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In this paper, we first derive closed-form formulas for mortality-interest durations and convexities of the prices of life insurance and annuity products with respect to an instantaneously proportional change and an instantaneously parallel movement, respectively, in μ* (the force of mortality-interest), the addition of μ (the force of mortality) and δ (the force of interest). We then build several mortality-interest duration and convexity matching strategies to determine the weights of whole life insurance and deferred whole life annuity products in a portfolio and evaluate the value at risk and the hedge effectiveness of the weighted portfolio surplus at time zero. Numerical illustrations show that using the mortality-interest duration and convexity matching strategies with respect to an instantaneously proportional change in μ* can more effectively hedge the longevity risk and interest rate risk embedded in the deferred whole life annuity products than using the mortality-only duration and convexity matching strategies with respect to an instantaneously proportional shift or an instantaneously constant movement in μ only.

This chapter deals with Dewey’s definition and discussion of the concept of interest in the 1932 Ethics. Part I of the chapter reconstructs the definition of interest as integration. The integrative force of interest is at work in at least two directions. According to Dewey, interests are able to integrate (1) the objective and subjective dimensions of conduct and experience, and (2) the pre-reflexive dimensions of needs, impulses, desires, emotions, and human reflexive capacity. Part II analyzes how the definition of interest as integration leads Dewey to a radical critique of the self-interest versus disinterestedness dichotomy. This double dismissal hinges on a radical rethinking of the relationship between self and interest. Every interest is an interest of the self, but not every interest is self-interest; disinterestedness does not mean lack of interest, but rather a particular type and quality of interest. Part III highlights how Dewey in the 1932 Ethics systematizes and analytically develops some insights about interest introduced in his psychological, pedagogical, and political essays.

St. Petersburg paradox, formulated by N. Bernoulli in the early 18th century, led to defining the utility function (D. Bernoulli, G. Cramer) as a way to resolve the paradox and played an important role in the development of decision making theory. In the 20th century, the paradox attracted the attention of many researchers, including Nobel Prize winners P. Samuelson, R. Aumann, L. Shapley. N. Bernoulli assumed that payments grow exponentially with the coin toss number. The growth rate of payments is higher than the exponential one in the generalized St. Petersburg paradox. The utility functions of Bernoulli and Cramer don't lead to the resolution of the paradox in this case. In 1934, K. Menger showed the necessity and sufficiency of the boundedness of the utility function for resolving of the generalized St. Petersburg paradox. A brief overview of the subject matter is given, as well as the autors' approach to resolving the classical paradox, based on discounting cash flows, in which the time intervals between consecutive coin tossings play a special role. The adaptation of the proposed approach to the generalized St. Petersburg paradox is also described. The proposed approach is an alternative to the traditional based utility function. It allows to solve, in particular, the inverse problem: to find (ambiguous solution) the moments of possible payments according to the set sizes of payments, the force of interest and the price of the game.

We introduce a pricing framework for a guaranteed annuity option (GAO) where both the interest and mortality risks are correlated. We assume that the short rate and the force of mortality follow the Cox–Ingersoll–Ross (CIR) and Lee–Carter models, respectively. Employing the change of measure technique, we decompose the pure endowment into the product of the bond price and survival probability, thereby facilitating the evaluation of the annuity expression. With the aid of the dynamics of interest and mortality processes under the forward measure, we construct an algorithm based on comonotonicity theory to estimate the quantiles of survival probability and annuity rate. The comonotonic upper and lower bounds in the convex order are used to approximate the annuity and GAO prices and henceforth avoiding the simulation-within-simulation problem. Numerical illustrations show that our algorithm gives an efficient and practical method to estimate GAO values.

This paper introduces the Fourier Space Time-Stepping algorithm to the valuation of variable annuity (VA) contracts embedded with guaranteed minimum withdrawal benefit (GMWB) riders when the underlying fund dynamics evolve under the influence of a regime-switching model. Mortality risk is introduced to the valuation framework by incorporating a two-factor affine stochastic mortality model proposed in Blackburn and Sherris (2013). The paper considers both, static and dynamic policyholder withdrawal behaviour associated with GMWB riders and assesses how model parameters influence the fees levied on providing such guarantees. Our numerical experiments reveal that the GMWB fees are very sensitive to regime-switching parameters; a percentage increase in the force of interest results in significant decrease in guarantee fees. The guarantee fees increase substantially with increasing volatility levels. Numerical experiments also highlight an increasing importance of mortality as maturity of the VA contract increases. Mortality has less impact on shorter maturity contracts regardless of the policyholder's withdrawal behaviour. As much as mortality influences pricing results for long maturities, the associated guarantee fees are decreasing functions of maturities for the VA contracts. Robustness checks of the Fourier Space Time-Stepping algorithm are performed by making numerical comparisons with several existing valuation approaches.

In this paper, we use a semi-Markovian model to compute the conditional higher moments of any order of the present value of cash flows generated by an investment, taking into account the state of the market. With the force of interest following a stochastic process, we give an example to illustrate our results.

Considering stochastic behavior of interest rates in financial market, we construct a new class of interest models based on compound Poisson process. Different from the references, this paper describes the randomness of interest rates by modeling the force of interest with Poisson random jumps directly. To solve the problem in calculation of accumulated interest force function, one important integral technique is employed. And a conception called the critical value is introduced to investigate the validity condition of this new model. We also discuss actuarial present values of several life annuities under this new interest model. Simulations are done to illustrate the theoretical results and the effect of parameters in interest model on actuarial present values is also analyzed.

Conceptualized in terms of rational calculation and market logic, entrepreneurship appears as a straightforward matter of reckoning risks and rewards, assessing the prospects for profit. When viewed through an ethnographic lens, however, more mysterious – even ‘magical’ – aspects come into focus. This article explores the magical aspects through three case studies, each revealing a distinct process: symbolic efficacy, the conversion of words into things, and the ability to peer into the future. While exposing these otherwise hidden aspects, the article simultaneously explores a view of profit as arising from the capture – in different ways – of freely flowing, non-commoditized culture. The entrepreneur taps into cultural flows that are propelled by the force of interest, and, by capturing the flows, turns the culture into commodities capable of yielding profit. In these cases we glimpse the truth of Marcel Mauss's claim many years ago: ‘Though we may feel ourselves to be very far removed from magic, we are still very much bound up with it’ (Mauss 1972: 178).

In this paper, we introduce a class of stochastic interest model driven by a compound Poisson process and a Brownian motion, in which the jumping times of force of interest obeys compound Poisson process and the continuous tiny fluctuations are described by Brownian motion, and the adjustment in each jump of interest force is assumed to be random. Based on the proposed interest model, we discuss the expected discounted function, the validity of the model and actuarial present values of life annuities and life insurances under different parameters and distribution settings. Ournumerical results show actuarial values could be sensitive to the parameters and distribution settings,which shows the importance of introducing this kind interest model.

This work considers a perturbed risk process with investment, where the investments are either into invested in risky and risk-less assets. A third order differential equation for the ruin probability is derived from the resulting integrodifferential equation. This equation is further decomposed into two equations describing the contributions of the claim and oscillation to the ruin probability. These two equations are solved separately using suitable transformations as well as theory of Kummer confluence hypergeometric equations.We further investigated these solutions and were able to conclude that the higher the fraction of investment into risky assets, the lower the ruin probability, provided all other parameters are kept constant. Keywords: Risk Reserve; Ruin Probability; Interest; Stochastic Investment; Exponential distribution; Kummer hyper-geometric equation AMS 2010 Mathematics Subject Classification : 60J25; 60J60

In this paper, we consider the dividends and equity issuances control problem in the perturbed dual model under a penalty for ruin. Transaction costs are incurred by these business activities: dividend is taxed and fixed costs are generated by equity issuance. The objective is to maximize the expected present value of dividends minus the discounted costs of equity issuances and the discounted penalty until the ruin time. We find the joint optimal dividend and equity issuance strategy by solving the control problems of two categories of suboptimal model. Furthermore, we derive the explicit closed solutions for the value functions and the optimal strategies when the jumps are mixed exponential. In particular, we investigate the effects of the penalty, the proportional and fixed transaction costs, the expense rate and the force of interest on the optimal strategies by numerical calculations, and give some interesting economic insights.

In this paper, we investigate the optimal asset allocation in the distribution phase for an equity-linked annuity scheme of a DC pension plan. We extend previous research to consider the interest rate risk. To prevent a shortfall on the annuity payment in case of poor investment performances, a minimum guarantee on the annuity payment is included in our model. We show that the optimal asset allocation could be represented by a weighted average of two portfolios. The first portfolio is the so-called growth optimal portfolio, which maximizes the expected growth of annuity payment over the planning horizon. The second portfolio is the replicating portfolio, which replicates the returns on a level life annuity. The optimal portfolio weights on these two parts are decided by the coverage ratio of the fund size to the present value of guaranteed annuity payments. Numerical examples show that a high level of guarantee effectively reduces the uncertainty of future annuity payments but loses the opportunities for fund growth. The demand for stocks decreases with the level of guarantee, while the demand for bonds is insensitive to the level of guarantee.

This paper provides clear-cut evidence that the ECB follows an asymmetric monetary policy rule concerned more with inflation rather than sustaining economic growth. The driving force of interest rate cuts is the fall of the inflation rate below its target level. The paper evaluates the implications of the above policy on real activity by simulating a small New Keynesian model. This clearly indicates that the reaction of the ECB to negative output deviations and/or to financial stress conditions in the low inflation regime fails to reduce the adverse effects of negative demand and financial sector shocks on economic activity.

In this paper, we study optimal dividend problems in a diffusion risk model for two different cases depending on whether reinsurance is incorporated. In either case, the dividend rate is bounded above by a constant, and the company earns investment income at a constant force of interest. Unlike existing approaches in the literature dealing with optimal problems with interest, we allow the force of interest to be greater than the discount factor, and we use a different method to solve the corresponding Hamilton---Jacobi---Bellman (HJB) equation instead of introducing a confluent hypergeometric function. We conclude that the optimal dividend policy is of a threshold type and show that the corresponding dividend barrier is nondecreasing in the dividend rate bound. In cases where there is no reinsurance, we construct an auxiliary reflecting control problem to find the nonzero dividend barrier. If proportional reinsurance is purchased, the optimal reinsurance strategy looks somewhat strange. The optimal retention level of risk first increases monotonically with risk reserve to some possible value (less than $$1$$1) and then stays at level $$1$$1 for a while or, if $$1$$1 has been reached, finally, it decreases to 0.

In portfolios of life annuity contracts, the payments made by an annuity provider (an insurance company or a pension fund) are driven by the random number of survivors. This paper aims to provide accurate approximations for the present value of the payments made by the annuity provider. These approximations account not only for systematic longevity risk but also for the diversifiable fluctuations around the unknown life table. They provide the practitioner with a useful tool avoiding the problem of simulations within simulations in, for instance, Solvency 2 calculations, valid whatever the size of the portfolio.

This paper studies the asymptotic behavior of the finite-timeruin probability in a jump-diffusion risk model with constant force of in-terest, upper tail asymptotically independent claims and a general count-ing arrival process. Particularly, if the claim inter-arrival times follow acertain dependence structure, the obtained result also covers the case ofthe infinite-time ruin probability. 1. IntroductionIn this paper, we consider the asymptotic ruin probabilities in a generalizedjump-diffusion risk model with constant force of interest, where the claim sizes{X i ,i≥ 1} are a sequence of nonnegative, but not necessarily independent,random variables (r.v.s) with distributions F i , i≥ 1, respectively, while theclaim arrival process {N(t),t≥ 0} is a general counting process, independentof {X i ,i≥ 1}. Hence, the aggregate claim amount up to time t≥ 0 isS(t) = N X (t)i=1 X i with S(t) = 0 if N(t) = 0. Assume that the total amount of premiums accu-mulated up to time t≥ 0, denoted by C(t), is a nonnegative and nondecreasingstochastic process with C(0) = 0 and C(t)

This article considers the pricing of equity-indexed annuity (EIA). By employing the change of measure technique, we derive the closed-form solutions for the prices of both point-to-point and annual reset equity-indexed annuities. We also provide numerical results to illustrate the method and computational efficiency of our simulation scheme and the effects of various model parameters on the participation rate.

In this paper, we apply the linear hazard transform to mortality immunization. When there is a change in mortality rates, the respective surplus (negative reserve) changes for life insurance and annuity policies lead to oppositive sign changes, which provides mortality hedging strategies with a portfolio of life insurance and annuity policies. We first show that by the strategy of matching duration of the weighted surplus at time 0, the surplus changes at time 0 for both portfolios P T P (the n -year term life insurance and the n -year pure endowment) and P W A (the n -payment whole life insurance and the n -year deferred whole life annuity) in response to a proportional or parallel shift in the underlying force of mortality are always negative. Next, we prove that the term life insurance, the whole life insurance and the deferred whole life annuity cannot always form a feasible portfolio (feasibility means that all the weights of the product members of a portfolio are positive) by the strategy of matching two durations or one duration and one convexity of the weighted surplus at time 0. Finally, numerical examples including figures and tables are exhibited for illustrations.

Risk analysis in actuarial science has shifted its focus from diversifiable risk to systematic risk in the last 20 years or so. This article contributes further in this direction by proposing the concept of annuity rate to take account of systematic risk inherent in annuity products. The annuity rate is the conditional expectation of the annuity’s future payments, given the future paths of mortality and interest rates. We provide an empirical study to investigate the impact of the two systematic risk factors on the distribution of the annuity rate. In particular, we adopt the Lee-Carter and the Cairns-Blake-Dowd models for mortality risk, and the one-factor and two-factor CIR models for interest risk. Monte Carlo simulation is used to provide numerical illustrations of sensitivity analysis of the annuity rate and of risk assessment of a guaranteed annuity option.

The paper deals with the problem of estimating the actuarial present value of the continuous whole life and n-year term life annuities. We synthesize nonparametric estimators of these statuses of life annuity. The main parts of their asymptotic mean square errors for these estimators and their limit distributions are found. By individuals' death moments, both parametric and nonparametric estimates are constructed for the models of the whole and n-year term life insurance. The asymptotic normality and mean square convergence of the proposed estimators are proved. The simulations show that the empirical mean square errors of life annuity estimates decrease when the sample size increases. Also, when the model distribution is changed, the nonparametric estimates are more adaptable in comparison with parametric estimates, oriented on the best results only for the given distributions.

Recursive moments, joint moments, moments generating functions, distribution functions, stop-loss premiums and risk measures have been found for the univariate compound renewal sums with discounted claims, for a constant force of real interest. More recently, moments and joint moments have also been found when the force of interest is stochastic. In this paper, we extend some of the preceding results to the bivariate compound renewal sums with discounted claims by first presenting a lemma that gives the conditional joint distribution of the occurrence times of the claims given the number of claims of each type up to time t, result that will be used to get the second moment, the first joint moment and other quantities related to our bivariate risk process.

It is well known that the expected present value of a life annuity is smaller than the present value of an annuity certain with term equal to life expectancy. This result can be viewed as a consequence of the lack of interchangeability between the present value operator and the mathematical expectation operation. However, we prove that the median and present value operators can be interchanged; that is, the median of the present value of a life annuity equals the present value of an annuity certain whose term is the median additional years of life. At young ages and through late middle age, median additional years of life exceed life expectancy. Therefore, the median value of a life annuity exceeds an annuity certain paid to life expectancy which exceeds the expected present value of a life annuity - the fair price of a life annuity.

A life annuity contract is an insurance instrument which pays pre-scheduled living benefits conditional on the survival of the annuitant. In order to manage the risk borne by annuity providers, one needs to take into account all sources of uncertainty that affect the value of future obligations under the contract. In this paper, we define the concept of annuity rate as the conditional expected present value random variable of future payments of the annuity, given the future dynamics of its risk factors. The annuity rate deals with the non-diversifiable systematic risk contained in the life annuity contract, and it involves mortality risk as well as investment risk. While it is plausible to assume that there is no correlation between the two risks, each affects the annuity rate through a combination of dependent random variables. In order to understand the probabilistic profile of the annuity rate, we apply comonotonicity theory to approximate its quantile function. We also derive accurate upper and lower bounds for prediction intervals for annuity rates. We use the Lee–Carter model for mortality risk and the Vasicek model for the term structure of interest rates with an annually renewable fixed-income investment policy. Different investment strategies can be handled using this framework.

An equity-indexed annuity (EIA) contract offers a proportional participation in the return on a specified equity index, in addition to a guaranteed return on the single premium. In this paper, we discuss the valuation of equity-indexed annuities under stochastic mortality and interest rate which are assumed to be dependent on each other. Employing the method of change of measure, we present the pricing formulas in closed form for the most common product designs: the point-to-point and the annual reset. Finally, we conduct several numerical experiments, in which we analyze the relationship between some parameters and the pricing of EIAs.

This paper presents an extension of the classical compound Poisson risk model for which the inter-claim time and the forthcoming claim amount are no longer independent random variables (rv's). Asymptotic tail probabilities for the discounted aggregate claims are presented when the force of interest is constant and the claim amounts are heavy tail distributed rv's. Furthermore, we derive asymptotic finite time ruin probabilities, as well as asymptotic approximations for some common risk measures associated with the discounted aggregate claims. A simulation study is performed in order to validate the results obtained in the free interest risk model.

Guaranteed annuity options are options providing the right to convert a policyholder’s accumulated funds to a life annuity at a fixed rate when the policy matures. These options were a common feature in UK retirement savings contracts issued in the 1970’s and 1980’s when interest rates were high, but caused problems for insurers as the interest rates began to fall in the 1990’s. Currently, these options are frequently sold in the US and Japan as part of variable annuity products. The last decade the literature on pricing and risk management of these options evolved. Until now, for pricing these options generally a geometric Brownian motion for equity prices is assumed. However, given the long maturities of the insurance contracts a stochastic volatility model for equity prices would be more suitable. In this paper explicit expressions are derived for prices of guaranteed annuity options assuming stochastic volatility for equity prices and either a 1-factor or 2-factor Gaussian interest rate model. The results indicate that the impact of ignoring stochastic volatility can be significant.

We consider the compound Poisson risk model with a constant force of interest and a threshold strategy. Under such a strategy, no dividends are paid if the insurer's surplus is below a certain threshold level. When the surplus is above the threshold level, part of the premium income and all of the interest income are paid out as dividends. The integro-differential equations for the Gerber–Shiu discounted penalty function and the expected discounted dividends are derived and solved. Closed-form expressions are given when the claim size is exponentially distributed. Numerical presentations are also provided for the case of exponential individual claim to illustrate the influence of force of interest and the safety loading on the expected discounted dividends.

The rapid emergence of centenarians has highlighted the importance of survival probabilities at extreme ages and has motivated actuaries to look for alternative ways to close of life tables in place of assigning a death probability of 1 at an arbitrarily chosen age. Using the asymptotic results of modern extreme value theory, we propose a model, which we call the threshold life table, to extrapolate survival distributions to extreme ages and to determine the appropriate end point of a life table. By combining the threshold life table with the Lee-Carter model for stochastic mortality forecasting, we consider applications to the valuation of a life annuity portfolio and to the prediction of the highest attained age. We illustrate the theoretical results using U.S., Canadian, and Japanese mortality data.

These Lecture Notes aim at introducing technical and financial aspects of the life annuity products, with a special emphasis on the actuarial valuation of life annuity benefits. The text has been planned assuming as target readers:- advanced undergraduate and graduate students in Economics, Business and Finance;- advanced undergraduate students in Mathematics and Statistics, possibly aiming at attending, after graduation, actuarial courses at a master level;- professionals and technicians operating in insurance and pension areas, whose job may regard investments, risk analysis, financial reporting, and so on, hence implying communication with actuarial professionals and managers.Given the assumed target, the use of complex mathematical tools has been avoided.We assume that the reader has attended courses providing basic notions of Financial Mathematics (interest rates, compound interest, present values, accumulations, annuities, etc.) and Probability (probability distributions, conditional probabilities, expected value, variance, etc). As mentioned, Mathematics has been kept at a rather low level. Indeed, all topics are presented in a “discrete” framework, thus not requiring analytical tools like differentials, integrals, etc.The Lecture Notes are organized as follows. Guarantees and options in life contingency products, and in life annuities in particular, are sketched in Chap. 1. In Chap. 2 the basic actuarial aspects are introduced with reference to conventional life annuity products, starting from expected present value definitions, then moving to premium and reserve calculations.A more general framework is proposed in Chap. 3, in order to introduce a wide range of life annuity products. The guarantee structure implied by various products is analyzed in Chap. 4, while Chap. 5 focusses on the time profile of the annuity benefits.Options and riders which can be added to life annuity products are dealt with in Chap. 6, while Chap. 7 focusses on annuity rates adopted to determine the annuity premiums. Cross-subsidy mechanisms working in life annuity portfolios are addressed in Chap. 8, where special attention is placed on “tontine” schemes.Strategies, which can be adopted to get the post-retirement income and, to some extent, can constitute alternatives to the immediate full annuitization of resources available at retirement, are described in Chap. 9.Long-term care insurance (LTCI) is briefly addressed in Chap. 10, with a special focus on life annuity benefits combined with LTCI benefits. Finally, Chap. 11 concludes with suggestions for further reading.

In this paper we obtain the exact asymptotics of the ruin probability for the integrated Gaussian process with force of interest. The results obtained are consistent with those obtained for the case in which there is no force of interest.

In the classical risk model with initial capital u, let τ(u) be the time of ruin, X +(u) be the risk reserve just before ruin, and Y +(u) be the deficit at ruin. Gerber and Shiu (1998) defined the function m δ(u) =E[e−δ τ(u) w(X +(u), Y +(u)) 1 (τ(u) < ∞)], where δ ≥ 0 can be interpreted as a force of interest and w(r,s) as a penalty function, meaning that m δ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for m δ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired m δ(u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.

Optimal investment in a nonrenewable resource project occurs when the rate of increase of the project's forward value falls to the force of interest. This stopping rule yields a financial interpretation of resource quality as being a property of the project rather than of individual units of reserves. It also leads to re-interpretations of (a) rent as the present value of the project rather than of units of reserves and (b) Hotelling's insight as, not a rule for the path of rents, but an equilibrium algorithm for price. The analysis is extended to sequential development of pesticides, antibiotics, and forests. Copyright 2007, Oxford University Press.

This paper aims to introduce jump tests to the actuarial community. In actuarial science, semimartingales are extensively used in the models for interest rates, options, variable annuities and equity-linked annuities. Those models usually assume without justification that the underlying asset process follows a continuous stochastic process such as a geometric Brownian motion, for the market data sometimes tell a different story. Choosing between a continuous model and a model with jumps is not only important for pricing of insurance products but also crucial for implementing other post-sales risk management measures such as dynamic liability hedging. A test for jumps allows actuaries to rigorously test whether the underlying asset process has jumps, which is the first critical step in model selection. The ability to conduct the test should thus belong to the repertoire of every expert and practitioner working in this field. In this paper, we review several major tests for jumps, describe their advantages and disadvantages, and offer suggestions for their implementation. We also implement several tests using real data, enabling practitioners to apply these tests in their work.Keywords: Asset price; Black–Scholes; equity-linked annuity; variable annuity

It is considered the classical risk model with mixed exponential claim sizes. Using known results it is obtained the explicit expression of the GerberShiu discounted penalty function ψ(x,δ) = E e −δT 1(T < ∞) , by some infinite series. Here δ > 0 is the force of interest, x – the initial reserve and T – ruin time. The dependance of the discounted penalty function on the main parameters x, θ, λ, δ, α, σ, ν is presented in diagrams, where λ > 0 is the parameter of Poisson process, θ > 0 is the safety loading coefficient, 0 ≤ α ≤ 1 and σ, ν > 0 are the parameters of the mixed exponential distribution

This paper examines the merger of two stock companies under the premise, due to Bruno de Finetti, that the companies pay out dividends to their shareholders in such a way so as to maximize the expectation of the discounted dividends until (possible) ruin or insolvency. The aggregate net income streams of the two companies are modeled by a bivariate Wiener process. Explicit results are presented. In particular, it is shown that if for each company the product of the valuation force of interest and the square of the coefficient of variation of its aggregate net income process is less than 6.87%, the merger of the two companies would result in a gain.

We consider the Sparre Andersen model modified by the inclusion of interest on the surplus. Approximation for the ultimate ruin probability is derived by rounding. And upper bound and lower bound are also derived by rounding-down and rounding-up respectively. According to the upper bound and lower bound, we can easily obtain the error estimation of the approximation. Applications of the results to the compound Poisson model are given.

The comprehensive development and application of progressive levels of generalization of the concept of a classic ordinary annuity from an interest theory perspective with either discrete or continuous compounding (first level) is the focus of this article. The second level incorporates the effect of inflation, or cost-of-living, on the annuity rent specified by a second interest rate. Next, we allow the principal to fund multiple investment economies, each with its own cost-of-living interest rate (third level). The fourth level recognizes the need for different payment spans and nonuniform economy-dependent annuity rents and its utility is illustrated with a multinational corporation capital investment example. At the fifth level, cost-of-living interest rates are allowed to vary over time. A retirement example illustrates application of the most generalized annuity formula derived. Finally, useful tables presented throughout the article summarize a total of 29 annuity formulas.

We consider the Sparre Andersen model modified by the inclusion of interest on the surplus. Exponential type upper bounds for the ultimate ruin probability are derived by martingale and recursive techniques. Applications of the results to the compound Poisson model are given. Numerical comparisons of upper bounds derived by each technique are presented.

In this paper we investigate the consequences on the pricing of insurance contingent claims when we relax the typical independence assumption made in the actuarial literature between mortality risk and interest rate risk. Starting from the Gaussian approach of Liu et al. (2014), we consider some multifactor models for the mortality and interest rates based on more general affine models which remain positive and we derive pricing formulas for insurance contracts like Guaranteed Annuity Options (GAOs). In a Wishart affine model, which allows for a non-trivial dependence between the mortality and the interest rates, we go far beyond the results found in the Gaussian case by Liu et al. (2014), where the value of these insurance contracts can be explained only in terms of the initial pairwise linear correlation.

The aging problem became more important and serious in China in recent years. The reverse mortgage is an innovation model to support the aged people. In this study, the accumulation function model of interest force with a Wiener process and a negative-binomial distribution is proposed as the basis for the reverse mortgage. With the proposed model, a lump sum pricing model, annuity pricing model, linear increasing annuity pricing model for single-life and double-lives are provided. All of the models can be improved to solve the problem that the actuarial pricing models of reverse mortgage only could be calculated by the fixed interest rate. LS : The amount of the loan which is got by one borrower with a method of payment with the stochastic interest rate. ' PMT : The amount of the loan which is got by one borrower at the beginning of each year with the method of annuity under the stochastic interest rate. ' PMTA : The amount of the loan which is got by one borrower at the beginning of the first year by linear increasing annuity with the stochastic interest rate. Q: The increasing amount every year of the loan when get the loan by linear increasing annuity. d: The increasing ratio every year of the loan when get the loan by equal ratio increasing annuity. xy LS ' : The amount of the loan which is got by two borrowers by a sum of loan ceiling with the stochastic interest rate. xy PMT ' : The amount of the loan which is got by two borrowers at the beginning of each year by annuity with the stochastic interest rate. xy

In order to establish continuous life insurance actuarial model theoretical system based on credibility measure, this paper defined the force of mortality based on credibility measure, and deduced the expressions of future lifetime distribution function, survival function, density function and actuarial notation by the force of mortality. Accordingly, it established models of continuous life functions based on credibility measure.

In this chapter, we review the differential calculus that you may be familiar with from previous studies. The material can however be considered as a direct continuation of Chapter 2 and so you should not worry if you have little or no prior experience of calculus. We begin by refining what we mean by continuity, a concept introduced in the previous chapter, and give a discussion of limits. Mathematical derivatives are then discussed within the context of these two ideas and from the perspective of calculating the gradients of functions. We then proceed to demonstrate how to obtain the derivatives of the common classes of functions discussed in the previous chapter, before moving on to a discussion of how to work with more complicated functions.

In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest on the surplus. The ‘penalty’ is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process.

Money has a time value; if we invest $1 today, we expect to get back more than $1 at some future time as a reward for lending our money to someone else who will use it productively. Suppose that we invest $1, and a year later we get back $(1 + i). The amount invested is called the principal, and we say that i is the effective rate of interest per year. Evidently, this definition depends on the time unit we choose to use. In a riskless world, which may be well approximated by the market for good quality government bonds, i will be certain, but if the investment is risky, i is uncertain, and our expectation at the outset to receive $(1 + i) can only be in the probabilistic sense. We can regard the accumulation of invested money in either a retrospective or prospective way. We may take a given amount, $X say, to be invested now and ask, as above, to what amount will it accumulate after T years? Or, we may take a given amount, $Y say, required in T years' time (to meet some liability perhaps) and ask, how much we should invest now, so that the accumulation in T years' time will equal $Y? The latter quantity is called the present value of $Y in T years' time. For example, if the effective annual rate of interest is i per year, then we need to invest $1/(1 + i) now, in order to receive $1 at the end of one year. In standard actuarial notation, $1/(1 + i) is denoted v, and is called the discount factor. It is immediately clear that in a deterministic setting, accumulating and taking present values are inverse operations. Keywords: accumulation; annuity; discounting; expected present value; force of interest; interest; present value

This article has no abstract. Keywords: international actuarial notation; actuarial symbols; actuarial history; actuarial mathematics