Value Of Annuity Research Articles

Pension funds with longevity risk: an optimal portfolio insurance approach

We present a unified framework designed to provide an optimal investment strategy for members of a defined contribution pension plan. Our model guarantees a minimum retirement savings level, expressed as a target annuity, by assuming uncertainty in interest rates, labor income, and mortality during the accumulation phase. To protect the accumulated retirement capital against both investment and longevity risks, the present value of the guaranteed lifetime annuity is regarded as the baseline wealth to hold upon reaching the retirement date, while a purpose-oriented proportion portfolio strategy is employed to invest the residual wealth. By applying standard dynamic programming techniques, we determine a closed-form solution to the stochastic control problem with the objective of maximizing the expected utility of the final surplus, defined as the difference between the accumulated wealth and the target annuity value. The theoretical findings are bolstered by a comprehensive numerical analysis designed to assess the impact of longevity on investment policies, highlighting the suitability of our proposal for managing defined contribution schemes.

Uniform Distribution of Deaths, Fractional Independence, and Negative Payment-Frequency*

Under the assumption of a uniform distribution of deaths (UDD), the expected present value of a life annuity with level payments payable m times a year can be expressed in terms of that payable once a year, together with some quantities that depend on interest only. By studying the expected annuity cashflows in the year of death, we determine the financial meaning of these interest-only quantities. Our analysis can be extended to the more general assumption of fractional independence (FI). Under the FI assumption, we express premium and reserve formulas for insurance with premiums payable m times each year in terms of those with premiums payable only once each year. We also discuss an intriguing concept called negative payment-frequency.

Analysis Of Future Health Insurance Premium Reserve Calculations Using The Commissioners Method With Woolhouse Formula

A premium reserve is the amount of funds that the insurance company needs to prepare for the payment of liability benefits when a claim occurs. One method of calculating premiums is the commissioners method which is an extension of the prospective reserve method.The main factor in the actuarial calculation is the mortality rate that can be determined using the WOOLHOUSE mortality law. This study aims to determine the size of the life insurance premiums reserve using the commisioners method and the WOLHOUSE mortality laws. The calculation of the premium reserve is linked to the determination of a person's chances of survival over a period of years of WOALHOUSE law which is then used to find the value of the insurance, the initial annuity value, the annual premium, the modified premiums based on the Commissioners method, and the amount of the premie reserve at the end of the year. The study was conducted on a 25-year-old woman who was enrolled in a dwiguna life insurance program with a 25 year liability period and a 23 year premium payment period at a 4% interest rate with a liability of Rs. 500,000,000. The results of the analysis showed that the large reserves of premiums obtained at the end of the period of liability with the Commissioners method and the WOOLHOUSE mortality law are equal to the given reservation value, so that the insurance company is prepared to give the amount of reservation promised to the policyholder.

ANALISIS PREMI TAHUNAN INDIVIDU DENGAN MANFAAT YANG DIBAYARKAN PADA AKHIR TAHUN KEMATIAN

Life insurance is an insurance product that provides guarantees in the form of benefits to the insured's family if the insured dies in the future. The amount of this benefit is influenced by the amount of the annual premium paid by the insurer to the insured. The researcher will calculate the amount of the individual's annual premium on endowment insurance. The researcher will use a fixed interest rate to calculate the discount factor and the Gompertz distribution approach to calculate the insurer's chance of survival. Previously, the researcher will estimate the Gompertz distribution parameters using the Maximum Likelihood Method (MLE) based on the 2011 Indonesian Mortality Table (female). Furthermore, the researcher will calculate the insurer's chance of survival using the Gompertz distribution to obtain the actuarial present value of the initial term annuity and the actuarial present value of the endowment life insurance. In addition, the researcher will calculate the comparison of the present value of the endowment life insurance to the present value of the life annuity. The last step is that the researcher will multiply the comparison of this present value by the benefits received to calculate the amount of the individual's annual premium. The calculation results show that the comparison of the present value of dual-purpose life insurance to the present value of a living annuity is getting smaller as the policy coverage period increases. As a result, the amount of the individual's annual premium will also be smaller. This shows that the amount of the annual premium is influenced by interest rate factors, the insurer's life chances, and the length of the coverage period.

Some Applications of Series Theory in Solving Economic Problems

The paper provides a formula for finding the present value of the payments stream, annuity. The application of the formula of the sum of an infinitely decreasing geometric progression to find the present value of infinite annuity is considered. The concepts of convergence, divergence of series, a necessary condition for convergence, and a sign of comparison are also given in this paper.

Annuities for Lost Household Services Using Healthy Life Tables

Abstract Forensic economists have used Healthy Life Expectancy: 2018 Tables (Kurt V. Krueger, 2020) in valuing the loss of household or personal care services due to injury or death of their supplier. The tables are used to decide how far into the future to project the loss. We focus on two easily avoided errors that may arise in that decision. First, the present value (PV) of a healthy life annuity (HLA ending when death occurs) falls short of the PV of an annuity certain for the healthy life expectancy (HLE), at any positive interest rate. Second, services are lost only during the combined healthy life of the supplier and life of the recipient. The expected PV of the supplier’s healthy life and the recipient’s whole life falls short of either. The paper illustrates the size of these potential errors for various ages of the two people and net interest rates of 0-3%; the errors can be significant. We provide an Excel workbook at the Journal of Forensic Economics website that allows the user to avoid both errors for calculating the PV of relevant (healthy) life annuities, for any of the demographic groups covered in the HLE tables. The paper gives formulas used for the present values of the life tables.

Mathematical Derivations of Actuarial Present Value for the Fully Continuous Whole Life Assurances from the Theoretical Market Price

Many current mortality tables are computationally Makehamised but technically devoid of continuous key life table functions such as and because of the underlying sophisticated mortality functions and the complex methods of computation which usually do not incorporate the market price. The computationally advanced Moore’s model, which does not endorse the market pricing mechanisms offers one of the most complex analytical results that seems not user friendly in actuarial literature. As a result of the Moore’s complexity, we offer the Gradshteyn and Ryzhik’s analytic integral as an alternative solution to examine the contingency issues involving the estimation of actuarial present values of the continuous whole life annuities. In this study, the objective are to (i) estimate the parameters of GM (1,2) through the method of equidistant points according to the natural order of human age (ii) Apply the mean value theorem to construct the survival probability function under the framework of policy modifications (iii) Develop a closed form pricing formula for the continuous whole life annuity and continuous whole life insurance. The Gradshteyn and Ryzhik’s analytic integral has eliminated the Moore’s complexities associated with symbolic computations such that the continuity assumption applied allows us to compute the continuous life annuity with expedience. Computational evidence further shows that the continuous life annuity reduces with age confirming that life annuities is a decreasing function.

Valuing Constant-Growth Perpetuities with Finance Intuition (Instead of Infinite Series)

The formula for calculating the present value of a finite level annuity is one of the simplest equations in corporate finance. It is PV = PMT/r, where PV is the present value, PMT is the payment, and r is the periodic interest rate. However, corporate finance instructors and textbooks rarely discuss the derivation of this equation. This is because the typical derivation requires a knowledge of calculus beyond that of most business majors. In this paper, I demonstrate that it is possible to use basic ideas from finance – how competitive markets set the price of an asset equal to the asset’s market value – at the same place in the derivation that usually requires taking limits.

On stochastic modelling of actuarial compensation for loss of future earnings in Turkey

Abstract This article presents an actuarial perspective to estimate the economic loss for compensations resulting from a wrongful act. The main aim of this study is to propose a stochastic approach for modelling discount rates and survival probabilities which are used to calculate the present value of the future loss of earnings. We adopted the Wilkie model as an economic scenario generator to forecast future stochastic discount rates and fitted the Lee–Carter and the Heligman–Pollard models to forecast future age-specific survival probabilities using Turkish data. We discussed the impact of stochastic modelling on annuity values and compensation amounts through sensitivity analyses. We also compared the compensation amounts for various cases with the benchmark case representing current practice in Turkey. The results show that the compensation amounts are substantially affected by the financial assumptions, particularly in the case of an unstable economy. The major finding is that the uncertainty in future mortality rates is largely outweighed by the uncertainty associated with future financial and economic variables in the Turkish case.

Valuation of general GMWB annuities in a low interest rate environment

Variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) entitle the policy holder to periodic withdrawals together with a terminal payoff linked to the performance of an equity fund. In this paper, we consider the valuation of a general class of GMWB annuities, allowing for step-up, bonus and surrender features, taking also into account mortality risk and death benefits. When dynamic withdrawals are allowed, the valuation of GMWB annuities leads to a stochastic optimal control problem, which we address here by dynamic programming techniques. Adopting a Hull-White interest rate model, correlated with the equity fund, we propose an efficient tree-based algorithm. We perform a thorough analysis of the determinants of the market value of GMWB annuities and of the optimal withdrawal strategies. In particular, we study the impact of a low/negative interest rate environment. Our findings indicate that low/negative rates profoundly affect the optimal withdrawal behaviour and, in combination with step-up and bonus features, increase significantly the fair values of GMWB annuities, which can only be compensated by large management fees.

Some Notes on Payments Streams

The article discusses such concepts as the payment’s streams and financial annuities, credit accounts. The generalizing characteristics of payments streams, the increased amount of constant annuity postnumerando are given. The formulas of the incremented sum are considered, as well as the current value of constant annuities postnumerando.

Measurement and Impact of Longevity Risk in Portfolios of Pension Annuity: The Case in Sub Saharan Africa

Longevity is without a doubt on the rise throughout the world due to advances in technology and health. Since 1960, Ghana’s average annual mortality improvement has been about 1.236%. This poses serious longevity risks to numerous longevity-bearing assets and liabilities. As a result, this research investigates the effect of mortality improvement on pension annuities related to a particular pension scheme in Ghana. Different stochastic mortality models (Lee–Carter, Renshaw–Haberman, Cairns–Blake–Dowd, and Quadratic Cairns–Blake–Dowd) are used to forecast mortality improvements between 2021 and 2030. The results from accuracy metrics indicate that the quadratic Cairns–Blake–Dowd model exhibits the best fit to the mortality data. The findings from the study demonstrate that mortality for increasing ages within the retirement period was declining, with increasing improvement associated with increasing ages. Furthermore, the forecasts were used to estimate the associated single benefit annuity for a GHS 1 per annum payment to pensioners, and it was discovered that the annuity value expected to be paid to such people was not significantly different regardless of the pensioner’s current age.

Guessing, math, or something else? Lay people's processes for valuing annuities

AbstractResearchers have long been trying to understand why individuals dislike annuities. Here, we investigate if the process individuals use to assess the financial value of annuities may lead them to inaccurately value annuities. In Study 1, participants were asked to assess the monthly payments associated with a specific annuity lump sum or the annuity lump sum associated with a specific monthly payment. They were then asked to describe how they arrived at their answers. We find that when making this assessment, 42% of participants report attempts at using math, with some even describing mathematical formulas. Most other participants reported guessing instead. Reporting attempts at math is more common among participants with higher financial literacy and numeracy. Reported attempts at math, financial literacy, and numeracy predict arriving at more realistic financial values for annuities, as well as incorporating assessments of life expectancy in the math. Based on this process knowledge, we then designed an experiment in Study 2 and tested the effect of presenting information about life expectancy, providing feedback about payouts or their combination. We find that we can thereby change the assessed financial value of annuities and increase participants' interest in annuities, especially among participants that reported attempts at using math. Understanding the processes individuals use to assess the value of annuities informs theory and practice.

MULTI-STATE MODEL FOR CALCULATION OF LONG-TERM CARE INSURANCE PRODUCT PREMIUM IN INDONESIA

Long Term Care (LTC) insurance is a type of health insurance. One of the LTC products is Annuity as A Rider Benefit. This insurance provides benefits for medical care costs during the term and death benefits if the insured dies. This insurance product can be modeled with a multi-state model. The multi-state model is a stochastic process in which the subject can switch states at a specified number of states. This paper discusses the calculation of LTC insurance premiums with the Annuity as A Rider Benefit product using a multi-state model for critically ill patients in Indonesia. The state used consisted of eight states, namely healthy, cancer, heart disease, stroke, died from the illness from each disease, and died from others. The premium calculation also utilized Markov chain transition probabilities. The data used were data on Indonesia's population in 2018, data on the prevalence of cancer, heart disease, stroke, and Indonesia's 2019 mortality table. The stages of this study were calculating the net single premium value, benefit annuity value, and insurance premium value. The case study was conducted on a 25 years old male in good health following LTC insurance with a coverage period of 5 years. It was known that the compensation value for someone who dies was IDR 100,000,000 and the interest rate used was 5%. The calculation results obtained an annual premium of IDR 5,308,915 which was then varied based on gender and varied interest. Insurance premiums for men were more expensive than for women since men had a greater chance of dying. Then, the higher the interest rate taken; the lower premium paid. This was because the interest rate is a discount variable.

Less-expensive long-term annuities linked to mortality, cash and equity

AbstractThis paper proposes a shift in the valuation and production of long-term annuities, away from the classical risk-neutral methodology towards a methodology using the real-world probability measure. The proposed production method is applied to three examples of annuity products, one having annual payments linked to a mortality index and the savings account and the others having annual payments linked to a mortality index and an equity index with a guarantee that is linked to the same mortality index and the savings account. Out-of-sample hedge simulations demonstrate the effectiveness of the proposed less-expensive production method. In contrast to classical risk-neutral production, which revolves around the savings account as reference unit, the long-term best-performing portfolio, the numéraire portfolio of the equity market, is employed as the fundamental reference unit in the production of the annuity. The numéraire portfolio is the strictly positive, tradable portfolio that when used as denominator or benchmark makes all benchmarked non-negative portfolios supermartingales. Under real-world valuation, the initial benchmarked value of a benchmarked contingent claim equals its real-world conditional expectation. The proposed real-world valuation and production can lead to significantly lower values of long-term annuities and their less-expensive production than suggested by the risk-neutral approach.

Polynomial Annuities

We use a payment pattern of the type {1k,2k,3k,…} to generalize the standard level payment and increasing annuity to polynomial payment patterns. We derive explicit formulas for the present value of an n-year polynomial annuity, the present value of an m-monthly n-year polynomial annuity, and the present value of an n-year continuous polynomial annuity. We also use the idea to extend the annuities to payment patterns derived from analytic functions, as well as to payment patterns of the type {1r,2r,3r,…}, with r being an arbitrary real number. In the process, we develop possible approximations to k! and for the gamma function evaluated at real numbers.

Modification of the Lee-Carter mortality model and its application in the pension scheme

In this paper, a new approach to mortality forecasting is proposed based on an improved model of the Lee-Carter type. The standard Lee-Carter model and its modified version were introduced and compared using mortality data for Poland and some other European countries. Forecasts of log-central age-specific death rates were then derived and used to predict death probabilities and life expectancies for males and females in Poland, which are the main parameters of the so-called dynamic life tables (also known as mortality tables). The application of the proposed methodology in calculations of the present values of future pension annuities is presented in the article. Scenarios of monthly pensions obtained with the use of dynamic life tables were considered and compared with analogous scenarios based on the static (period) life tables published every year by the Central Statistical Office of Poland.

Optimal fee structure of variable annuities

We study the design of fee structures of variable annuities as a stochastic control problem, in which an insurer is allowed to choose the fee structure in any form that satisfies the budget constraint, and seeks an optimal one to maximize its business objective. Under the no surrender assumption, we show that the optimal fee structure is of barrier type with a time-dependent free boundary. The insurer's optimal strategy is to charge fees if and only if the account value of variable annuities hits the free boundary from below.

Optimal Fee Structure of Variable Annuities

We study the design of fee structures of variable annuities as a stochastic control problem, in which an insurer is allowed to choose the fee structure in any form, and seeks an optimal one to maximize her expected discounted net profit. We obtain a semi-explicit characterization result for the optimal fee structure which is of barrier type with a free boundary. The insurer's optimal strategy is to charge fees if and only if the account value of variable annuities hits the free boundary from below, which differs from the constant fee structure that is commonly used in the industry, and other state- and time-dependent fee structures that have been proposed in the literature.

Mandatory annuitization and money's worth: evidence from Singapore

AbstractThis paper examines the impact of uncertainties in the future trends of mortality on annuity values in Singapore's compulsory purchase market. We document persistent population mortality improvement trends over the past few decades, which underscores the importance of longevity risk in this market. Using the money's worth framework, we find that the life annuities delivered expected payouts valued at 1.019–1.185 (0.973–1.170) per dollar of annuity premium for males (females). Even in a low mortality improvement scenario, the annuities provide an expected value exceeding 0.950. This suggests that participants in the national annuity pool have access to attractively priced annuities, regardless of sex, product, and premium invested.