Reversionary Annuities Research Articles

THE BENEFITS OF FAMILY ANNUITY CALCULATION WITH VINE’S COPULA AND FUZZY INTEREST RATE

One example of a multiple life annuity product (covering more than one person) is a reversionary annuity, which is a life annuity product for two or more annuitants whose annuity payments will begin after one of the annuitants specified in the contract dies first until the other annuitant also dies. This type of annuity is modified into a family annuity consisting of husband, wife, and child. The marginal distribution is constructed from a combined model of several mortality models such as Heligman-Pollard, Costakis, and Kannisto-Makeham models to capture mortality at young and old ages.This study takes this dependency into account when modeling the joint distribution of remaining life expectancy between the parties. The joint distribution of remaining lifetime between annuitants is modeled with a Vine’s copula constructed from the marginal distribution of each annuitant. This research also takes account the actuarial margin rate using BI-7-day (reverse) repo rate data estimated with fuzzy sets. The annuity benefits calculation is assumed with some Kendall's tau () values. The result shows the value of annuity benefits increases as the value of increases.

Adam Smith's reversionary annuity: money's worth, default options and auto-enrollment

When Adam Smith – author of Wealth of Nations (1776) and Theory of Moral Sentiments (1759) – was elected a professor at the University of Glasgow in 1751, he also joined an annuity ‘scheme’ that was unique for its time. The Scottish Ministers’ Widows’ Fund, as it was known, offered members of the Presbyterian Church as well as the university a choice of levels at which to contribute investment savings, ranging from 2 to 10 percent of their wages. The life-contingent benefits were in the form of a reversionary annuity to a spouse and/or lump sum death benefit to children. This article (i) describes the scheme in financial and actuarial terms, (ii) values Smith's reversionary annuity and (iii) examines the choices made by individual participants. The specific research contribution is to compile the archival data to measure the extent of insurance anti-selection and to demonstrate that debates around choice architecture, default options and auto-enrollment, which infuse the literature in the twenty-first century, were prevalent in the mid eighteenth. For the record, Adam Smith actively contributed at the highest allowed rate, but it wasn't a ‘good’ investment for him, either ex ante or ex post. As for why, one must read the article.

Adam Smith's Reversionary Annuity: Money's Worth, Default Options and Auto Enrollment

Adam Smith's Reversionary Annuity: Money's Worth, Default Options and Auto Enrollment

Penentuan Premi Tunggal Bersih pada Reversionary Annuity untuk Pasangan Suami Istri dengan Model Frank’s Copula

One of the multi-life annuity products is a reversionary annuity, a life annuity product for two insured people. The annuity payment for this product will begin after one of the insured specified in the contract dies first until the other insured dies as well. The calculation of the annuity premium is usually done by assuming independence between the random variable of remaining life-times of the insured parties. However, this is not relevant to the actual situation because the husband and wife are interrelated with their lives. This study considered this relationship when modelling the joint distribution of the remaining life-times between husband and wife. Frank’s copula was used to model the joint distribution of the remaining life-times of husband and wife. It was built from marginal distribution, which was assumed to follow the mortality value in the 2019 Indonesian Mortality Table IV. The mortality value at non-integer ages was assumed to follow a uniform distribution (uniform distribution of death). Furthermore, the joint survival distribution modelled by copula was constructed exactly when husband and wife are married. This study also considered the net single premium of reversionary annuity for several beneficiary cases. In general, the results of the calculation of the net single premium with Frank’s copula produced a lower value than the calculation of the assumption of independence for all beneficiary cases. In addition, the purchase time of an annuity also affects the net single premium, which increases up to a certain point of marriage age and then decreases thereafter.

Spouses’ Dependence across Generations and Pricing Impact on Reversionary Annuities

This paper studies the dependence between coupled lives, i.e., the spouses’ dependence, across different generations, and its effects on prices of reversionary annuities in the presence of longevity risk. Longevity risk is represented via a stochastic mortality intensity. We find that a generation-based model is important, since spouses’ dependence decreases when passing from older generations to younger generations. The independence assumption produces quantifiable mispricing of reversionary annuities, with different effects on different generations. The research is conducted using a well-known dataset of double life contracts.

Love and death: A Freund model with frailty

We introduce new models for analyzing the mortality dependence between individuals in a couple. The mortality risk dependence is usually taken into account in the actuarial literature by introducing special copulas with continuous density. This practice implies symmetric effects on the remaining lifetime of the surviving spouse. The new model allows for both asymmetric reactions by means of a Freund model, and risk dependence by means of an unobservable common risk factor (or frailty). These models allow for distinguishing in the lifetime dependence the component due to common lifetime (frailty) from the jump in mortality intensity upon death of spouse (Freund model). The model is applied to the pricing of insurance products such as joint life policy, last survivor insurance, or contracts with reversionary annuities. A discussion of identification is also provided.

Evolution of Coupled Lives' Dependency Across Generations and Pricing Impact

This paper studies the dependence between coupled lives - both within and across generations - and its effects on prices of reversionary annuities in the presence of longevity risk. Longevity risk is represented via a stochastic mortality intensity. Dependence is modelled through copula functions. We consider Archimedean single and multi-parameter copulas. We and that dependence decreases when passing from older generations to younger generations. Not only the level of dependence but also its features - as measured by the copula - change across generations: the best-fit Archimedean copula is not the same across generations. Moreover, for all the generations under exam the single-parameter copula is dominated by the two-parameter one. The independence assumption produces quantifiable mispricing of reversionary annuities. The misspecification of the copula produces different mispricing effects on different generations. The research is conducted using a well-known dataset of double life contracts.

Evolution of Coupled Lives' Dependency Across Generations and Pricing Impact

This paper studies the dependence between coupled lives - both within and across generations - and its effects on prices of reversionary annuities in the presence of longevity risk. Longevity risk is represented via a stochastic mortality intensity. Dependence is modelled through copula functions. We consider Archimedean single and multi-parameter copulas. We find that dependence decreases when passing from older generations to younger generations. Not only the level of dependence but also its features - as measured by the copula - change across generations: the best-fit Archimedean copula is not the same across generations. Moreover, for all the generations under exam the single-parameter copula is dominated by the two-parameter one. The independence assumption produces quantifiable mispricing of reversionary annuities. The misspecification of the copula produces different mispricing effects on different generations. The research is conducted using a well-known dataset of double life contracts.

The emergence of profit in life insurance

Traditionally, the mortality tables used in life insurance have margins of safety built into them, and profit can, therefore, be expected to emerge over the life of a portfolio of business. In this paper life insurance policies are modelled by means of time-inhomogeneous Markov chains, and the paper examines some of the stochastic properties of the gains attributable to the various forces of transition. A reversionary annuity serves as an illustrating example.

The mathematician Rehuel Lobatto advocates life insurances in the Netherlands in the period 1830-1860.

Summary In 1807 the first life insurance society was established in The Netherlands. In the second half of the century, life insurance societies underwent considerable expansion. During the intervening period, the lines had to be laid along which this new phenomenon was to develop in the future: between 1827 and 1830, the government started discussing the nature of its responsibility in this field and the kind of policy to be developed, and in 1830, a book on the organization of life insurance societies, the calculation of life annuities and widows' fund premiums was published, written by the mathematician Rehuel Lobatto. This book played an important role in the government's discussion. Royal Decrees which prescribed government approval for the establishment of life assurance societies were promulgated in 1830, 1833 and 1840. In 1832, Lobatto became the government's scientific adviser on the assessment of the calculations performed by these societies, and in the same year, he was also appointed adviser to the first life insurance society. From 1832 until his death in 1866 he advised the company on the use of life tables for life as well as for reversionary annuities, and he calculated the premiums based on these life tables. Another decree was promulgated in 1864 prescribing exactly which life tables were to be used. Because Lobatto probably played a part in this decree, he was responsible for a very ‘conservative’ government policy, which was no longer adequate in the second half of the century.

An Approximation to the Value of the Reversionary Annuity ax+t—ax:x+1 with Consequent Simplification in the Valuation of Last-Survivor Annuities

An Approximation to the Value of the Reversionary Annuity <i>a</i>_<i>x+t</i>—<i>a</i>_<i>x:x</i>+1 with Consequent Simplification in the Valuation of Last-Survivor Annuities

An Approximation to the Values of Joint-Life and Reversionary Annuities Involving the Use of Two Tables of Mortality

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Scottish Bankers' Mortality and Marriage Experience, 1903–1923

Synopsis of PaperInvestigation into experience from 1903 to 1923 of five Scottish Banks' Widows' Funds; Bachelors, Married Men + Widowers, Wives and Widows, dealt with separately for mortality (total exposed to risk 99,354, 1305 deaths); Bachelors and Widowers separately for marriage experience (757 marriages).Description pf methods of obtaining Exposed to Risk, etc., with special reference to exclusion of War period for younger male lives.Males:—Low mortality at younger ages ; same improvement not maintained at older ages ; Bachelor mortality much higher than for Married Men at all ages. Females :—Exceptionally light mortality at younger ages ; improvement continued to some extent to oldest ages. Comparisons made with Hewat's 1893 experience.Marriage:—Rates for Bachelors less than Hewat's at early ages, higher at principal marrying ages, and then again lower. Rates for Widowers much higher than for Bachelors. Difference between ages of husband and wife at marriage of Bachelors three years less than by Hewat.Comparisons of Monetary Values—Whole Life Annuities and Annuities to Widows by first and second Marriages of Bachelors, etc.—with values by Hewat and Huie.Valuation of Fund by new tables compared with valuation by Hewat and Huie ; reserves very similar for old-established Fund ; liabilities for young Bachelor entrants lighter.Collective method, based on deaths in period. Liability in Valuation substantially lower than by Reversionary Annuity method; examination of difference, traced mainly to effects of War on marriage status. Discussion of the two methods, with references to Schjoll's alternative Collective method, based on living at end of period.Tables at 3, 3½, and 4 per cent.; all tables given by Hewat and Huie relating to payments on marriage and to widows' annuities. Collective Method ; tables at 3½ per cent.

On the Value of the Option of Second Nomination in Reversionary Annuities

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No. 2.—On Reversionary Annuities

No. 2.—On Reversionary Annuities

On the Value of a Reversionary Annuity Payable oftener than Once a Year

At one of the meetings of this Society last session a paper was read on this subject by Mr. Meikle, which gave rise to considerable discussion on the question; and as no definite result was arrived at, I believe a further investigation of the problem will not be without interest. Accordingly, I propose to go over the various formulas given for the value of this annuity, and offer some remarks upon each, stating wherein they are essentially different, and what appears to me to be the exact interpretation of each.

Reduction of Formulæ for Annuities and Assurances Investigated by Mr. Sprague on the Common hypothesis of Equal Decrements in each Year of Life

In 1869 Mr. Sprague read before the Institute an interesting paper “On the value of Reversionary Annuities payable half-yearly, quarterly, &amp;c., according to the conditions which prevail in practice.” This valuable paper is given in the Journal, vol. xv, page 126. The conditions which prevail in practice, as here for the first time introduced and mathematically investigated by Mr. Sprague, are the following:—For a yearly reversionary annuity on a life x after the death of another life y, the first payment of the annuity is supposed to be made just one year after the day of the death of y, provided that x be then living; and for a like annuity payable by m instalments in each year, the first payment of an instalment is supposed to be made just of a year after the day of the death of y. Also, in each case, it is further supposed that whenever x shall die, after surviving y, the annuity shall be completed up to the day of the death of x by the supplementary payment of the proportion of the annuity estimated in respect of the fractional interval that may then have elapsed since the last yearly payment or instalment was received.

On the value of Reversionary Annuities payable half-yearly, quarterly, &amp;c, according to the conditions which prevail in practice

The ordinary formula for the value of a reversionary annuity, ax–axy, is not strictly applicable to the cases which occur in practice. For it assumes that the first payment of the annuity is made at the end of the year in which y dies, and the last payment at the end of the year before that in which x dies. But the conditions which prevail in practice are different; for whenever a reversionary annuity is granted, it will run from the death of y, and the first payment will be made twelve months after that death, i.e., on the average, six months later than is assumed in the above formula. On the other hand, the reversionary annuity will in practice be payable up to the day of the death of x; or, on the average, a full year's payment will be made six months later than the last payment is made according to the above formula.

On the Value of Reversionary Annuities Payable Half Yearly or Quarterly

On the Value of Reversionary Annuities Payable Half Yearly or Quarterly