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Working Papers

2023May
Age-Dependent Multi-Cohort Affine Mortality Model with Cohort Correlation

Yuxin Zhou, Len Patrick Garces, Yang Shen, Michael Sherris and Jonathan Ziveyi

Abstract: Continuous-time affine mortality models are useful in the analysis of age-cohort mortality rates as they yield a closed-form expression for survival curves which are consistent with the dynamics of latent factors driving mortality and are well-suited for finance and insurance applications. We extend and improve these mortality models by introducing age dependence of mortality rates and correlation between cohorts. We propose and compare two classes of age-dependent mortality models, namely the age-dependent coefficient model and the age-dependent factor model. Specifically, we assess both Gaussian and CIR-type models for each category of age-dependent models. Both categories of age-dependent models involve age and calendar time, which in turn specifies the cohort. Thus, our models admit an analytical form for the instantaneous correlation between mortality rates of different cohorts. Moreover, we propose two improvements to the parameter estimation process. First, to improve the estimation of cohort correlations, we regularise the parameter estimation by adding a penalty term which penalises larger differences between empirical and estimated correlations. Second, we develop and assess a method to include incomplete cohorts into the Kalman filtering algorithm for parameter estimation. We calibrate the mortality models to data from multiple countries which include Australia, Denmark, UK, and USA to assess and compare in-sample fit and forecasting performance. By incorporating age dependence and using incomplete cohort mortality data, we improve the goodness of fit and produce more reasonable out-of-sample forecasts of survival probabilities. We also show that regularisation produces more realistic correlations between cohorts for varying age differences. Our results show that, under most circumstances, the correlation between cohorts decreases as the age difference increases.

Keywords: Mortality model, Age-dependent, Multi-cohort, Cohort correlation, Incomplete cohort, Affine, Regularisation

 

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2023Apr
An Augmented Variable Dirichlet Process Mixture Model for the Analysis of Dependent Lifetimes

Francesco Ungolo and Patrick J. Laub

Abstract: The analysis of insurance and annuity products issued on multiple lives requires the use of statistical models which account for lifetime dependence. This work presents a Dirichlet Process Mixture-based approach which allows to model dependent lifetimes within a group, such as married couples, accounting for individual as well as group-specific covariates. The model is analysed in a fully Bayesian setting, and illustrated to jointly model the lifetime of male-female couples in a portfolio of joint and last survivor annuities of a Canadian life insurer. The inferential approach allows to account for right censoring and left truncation, which are common features of data in survival analysis. The model shows improved in-sample and out-of-sample performance compared to traditional approaches assuming independent lifetimes, and offers additional insights into the determinants of the dependence between lifetimes and their impact on joint and last survivor annuity prices.

Keywords: Dependent lifetimes, Survival Analysis, Dirichlet Process, Bayesian analysis, Life insurance, MCMC, Mixture models

 

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2023Apr
A Dirichlet Process Mixture Regression Model for the Analysis of Competing Risk Events

Francesco Ungolo and Edwin R. van den Heuvel

Abstract: We develop a regression model for the analysis of competing risk events. The joint distribution of the time to these events is characterized by a random effect following a Dirichlet Process, explaining their variability. This entails an additional layer of flexibility of this joint model, whose inference is robust with respect to the misspecification of the distribution of the random effects. The model is analysed in a fully Bayesian setting, yielding a flexible Dirichlet Process Mixture model for the joint distribution of the time to events. An efficient MCMC sampler is developed for inference. The modelling approach is applied to the empirical analysis of the surrending risk in a US life insurance predictive performance of the surrending rates.

Keywords: Competing Risks, Survival Analysis, Dirichlet Processes, Bayesian analysis, Lapse risk, MCMC

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2023Mar
Data analysis
AffineMortality: An R package for estimation, analysis and projection of affine mortality models

Francesco Ungolo, Len Patrick Dominic M. Garces, Michael Sherris, and Yuxin Zhou

Abstract: This paper presents the AffineMortality R package which performs parameter estimation, goodness of fit analysis, simulation and projection of future mortality rates for a set of affine mortality models for use in pricing and reserving. The computational routines build on the univariate Kalman Filtering approach of Koopman & Durbin (2000) along other numerical methods to enhance the robustness of the results. This paper provides a discussion of how the package works in order to effectively estimate and project the models, and describes the available functions. Illustration of the package for mortality analysis of the US HMD dataset is provided.

Keywords: Longevity Risk, Kalman Filter, State-space models, Affine mortality

 

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2023Mar
Nonlinear Means-Tested Pensions: Welfare and Distributional Analyses

Daniel Wheadon, Gonzalo Castex, George Kudrna and Alan Woodland

Abstract: Several countries, including Australia, have a means-tested public age pension. Means testing the age pension can reduce the overall fiscal burden relative to a universal pension, but can also distort households’ incentives to work and save. Policymakers can influence the sizes of these distortions by adjusting the structure of the pension function (e.g., the withdrawal rate of the pension). In contrast with the standard piece-wise linear means test, we introduce a class of non-linear means tests that contain the standard linear test as a special case and allow for progressive or regressive tests in which the withdrawal rate respectively increases or decreases as means increase. To identify the socially optimal nonlinear income-tested pension function, we develop an overlapping generations model of a small open economy with heterogeneous agents with stochastic wage and mortality profiles. We find that the optimal nonlinear income test is strongly regressive with a low average withdrawal rate as income increases.

Keywords: Population aging, Sustainability, Social security, Means testing, Redis- tribution, Overlapping generations, Dynamic general equilibrium.

 

 

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2023Mar
George Kudrna
The Economy-wide Effects of Mandating Private Retirement Incomes

George Kudrna

Abstract: This paper investigates the economy-wide effects of mandating private pensions. Drawing on Australia’s Superannuation Guarantee (SG) legislation, which mandates contributions to private retirement (superannuation) accounts, our objective is to quantify the long-term effects of the SG mandate on households’ economic decisions, welfare, and macroeconomic and fiscal indicators. We begin with the partial equilibrium (PE) life-cycle analysis that considers private (liquid) and superannuation (illiquid) assets, highlighting the interactions of the SG mandate with income taxation, public pensions, and bequest re- distribution. We then develop a general equilibrium (GE) model that includes overlapping generations of heterogenous households, labor income and survival risks, and both types of household assets. The model is calibrated using Australian data and incorporates a detailed representation of government policy, including mandatory superannuation. The simulation results indicate that higher SG rates lead to significantly greater household wealth, output and consumption per capita, and household welfare across the skill distribution in the long run. These positive effects are due to (combination of) increased tax subsidies, more binding means testing reducing public pensions, redistribution of increased accidental bequests and also GE effects on factor prices (with higher gross wage rates).

Keywords: Private Pension, Social Security, Income Taxation, Labor Supply, Life- Cycle, General Equilibrium

 

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2023Feb
Estimation, Comparison and Projection of Multi-factor Age-Cohort Affine Mortality Models

Francesco Ungolo Len Patrick Dominic M. Garces, Michael Sherris and Yuxin Zhou

Abstract: Affine mortality models, developed in continuous time, are well suited to longevity risk applications including pricing and capital management. A major advantage of this mortality modelling approach is the availability of closed-form cohort survival curves, consistent with the assumed time dynamics of mortality rates. This paper makes new contributions to the estimation of multi-factor continuous-time affine models including the canonical Blackburn-Sherris, the AFNS and the CIR mortality models. We discuss and address numerical issues with model estimation. We apply the estimation methods to age-cohort mortality data from five different countries, providing insights into the dynamics of mortality rates and the fitting performance of the models. We show how the use of maximum likelihood with the univariate Kalman filter turns out to be faster and more robust compared to traditional estimation methods which heavily use large matrix multiplication and inversion. We present graphical and numerical goodness-of-fit results and assess model robustness. We project cohort survival curves and assess the out-of-sample performance of the models for the five countries. We con- firm previous results, by showing that, across these countries, although the CIR mortality model fits the historical mortality data well, particularly at older ages, the canonical and AFNS affine mortality models provide better out-of-sample performance. We also show how these affine mortality models are robust with respect to the set of age-cohort data used for parameter estimation. R code is provided.

Keywords: Longevity Risk, Kalman filter, State-space models, Affine mortality models

 

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2023Jan
Loretti Dobrescu
A Fast Upper Envelope Scan Method for Discrete-Continuous Dynamic Programming

Loretti I. Dobrescu and Akshay Shanker

Abstract: We introduce a fast upper envelope scan (FUES) method to solve and estimate dynamic programming problems with discrete and continuous choices. FUES builds on the standard endogenous grid method (EGM). EGM applied to problems with continuous and discrete choices, however, does not by itself generate the optimal solution, as the first order conditions used to retrieve the endogenous grid are necessary but not sufficient. FUES sequentially checks EGM candidate solution points and eliminates those not on the upper envelope of the value correspondence by only allowing discontinuities in the policy function at non-concave regions of the value correspondence. Unlike previous methods used to perform EGM in discrete-continuous dynamic models, FUES does not require the monotonicity of the policy functions. It is also computationally efficient, straightforward to implement, and for sufficiently large EGM grid sizes, guaranteed to recover the optimal solution.

Key Words: discrete and continuous choices, non-convex optimization, Euler equation, computational methods, dynamic programming

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2023Jan
Ageing data
Disability and Morbidity among US Birth Cohorts, 1998-2018: A Multidimensional Test of Dynamic Equilibrium Theory

Tianyu Shen and Collin Payne

Abstract: A substantial body of prior research has explored patterns of disability-free and morbidity-free life expectancy (LE) among older populations. However, these distinct facets of later-life health are almost always studied in isolation, even though they are very likely to interact with each other. Using data from the US Health and Retirement Study (HRS) and a multistate life table approach, this paper explores the interactions between disability, morbidity, and mortality among four successive US birth cohorts, born from 1914-1923 to 1944-1953. These 10-year cohorts are compared in the periods 1998-2008 and 2008-2018. The LE and health expectancies (HEs) are calculated via demographic microsimulation, and are modelled separately by sex, educational attainment and race/ethnicity. We find little compression of disability but a substantial expansion of morbidity across cohorts in each of the three age ranges. Investigating interactions between morbidity and disability, we find that disability-free life expectancy (DFLE) among those living with chronic morbidities has increased, but that at the population level DFLE is largely unchanged across successive cohorts. Investigating patterns in population sub-groups, we find that less advantaged populations (low educated and non- white groups) live substantially fewer years free of disabilities or chronic morbidities. Broadly, these patterns suggest that the link between chronic morbidities and disability has weakened over time in the US population. However, at the population level, successive cohorts are spending fewer years of life free of both chronic morbidities and disability.

This paper has been published in SSM-Population Health. For the peer-reviewed paper, please refer to https://doi.org/10.1016/j.ssmph.2023.101528

 

Suggested citation (APA): Shen, T., & Payne, C. F. (2023). Disability and morbidity among US birth cohorts, 1998–2018: A multidimensional test of dynamic equilibrium theory. SSM - Population Health, 101528. https://doi.org/https://doi.org/10.1016/j.ssmph.2023.101528

 

Keywords: Morbidity, Disability, Aging, Dynamic equilibrium, Health expectancy

 

 

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