On annuities under random rates of interest

In the article we consider accumulated values of annuities-certain with yearly payments with independent random interest rates. We focus on general annuities with payments varying in arithmetic and geometric progression which are important basic varying annuities (see Kellison, 1991). They are equivalent to the types studied recently by Zaks (2001). We derive, via recursive relationships, mean and variance formulae of the final values of the annuities. As a consequence, we obtain the moments related to the already discussed cases, which leads to a correction of main results from Zaks (2001).Finance mathematics; Annuity; Accumulated value; Random interest rate;

Financial protection for the elderly - contracts based on equity release and critical health insurance

Purpose: The purpose of the article is to present a new marriage contract combining the advantages of the reverse annuity contract and the critical health insurance contract. The content of the article focuses on the method of determining premiums and benefits related to this contract. Design/Methodology/Approach: The multistate model is used to describe the survival time of the spouses. The elements of transition probabilities matrix were estimated assuming that the stochastic process describing further life expectancy has Markov property. Empirical examples are presented on the basis of data concerning lung cancer taking into account the incidence, mortality and fatality rates from this disease. Findings: The obtained results indicate that older people can obtain additional financial resources which can improve the quality of life and raise its standard significantly. Moreover, additional financial protection can be ensured in case of a critical illness. Practical implications: The described contract can be used in practice as a form of providing additional financial resources supplementing the home budget of pensioners, especially in the event of a serious illness. The contract can also be used with the option of long-term care insurance. Despite the guaranteed free health care, such contracts can significantly increase the quality of life of patients during a chronic severe illness. Originality/Value: The model presented in the paper is original. It allows flexible modelling of cash flows to provide financial protection in case of low income during retirement and health deterioration of the elderly.peer-reviewe

Insights into developmental processes in anthers, ovaries, and ovules of Taraxacum belorussicum (Asteraceae-Cichorioideae) using DIC optics

This study represents the first report on the embryological characteristics of triploid male-sterile dandelion Taraxacum belorussicum (section Palustria) from Poland. While this taxon is considered to be a male-sterile species, we found that the investigated individuals produced pollen. Irregular tetrads, triads and diads with microspores of unequal size were observed in the pollen loculi as a result of disturbed meiotic division, while anthers’ tapetum did not show structural disorders. Possible reasons for the plasticity in the expression of male sterility, as well as the role of pollen in apomicts, are discussed. Flowers of the examined individuals contained well-developed nectaries. The course of embryological processes in the ovules indicated an apomictic mode of reproduction in T. belorussicum. We observed meiotic diplospory of the Taraxacum type, in which first meiotic division starts but results in nuclear restitution, while undisturbed second meiotic division gives rise to a dyad of unreduced megaspores (diplodyad). After three mitotic divisions of the chalazal megaspore, a seven-celled unreduced female gametophyte developed. The features of ovule anatomy and characteristics of a mature female gametophyte corresponded to these described in sexually reproducing dandelions

Marriage and Individual Equity Release Contracts with Dread Disease Insurance as a Tool for Managing the Pensioners’ Budget

In many countries around the world, demographic and civilization changes have brought about the phenomenon of aging societies. This phenomenon affects the economy, especially the pension and health care systems, causing difficulties in their financing. The implementation of a policy that would effectively manage the problem of the longevity risk is thus required. Using housing resources and private health insurance to improve retirees’ living standards may serve this purpose. The instruments we propose comprise two variants of contracts: the first for a marriage, the second for an individual client. We analysed the cash flow in both the cases. The results suggest that the amount of cash flows related to reverse equity and dread disease insurance benefits depends on the spouse’s economic status, age, and health conditions. The benefits of the two variants of the contract vary. This paper examines numerous strategies for selecting the type of the contract, taking into consideration the abovementioned factors

Marriage reverse annuity contract taking into account the dependence between the future lifetime of spouses

Społeczeństwo żyje coraz dłużej, a znaczny spadek umieralności w stosunku do lat 60.-70. XX wieku obserwuje się u osób w wieku emerytalnym. Emerytura z tytułu ubezpieczeń społecznych jest niska i może nie być wystarczająca, aby godnie przeżyć starość. Ważną kwestię stanowi możliwość pozyskania dodatkowych środków finansowych, aby zachować dotychczasowy standard. Jednym z takich rozwiązań może być tzw. renta hipoteczna (reverse annuity contracts), czyli dożywotnia renta, którą właściciel mieszkania może otrzymywać w zamian za przeniesienie praw własności na spółkę, gwarantując sobie w akcie notarialnym prawo do mieszkania w nim do śmierci. W sytuacji gdy właścicielami nieruchomości są małżonkowie, istnieje możliwość wypłacania tzw. hipotecznych rent małżeńskich. Celem artykułu jest wyznaczenie wielkości świadczenia w przypadku, gdy renta jest wypłacana do momentu śmierci pierwszego z małżonków (status wspólnego życia) oraz gdy renta jest wypłacana do śmierci drugiego małżonka (status ostatniego przeżywającego), przy bardziej realnym założeniu, że przyszłe czasy trwania życia małżonków są zależnymi zmiennymi losowymi. Porównano wysokość świadczeń w przypadku zależności i niezależności dalszego trwania życia małżonków. Uwzględniono również częstotliwość wypłacania renty. Prawdopodobieństwa trwania życia wyznaczono na podstawie danych GUS z wykorzystaniem modelu Markowa. W obliczeniach zastosowano stopę procentową określoną modelem Svenssona.People live longer, and a significant decrease in mortality compared to 60-70 years of the 20th century, is observed in people at retirement age. The social insurance pensions are low and may be insufficient to survive with dignity. In this context, an important issue is the possibility of obtaining additional financial resources. One of those is the so-called a reverse annuity contract. In return for the transfer of the ownership onto the company, an owner is guaranteed by a notarial act the right to live in property until his death and whole life annuity. Since in many cases the property owners are couples, therefore an important issue is enabling the marriage reverse annuity contract. This product is not offered on the Polish market. The aim of the article is to calculate the benefit in two cases, when both spouses are alive (the status of the common life), and then when one of them dies (the status of the last surviving). The calculations are made under the more realistic assumption of the dependence of future lifetime of spouses. The value of benefits is compared in both cases i.e. independence and dependence of future lifetime of spouses. Moreover the frequency of payment is taken into consideration. The surviving probability is calculated on the basis of data follows from Main Statistical Office. For this purpose it is used the Markov process. The Svensson interest rate model is applied to calculate the benefits of reverse annuity contract

Analiza ryzyka renty hipotecznej i odwróconego kredytu hipotecznego

Renta hipoteczna oraz odwrócony kredyt hipoteczny umożliwiają właścicie- lowi nieruchomości otrzymywanie comiesięcznego świadczenia w zamian za przekazanie nieruchomości firmie zainteresowanej jej nabyciem. Produkty te wzbudzają coraz większe zainteresowanie i stają się coraz bardziej popularne, szczególnie w obliczu starzenia się społeczeństwa polskiego i obserwowanego wzrostu przeciętnego dalszego trwania życia. Renta hipoteczna jest dostępna w Polsce od 2005 r., natomiast kredyt hipoteczny nie jest obecnie jeszcze oferowany przez żadną instytucję finansową w Polsce, choć jest produktem konkurencyjnym dla renty hipotecznej. W ogólnie dostępnych mediach podaje się jedynie informacje o ich zaletach. Nie wspomina się natomiast o wadach produktów ani o ryzyku, którym owe produkty są obarczone. Jedyną wadą, o której można przeczytać, jest to, że renta hipoteczna niesie za sobą ryzyko w przypadku bankructwa funduszu hipotecznego. Nie jest to jedyne ryzyko związane z nabyciem renty hipotecznej, jak również odwróconego kredytu hipotecznego. W obliczu zwiększającego się zainteresowania tymi produktami ryzyko jest bardzo ważnym problemem. Dlatego też w artykule podjęto się analizy ryzyka, połączonej z wywodem teoretycznym na temat ryzyka tych produktów oraz przykładami numerycznymi. Do obliczeń zastosowany jest aparat rent życiowych. Przyjęto stałą stopę procentowaną oraz funkcję czasu określoną modelem Svenssona. Do wyceny nieruchomości zastosowano tzw. metodę mieszaną, która obejmuje podejście porównawcze i metodę korygowania średniej ceny za m2 wycenianej nieruchomości

Reverse annuity contract and reverse mortgage on the Polish market

The reverse annuity contracts and reverse mortgages are different and competitive products, often mistakenly identified with each other. The purpose of the article is to compare the products, present their essence, the method of the calculating of the received benefits value and an overview of the advantages and disadvantages. Moreover, there is included the ability to pay the marriage annuity, when both spouses are alive, and when one of them dies. The subject of the study is the amount of the annuity, when the future lifetimes of married couples are independent. There are considered the different cases of annuity due to the duration and the number of people joining the contract. The tool used to determine the value of benefits are the life and the certain annuities. The fixed interest rate and life tables from 2012 are considered in the calculations