We begin by examining the accumulated value functions of some annuities-certain. We then investigate the accumulated value of these annuities where the interest is a random variable under some restrictions. Calculations are derived for the expected value and the variance of these accumulated values and present values. In particular we will examine an annuity-due of k yearly payments of 1. Then we will consider an increasing annuity-due of k yearly payments of 1, 2, ⋯ , k. And finally, we examine a decreasing annuity-due of k yearly payments of n, n - 1, ⋯ , n - k + 1, for k ≤ n. Finally we extend our analysis to include a contingent annuity. That is an annuity in which each payment is contingent on the continuance of a given status. Specifically, we examine a life annuity under which each payment is contingent on the survival of one or more specified persons. We extend our methods from the previous sections to derive the formula of the expected value for the present value of the life annuities of a future life time at a random rate of interest.

Thesis (MComm)--Stellenbosch University, 2013. / In this thesis, based on historical data, a comparative study is conducted of various annuity strategies for South African males who retired during the 30 years from 1960 to 1989. To this end, the present values of the monthly cash ows provided by di erent annuity strategies are calculated and compared in order to ascertain which strategy would have provided the largest nancial bene ts. In contrast to previously held general beliefs, the calculations demonstrate that pure living annuity strategies are superior to composite annuity strategies, which in turn outperform switching annuity strategies, whereas pure life annuities yield the lowest return.

The thesis is concerned with the problem of sustainable spending towards the end of the human life cycle, which is a substantial quantitative problem in the pension framework. We gradually build a model, which coherently links the three key factors of retirement planning: uncertain length of human life, uncertain investment returns and spending rates. Within the framework of our intuitive model, we apply the method of moment matching to derive an approximation for the probability of individual's retirement ruin. The accuracy of presented approximation is analyzed via extensive Monte Carlo simulations. A numerical case study using Czech data is provided, including calculated values for the probability of ruin and maximal sustainable spending rate under various combinations of wealth-to-spending ratios and investment portfolio characteristics.

The aim of this diploma thesis is describe the most common methods of financing pension plans, focusing on some of the methods of fund financing pension plans. To describe the individual methods, their numerical illustration and allow comparison, it is necessary to dispose of necessary instruments. Therefore in the thesis there are constructed the cohort life tables for the Czech Republic. The thesis also deals with the modelling life annuities in continuous time, in particular, with the shape of im- mediate pension anuity factor for Gompertz law of mortality. Namely, this factor is one of the parameters entering the calculation of the individual methods of fund fi- nancing for pension plans.

Il y a eu 723 000 transactions dans l’immobilier ancien en 2013. Les ventes en viager ontreprésenté entre 0,5% et 1% de ce montant. Le taux de propriétaires de plus de 60 ansdépasse les 70%. Les seniors possèdent 700 milliards d’euros dans l’immobilier. Deux acteurscomposent le viager : un acheteur et un vendeur. Du côté de l’offre, les retraités sont de plusen plus nombreux à vendre en viager puisqu’ils ont besoin de liquidités : « house rich, cashpoor »1. Avec l’allongement de la durée de vie, les seniors doivent financer les frais de santéet le coût de la dépendance. Du côté de la demande, les particuliers comme les institutionnelssont à la recherche de diversifications patrimoniales. On trouve deux fois moins d’acheteursque de vendeurs en viager. Les institutionnels ont investi dans l’immobilier commercial et lesparticuliers dans le résidentiel depuis les années 90. Qu’en est-il pour le viager ? Pourquoi cemode d’acquisition n’a-t-il pas encore séduit les investisseurs ? Nous avons identifié deuxfreins majeurs : l’un financier, l’autre juridique. Nous avons simulé un portefeuille de 300viagers réels en utilisant 3 tables de mortalité. La modélisation de notre base de données apermis de trouver un faible taux de rendement interne sur l’espérance de vie du vendeur.Nous avons trouvé des TRI allant de 1,80% à 5,13% selon la table de mortalité retenue. Pourobtenir un taux de rendement interne de 5% sur l’espérance de vie du vendeur, en prenant lamoyenne des trois tables de mortalité, les investisseurs doivent faire baisser le montant de larente viagère de 17,55%.Nous avons recommandé différentes mesures en direction des pouvoirs publics afind’améliorer la liquidité du viager immobilier :-Déduire le paiement de la rente des autres revenus fonciers.-Déduire les intérêts d’emprunts ayant servi à financer le bouquet des autres revenus fonciers.-Reculer la durée de la clause résolutoire d’un à trois mois.-Ramener à 15 ans l’exonération des plus-values immobilières. / There were 723,000 transactions in existing property in 2013. Life annuity sales accounted forbetween 0.5% and 1% of this amount. The rate of home ownership among the over 60 agegroup exceeds 70%. Senior citizens own 700 million worth of real estate. Life annuity salesinvolve two players: a buyer and a seller. On the supply side, an increasing number ofpensioners are selling their property for life annuities since they need cash: «house rich, cashpoor». With longer life expectancy, senior citizens need to finance health and dependencycosts. On the demand side, both private and institutional investors seek asset diversification.However, there are twice as few buyers than sellers for life annuity property. Since thenineties, institutional investors have invested in commercial property, and private investors inresidential property. What is the situation for life annuity property sales ? We may wonderwhy this form of property acquisition has not so far attracted investors. We have identifiedtwo major hurdles: one financial, the other one legal. We have simulated a portfolio of 300real life annuity sales by using 3 mortality tables. The modeling of our data base enabled us toidentify a weak rate of return on the life expectancy of the seller. We found rates of internalreturn ranging from 1.8% to 5.13% according to the mortality table retained. In order toobtain a 5% rate of internal return on the life expectancy of the seller, taking the average ofthe three mortality tables, investors need to lower the amount of the life annuity by 17.55%.We have recommended different measures to the public authorities in order to improve theliquidity of property life annuities : deduct the payment of the annuity from other propertyincome, deduct the interests of loans used to fund the other property income mix and increasethe duration of the cancellation clause from one to three months.

Entre 1769 et 1789 le roi de France a émis des emprunts viagers et des tontines à des taux réels très supérieurs à ceux qu’il autorisait aux particuliers. Il l’a fait en connaissance de cause en offrant une prime de risque palliant la faiblesse de sa signature. Celle-ci était masquée par l’absence de table de mortalité et de tarif reconnus. Marginaux à l’origine, ces emprunts ont eu une première importance lors de la guerre de succession d’Espagne avec des rentes mixtes puis avec la consolidation des années 1720. Après une période de petits emprunts en classes d’âges, Ils sont revenus au premier plan des moyens d’endettement pendant la guerre de sept ans lorsque, en négligeant la table de Deparcieux, le roi emprunta en viager à un taux uniforme lors d’emprunts massifs, afin de capter les placements sur des têtes jeunes. Leur importance perdura après le conflit, et ils sont devenus un outil majeur des décennies suivantes, à l’impact financier croissant, par paix comme par guerre, au point de tenir le premier plan dans la dette publique à la veille de la révolution.Alors que genevois, génois et hollandais maitrisaient les règles rudimentaires de la mortalité et optimisaient avec plus ou moins d’efficacité leurs mises dans le viager de France, les nationaux ont ignoré les excellents apports académiques français dans ce domaine et, à l’exception des manieurs d’argent, ont eu un comportement bien moins efficace, aussi bien dans les emprunts publics que privés.Le viager a présenté les avantages et les inconvénients d’un marché trouble, où l’absence de règles affichées permet au roi comme aux particuliers de payer des primes de risque sans le montrer, mais où les personnes âgées sont lésées, et où ceux qui doivent revendre les contrats achetés ne peuvent le faire qu’à prix cassé.La progression du recours de l’état à ces emprunts n’est pas une marque d’incompétence mais d’une dégradation de son crédit pendant les trente ans qui précèdent la révolution, liée à un manque de ressource fiscale. / Between 1689 and 1789 France issued life annuities and tontines at true rates above what was permitted to private persons. This was made with plain knowledge by offering a risk premium palliating its weak creditworthiness, hidden by the absence of mortality table or accepted life annuities rates. Marginal at the beginning, these loans took a first importance during the war of Spanish succession with mixed annuities, then with the consolidation operations of the 1720s. After a period of small age-group loans, they came back at the forefront of ways to borrow during the seven years war when, neglecting Deparcieux’s life table, the king started to borrow at a uniform life rate in massive loans, trying to catch investments on young people. Their importance continued after this conflict, and they became a major tool for subsequent decades, with an increasing financial impact, by wartime like by peace, reaching the first rank of public debt at the eve of the revolution.Meanwhile Genevan, Genoese, and Dutch mastered the basic rules of mortality and optimized their investments in french life annuities with more or less efficiency, the nationals overlooked french first-class academic contributions in this field and, apart the business community, had far less efficient behaviours, as well for public or private loans.Life annuities had advantages and disadvantages of a murky market, where the lack of apparent rules allows the king or private person to pay risk premium without showing it, but where aged people suffer damage, and where those who have to assign their contract can do it only at a rock-bottom price.The progressive appeal of state at these toxic loans does not mean ineptitude but a heightening credit risk during the thirty years preceding the revolution, linked to a lack of tax based resource.