apply FPCR to calculate annuity pricing and compare it with the market annuity price.

Annuity puzzle: Evidence from a Swiss pension fund

AbstractWe analyze individual annuitization decisions at retirement in an environment with mandatory participation in a funded pension pillar and low average annuity price. Using administrative data from a large Swiss insurance company over the period 2011–2015, we document that, even in this favorable environment, only 42.7% of the retirees fully annuitize, against 45% taking full lump sum. We show that individual annuitization decisions strongly respond to financial incentives, measured by the Money Worth Ratio, and to the tax rates on annuity and lump‐sum payments. Lump‐sum payments are more common among French‐ and Italian‐speaking individuals. Using Survey of Health, Ageing and Retirement in Europe data, we show that French‐ and Italian‐speaking Swiss are more likely to leave inheritance—thus suggesting that preferences for bequests may affect annuitization decisions. Finally, we provide evidence of asymmetric information, using the unused observable test, to show that individuals living in high mortality municipalities are less likely to annuitize and more likely to cash out their pension wealth.

Publisher Correction: A new approximation of annuity prices for age–period–cohort models

Publisher Correction: A new approximation of annuity prices for age–period–cohort models

A new approximation of annuity prices for age–period–cohort models

A new approximation of annuity prices for age–period–cohort models

A black box approach to fitting smooth models of mortality

Actuaries have long been interested in the forecasting of mortality for the purpose of the pricing and reserving of pensions and annuities. Most models of mortality in age and year of death, and often year of birth, are not identifiable so actuaries worried about what constraints should be used to give sensible estimates of the age and year of death parameters, and, if required, the year of birth parameters. These parameters were then forecast with an ARIMA model to give the required forecasts of mortality. A recent article showed that, while the fitted parameters were not identifiable, both the fitted and forecast mortalities were. This result holds if the age term is smoothed with P-splines. The present article deals with generalized linear models with a rank deficient regression matrix. We have two aims. First, we investigate the effect that different constraints have on the estimated regression coefficients. We show that it is possible to fit the model under different constraints in R without imposing any explicit constraints. R does all the necessary booking-keeping ‘under the bonnet’. The estimated regression coefficients under a particular set of constraints can then be recovered from the invariant fitted values. We have a black box approach to fitting the model subject to any set of constraints.

Behavior of Canadian life annuity prices

AbstractWe examine the behavior of Canadian life annuity prices by measuring how quickly and fully they respond to changes in market interest rates. The price responses, though not immediate, start relatively quickly and become more complete over time. Over the whole sample period, the responses appear to be asymmetric – annuity providers generally raise prices faster when interest rates decline than reduce prices when the opposite occurs, which is disadvantageous to annuity customers. In addition, we find unusual annuity price behavior during the period of the 2008 financial crisis.

Coherent Mortality Forecasting with a Model Averaging Approach: Evidence from Global Populations

Accurate forecasts and analyses of mortality rates are essential to many economic and finance practices, such as the designing of pension schemes. Recent studies have proposed advanced mortality models with age coherent forecasts, such that long-term predictions will not diverge infinitely among ages. Despite their effectiveness, individual models are inevitably misspecified in the empirical analysis, which reduces the reliableness. In this article, we propose a model averaging approach (MAA) that allows for age-specific weights and asymptotically achieves age coherence. Relevant technical details are also provided. The proposed method balances both the in-sample fitting and out-of-sample forecasting, with a uniquely designed smoothness penalty to resolve the potential overfitting and thus avoid abrupt changes in the long-term mortality forecasting. Using a large empirical dataset of 10 countries including 0 to 100 ages and spanning 1950 to 2018, the outstanding forecasting performance of MAA is presented. This robustly holds in various sensitivity analyses and is supported by simulation evidence. A case study is further conducted to discuss the improved forecasting efficiency of MAA, as well as its usefulness in economic and finance applications such as the annuity pricing. The proposed MAA approach is therefore a useful tool in forecasting mortality data for other practices.

The impact of simultaneous shocks to financial markets and mortality on pension buy-out prices

AbstractIn this paper, we determine the fair value of a pension buyout contract under the assumption that changes in mortality can have an impact on financial markets. Our proposed model allows for shocks to occur simultaneously in mortality rates and financial markets, so that strong changes in mortality rates can affect interest rates and asset prices. This approach challenges the common but very strong assumption that mortality and market risk drivers are independent. A simulation-based pricing framework is applied to determine the buyout premium for a hypothetical fully funded pension scheme. The results of an extensive sensitivity analysis show how buyout prices are affected by changes in mortality and financial markets. Surprisingly, we find that the impact of shocks is similar whether or not these shocks occur simultaneously or not, although there are some differences in annuity prices and buyout premiums. We clearly see that the intensity and severity of shocks, and asset price volatility play a dominant role for buyout prices.

Proyeksi Tingkat Kematian di Indonesia Menggunakan Metode Holt-Winters Smoothing Exponential dan Moving Average

Inaccurate predictions would cause the insurance companies to incur huge losses and may lead to expensive premiums for which low-income consumers are unable to insure themselves. The ability to predict mortality rates accurately allows the insurance companies to take preventive steps to introduce new policies with reasonable prices. It is hoped that by carrying out mortality projections, losses caused by longevity risk in the life insurance industry would be minimized. This study used secondary data obtained from the World Health Organization (WHO) website in the Mortality and Global Health Estimates category with the sub-topic Life Table by Country Indonesia. In this paper, several models are used to predict the mortality rate in a case study population in Indonesia, namely the Moving Average and Exponential Smoothing forecasting methods. The results obtained are the best method for predicting mortality rates is by using the Exponential Smoothing method with the MAPE value of Exponential Smoothing is smaller than the MAPE value on the Moving Average. The results of this mortality projection will later be used to obtain the distribution of life expectations and the premium price of life annuities.

Pricing life annuities for impaired lives: the case of Portugal

Life annuities markets are underdeveloped in Portugal and other countries. This annuitization puzzle is explained by improvements in mortality at old ages and passive adverse selection, reasons that put lives with diminished life expectancies at an unfair disadvantage. Using data from the SHARE (Survey of Health, Ageing and Retirement in Europe) project, we assess the impact of some of the most serious and common diseases over a reference survival curve. Then we calculate the price of annuities for impaired lives, using adjusted survival, and compare them with those for the reference population. We show that applying the reference mortality to impaired lives is very unfair and that pricing annuities for lives weakened by disease in an accurate way is possible.

Forecasting mortality rates with a coherent ensemble averaging approach

AbstractModeling and forecasting of mortality rates are closely related to a wide range of actuarial practices, such as the designing of pension schemes. To improve the forecasting accuracy, age coherence is incorporated in many recent mortality models, which suggests that the long-term forecasts will not diverge infinitely among age groups. Despite their usefulness, misspecification is likely to occur for individual mortality models when applied in empirical studies. The reliableness and accuracy of forecast rates are therefore negatively affected. In this study, an ensemble averaging or model averaging (MA) approach is proposed, which adopts age-specific weights and asymptotically achieves age coherence in mortality forecasting. The ensemble space contains both newly developed age-coherent and classic age-incoherent models to achieve the diversity. To realize the asymptotic age coherence, consider parameter errors, and avoid overfitting, the proposed method minimizes the variance of out-of-sample forecasting errors, with a uniquely designed coherent penalty and smoothness penalty. Our empirical data set include ten European countries with mortality rates of 0–100 age groups and spanning 1950–2016. The outstanding performance of MA is presented using the empirical sample for mortality forecasting. This finding robustly holds in a range of sensitivity analyses. A case study based on the Italian population is finally conducted to demonstrate the improved forecasting efficiency of MA and the validity of the proposed estimation of weights, as well as its usefulness in actuarial applications such as the annuity pricing.

Variable annuity pricing, valuation, and risk management: a survey

Variable annuity is arguably the most complex individual retirement planning product in the financial market. Its intricacy stems from a variety of product features including investment options, guaranteed benefits, withdrawal options, etc. In many ways, variable annuities can be viewed as traditional life and annuity products at the next level of sophistication with added financial options. Despite a significant amount of publications by practitioners and academics on the subject matter, there have been few research papers that systematically exploit the basic principles underlying the operation of variable annuities. This survey paper aims to fill in the gap in the literature for an overview of state-of-the-art technology and recent trends in the development of variable annuities.

Pricing of variable long-term care annuities with guaranteed lifetime withdrawal and limited hospitalisation coverage benefits

Improvements in medical technology and longevity risk increase the popularity of healthcare insurance products. This article combines several well-known healthcare products and develops a theoretical model for a variable annuity product with some specific benefits. More precisely, the new product is a variable annuity product that accompanies with long-term care coverage, limited hospitalisation coverage and a guaranteed lifelong withdrawal benefit option. Under a non-arbitrage market (the geometric Brownian motion) investment, the fair lam sum premium of such insurance products has been evaluated. Using an MCMC simulation method, a numerical study has been conducted to illustrate the capability of the product.

A Deep Learning Integrated Cairns-Blake-Dowd (CBD) Sytematic Mortality Risk Model

Many actuarial science researchers on stochastic modeling and forecasting of systematic mortality risk use Cairns-Blake-Dowd (CBD) Model (2006) due to its ability to consider the cohort effects. A three-factor stochastic mortality model has three parameters that describe the mortality trends over time when dealing with future behaviors. This study aims to predict the trends of the model, kt(2) by applying the Recurrent Neural Networks within a Short-Term Long Memory (an artificial LSTM architecture) compared to traditional statistical ARIMA (p,d,q) models. The novel deep learning (machine learning) technique helps integrate the CBD model to enhance its accuracy and predictive capacity for future systematic mortality risk in countries with limited data availability, such as Kenya. The results show that Long Short-Term Memory network architecture had higher levels of precision when predicting the future systematic mortality risks than traditional methods. Ultimately, the results can be implemented by Kenyan insurance firms when modeling and forecasting systematic mortality risk helpful in the pricing of Annuities and Assurances.

Analysis of VIX-linked fee incentives in variable annuities via continuous-time Markov chain approximation

We consider the pricing of variable annuities (VAs) with general fee structures under a class of stochastic volatility models which includes the Heston, Hull-White, Scott, α-Hypergeometric, 3/2, and 4/2 models. In particular, we analyze the impact of different VIX-linked fee structures on the optimal surrender strategy of a VA contract with guaranteed minimum maturity benefit (GMMB). Under the assumption that the VA contract can be surrendered before maturity, the pricing of a VA contract corresponds to an optimal stopping problem with an unbounded, time-dependent, and discontinuous payoff function. We develop efficient algorithms for the pricing of VA contracts using a two-layer continuous-time Markov chain approximation for the fund value process. When the contract is kept until maturity and under a general fee structure, we show that the value of the contract can be approximated by a closed-form matrix expression. We also provide a quick and simple way to determine the value of early surrenders via a recursive algorithm and give an easy procedure to approximate the optimal surrender surface. We show numerically that the optimal surrender strategy is more robust to changes in the volatility of the account value when the fee is linked to the VIX index.

Joint and survivor annuity valuation with a bivariate reinforced urn process

We introduce a novel way of modeling the dependence of coupled lifetimes, for the pricing of joint and survivor annuities. Using a well-known Canadian data set, our results are analyzed and compared with the existing literature, mainly relying on copulas. Based on urn processes and a one-factor construction, the proposed model is able to improve its performances over time, in line with the machine learning paradigm, and it also allows for the use of experts' judgements, to complement the empirical data.