This book provides a comprehensive introduction to actuarial mathematics, covering both deterministic and stochastic models of life contingencies, as well as more advanced topics such as risk theory, credibility theory and multi-state models. This new edition includes additional material on credibility theory, continuous time multi-state models, more complex types of contingent insurances, flexible contracts such as universal life, the risk measures VaR and TVaR. Key Features: • Covers much of the syllabus material on the modeling examinations of the Society of Actuaries, Canadian Institute of Actuaries and the Casualty Actuarial Society. (SOA-CIA exams MLC and C, CSA exams 3L and 4.) • Extensively revised and updated with new material. • Orders the topics specifically to facilitate learning. • Provides a streamlined approach to actuarial notation. • Employs modern computational methods. • Contains a variety of exercises, both computational and theoretical, together with answers, enabling use for self-study. An ideal text for students planning for a professional career as actuaries, providing a solid preparation for the modeling examinations of the major North American actuarial associations. Furthermore, this book is highly suitable reference for those wanting a sound introduction to the subject, and for those working in insurance, annuities and pensions.

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... Employees as engine room for organisation growth and development are exposed to many serious perils, such as personal losses including incapacity, disability and death. Although individual cannot completely prevent the occurrence of these perils, but can make provision against their financial impacts (Black & Skipper, 2000; and Promislow, 2011). Of course, the function of insurance is to guard against such misfortunes by having the losses of the unfortunate few paid by contributions of many who are exposed to the similar peril. ...

... Employees as engine room for organisation growth and development are exposed to many serious perils, such as personal losses including incapacity, disability and death. Although individual cannot completely prevent the occurrence of these perils, but can make provision against their financial impacts (Black & Skipper, 2000;and Promislow, 2011). Of course, the function of insurance is to guard against such misfortunes by having the losses of the unfortunate few paid by contributions of many who are exposed to the similar peril. ...

... Much of our life is based on the belief that the future is largely unpredictable. Insurance is the mechanism for dealing with future unpredictable (Promislow, 2011). A commonly accepted definition of risk is that it is the possibility that something bad happen. ...

An understanding of occupational risks employees are exposed to in order to take proactive measures to make provisions for their financial impact cannot be overemphasized. This study examines availability and effectiveness of group life assurance policy in employees' place of work as well as determining the financial shocks they may be exposed to if such policy is not in place. The sample size of 533 was drawn from the following organisations: University of Benin, some selected banks in Benin City and manufacturing companies with a minimum of 20 staff strength. Cramer's V and ANOVA were used to analysis the data collected. The findings showed that: only 2.5% of the respondents used life insurance to reduce their level of exposures to occupational risks while the remaining 97.5% rely on unsustainable informal means; there is a strong relationship between employees' risk exposure and life insurance status (Cramer's V = 0.601); the relationship between employees' income and life insurance consumption is moderately strong (Cramer's V = 0.562, p < 0.05); and that informal risk mitigation is significantly varied with family system (F = 3.202, p < 0.05). On this basis, the study recommended that government should create incentives for the employers to put in place a group life assurance policy for their respective employees as part of benefits package in order to cushion the effect of economic hardship the dependants of working class are likely to experience.

... The purpose of the Sudoku game is using logical inference, starting from the puzzle form of Figure 39, to uncover those un-starred numbers in Figure Chistmas presents for the subsequent three years. 63 5* 7* 1* 9* 4* 3* 2* 9* 1* 3* 4* 5* 2* 7* 6* 8* 4* 2* 3* 7* 9* 5* 1* 9* 2* 4* 5* 7* 1* 5* 6* 4* 5* 4* 6* 8* 3* 1* 9* 2* 4* 6* 5* 1* 8* 3* 1* 3* 8* 4* 6* 2* 5* 7* 9* 5* 8* 2* 3* 1* 4* 6* 6 14 5* 7* 1* 8 18 9* 4* 3* 2* 9* 1* 3* 4* 5* 2* 7* 6* 8* 4* 8 15 2* 3* 6 19 7* 9* 5* 1* 3 12 6 13 9* 2* 1 1 4* 5* 8 3 7* 8 16 7 17 1* 9 7 3 6 5* 6* 2 2 4* 5* 2 4 4* 6* 7 5 8* 3* 1* 9* 2* 4* 6* 5* 9 9 1* 8* 7 10 3* 1* 3* 8* 7 8 4* 6* 2* 9 11 5* 7* 9* 5* 8* 2* 3* 1* 4* 6* Therefore, I literally ate and drank Sudoku during the entire period of those three years. ...

... Rather, be prepared for some easy way to pop up. Figure 3 for Puzzle 1 182 5* 6* 2* 1* 4* 1* 1* 4* 7* 2* 3* 3* 6* 8* 2* 4* 7* 5* 4 8 6* 2* 1 1 1* 2 7 4* 6 11 7 12 2 6 1* 3 13 1* 4* 7* 2 5 2* 4 10 4 9 2 4 3* 3* 2 2 Figure 6 for Puzzle 1 183 5* 4 8 6* 2* 1 1 7 14 3? 3? 1* 2 7 4* 6 11 7 12 2 6 1* 3 13 1* 4* 7* 2 5 2* 4 10 4 9 2 4 3* 3* 2 2 6* 8* 2* 4* 7* 2 3 5* 4 8 6* 2* 1 1 7 14 1* 2 7 7 35 3 25 4* 6 11 7 12 6 27 3 26 4 24 2 6 1* 3 13 6 30 1* 4* 7* 2 5 2* 8 34 6 28 4 10 4 9 2 4 3* 1 33 6 29 3* 2 2 4 23 1 22 6* 7 31 6 19 3 16 19 3 16 1* 3 13 6 30 1* 4* 8 36 7* 2 5 2* 8 34 6 28 4 10 4 9 2 4 3* 1 33 6 29 3* 2 2 4 23 1 22 6* 7 31 6 19 3 16 139b7 442b8 452b1 525g Figure 85. Figure 3 for Puzzle 2 185 2* 3* 8* 5* 7* 3* 1* 6* 8* 5* 2* 1* 2* 6* 4* 7* 3* 2* 3* 8* 2/3 5* 3/2 7* 5 1 3* 1* 6* 8* 5* 2* 8 3 1* 3 4 6 2 2* 5/8 6* 4* 8/5 7* 3* 2* 3* 9 53 5 32 1 28 7 18 4 44 6 58 8 49 4 52 6 38 1 40 9 59 8* 2 6 5* 7 15 9* 1* 2* 4* 3* 2* 8* 6* 9* 5* 1* 4* 8* 2* 1* 5* 7* 9* 5 14 2 7 1* 1 3 2* 4* 9 15 2 5 1 6 3* 9 13 8 16 2* 8* 5 9 6* 9 10 3? 3? 9* 5* 3? 1* 8 12 2 8 4* 1 1 5 2 8* 2* 7 11 1* 5* 7* 8 4 9* 5 14 2 7 1* 1 3 2* 4* 9 15 2 5 1 6 3* 9 13 8 16 2* 8* 5 9 6* 9 10 3? 9* 5* 3? 1* 8 12 2 8 4* 1 1 5 2 8* 2* 7 11 1* 5* 7* 8 4 No# Therefore, we can Take in Figure 89 317b5: 3c5b5→No#(95), but with the question mark left behind. ...

... For the puzzle of 23 3* 2 3 8* 6 15 5* 8 16 6* 3* 2 2 2* 3 1 1 10 4 14 5* 3* 2 5 7* 4 9 1 8 9 30 4 24 8 21 2* 7 29 1* 5 19 23 3* 2 3 8* 6 15 5* 8 16 6* 1? 3* 2 2 2* 3 1 1 10 4 14 5* 3* 2 5 7* 4 9 1 8 9 30 4 24 8 21 2* 7 29 1* 5 19 4* No3 No3 6* 1 1 9 31 2* 3 12 6 3 2 6 No3 4 5 1* 8 22 1 2 8* 9 8 3 7 2 7 4 6 6 4 5 18 7 28 4 23 3* 2 3 8* 6 15 5* 8 16 6* 3* 2 2 2* 3 1 1 10 4 14 5* 3* 2 5 7* 4 9 1 8 9 30 4 24 8 21 2* 7 29 1* 5 19 23 3* 2 3 8* 6 15 5* 8 16 6* 3* 2 2 2* 3 1 1 10 4 14 5* 3* 2 5 7* 4 9 1 8 9 30 4 24 8 21 2* 7 29 1* 5 19 3 13 6 26 6 25 4* 5 20 3 11 2 4 8 17 9 27 4* 6* 9 31 2* 3 12 6 4 2 6 4 5 No# 1* 8 22 3 3 8* 1 2 2 7 5 18 7 28 4 23 3* 2 3 8* 6 15 5* 8 16 6* 1 1 3* 2 2 2* 3 1 1 10 4 14 5* 3* 2 5 7* 4 9 1 8 9 30 4 24 8 21 2* 7 29 1* 5 19 3 13 6 26 6 25 4* 5 20 4* 7* 5* 7* 2* 6* 3* 8* 5* 2* 1* 4* 1* 7* 1* 5* 9* ...

Tsao, Hung-ping (2020). Mathematics of Hung-ping Tsao. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Wang, Lawrence K. and Tsao, Hung-ping (editors). Volume 2, Number 11, November 2020; 336 pages. Lenox Institute Press, Newtonville, NY, 12128-0405, USA. No. STEAM-VOL2-NUM11-NOV2020; ISBN 978-0-9890870-3-2. .............ABSTRACT: I would like to share some of my ideas in Number Theory, Actuarial Mathematics, Sudoku Solving and Optimization Teaching with college students and colleagues. ............KEYWORDS: Natural sequence, AP-sequence, Power-sum, Product-sum, Sorting, Combination, Permutation, Cycle, Subset, Binomial coefficient, Stirling number, Pascal triangle, Bernoulli coefficient, Eulerian number, Bell number, Ordered Bell polynomial, Eulerian Bell polynomial, Recursive formula, q-Gaussian coefficient, Life insurance, Life annuity, Interest, Mortality, Contingency, Premium, Reserve, Sudoku, Puzzle, Row, Column, Box, Unique solution, Flipflops chain, Residue.

... This sets a link between accounting and loan theory: The time-t outstanding balance is, in an amortization plan, the residual principal debt at time t; the IDF represents the contractual rate(s), the variation of the outstanding balance is the principal repayment, the cash flows are the instalments, and the product ry t−1 (r) is the interest charge (see also Kellison, 1991; Promislow, 2006). The idea of income as interest is unambiguous and already recognized in the relevant literature (see Forker and Powell, 2000, p. 237). ...

... This sets a link between accounting and loan theory: The time-t outstanding balance is, in an amortization plan, the residual principal debt at time t; the IDF represents the contractual rate(s), the variation of the outstanding balance is the principal repayment, the cash flows are the instalments, and the product r t y t−1 ( r) is the interest charge (see also Kellison, 1991; Promislow, 2006). The idea of income as interest is unambiguous and already recognized in the relevant literature (see Forker and Powell, 2000, p. 237). ...

  • Carlo Alberto Magni Carlo Alberto Magni

This paper presents a new way of measuring residual income, originally introduced by Magni (2000a,b,c, 2001a,b, 2003). Contrary to the standard residual income, the capital charge is equal to the capital lost by investors multiplied by the cost of capital. The lost capital may be viewed as (a) the foregone capital, (b) the capital implicitly infused into the business, (c) the outstanding capital of a shadow project, (d) the claimholders' credit. Relations of the lost capital with book values and market values are studied, as well as relations of the lost capital residual income with the classical standard paradigm; many appealing properties are derived, among which an aggregation property. Different concepts and results, provided by different authors in such different fields as economic theory, management accounting and corporate finance, are considered: O'Hanlon and Peasnell's (2002) unrecovered capital and Excess Value Created; Ohlson's (2005) Abnormal Earnings Growth; O'Byrne's (1997) EVA improvement; Miller and Modigliani's (1961) investment opportunities approach to valuation; Young and O'Byrne's (2001) Adjusted EVA; Keynes's (1936) user cost; Drukarczyk and Schueler's (2000) Net Economic Income; Fernández's (2002) Created Shareholder Value; Anthony's (1975) profit. They are all conveniently reinterpreted within the theoretical domain of the lost-capital paradigm and conjoined in a unified view. The results found make this new theoretical approach a good candidate for firm valuation, capital budgeting decision-making, managerial incentives and control.

... Penman, 2007). In the theory of financial contracts (and in actuarial sciences) the notion of interest is used since ancient times to represent the remuneration of the lender (Van de Mieroop, 2005) and is computed as the difference between the installment paid by the borrower and the principal repayment (Francis, 2004; Fabozzi, 2006; Promislow, 2006; Werner and Sotskov, 2006). The notion of return in capital budgeting is referred to a project: in a one-period project return is the difference between the end-of-period payoff and the initial outlay. ...

... This form stresses the role of the return rate (interest rate) as a driver of capital increase: it is usual in the construction of amortization tables, in the computation of project balances and in financial and insurance applications (Levi, 1964; Robichek and Myers, 1965; Teichroew, Robichek, Montalbano, 1965a,b; Hansen, 1972; Peccati, 1991; Promislow, 2006). The fundamental equation (1) alongside its equivalents eqs. ...

  • Carlo Alberto Magni Carlo Alberto Magni

This paper deals with the notion of residual income, which may be defined as the surplus profit that residues after a capital charge (opportunity cost) has been covered. While the origins of the notion trace back to the 19th century, in-depth theoretical investigations and widespread real-life applications are relatively recent and concern an interdisciplinary field connecting management accounting, corporate finance and financial mathematics (Peasnell, 1981, 1982; Peccati, 1987, 1989, 1991; Stewart, 1991; Ohlson, 1995; Arnold and Davies, 2000; Young and O'Byrne, 2001; Martin, Petty and Rich, 2003). This paper presents both a historical outline of its birth and development and an overview of the main recent contributions regarding capital budgeting decisions, production and sales decisions, implementation of optimal portfolios, forecasts of asset prices and calculation of intrinsic values. A most recent theory, the systemic-value-added approach (also named lost-capital paradigm), provides a different definition of residual income, consistent with arbitrage theory. Enfolded in Keynes's (1936) notion of user cost and forerun by Pressacco and Stucchi (1997), the theory has been formally introduced in Magni (2000a,b,c; 2001a,b; 2003), where its properties are thoroughly investigated as well as its relations with the standard theory; two different lost-capital metrics have been considered, for value-based management purposes, by Drukarczyk and Schueler (2000) and Young and O'Byrne (2001). This work illustrates the main properties of the two theories and their relations, and provides a minimal guide to construction of performance metrics in the two approaches.

... I later published thirteen papers (6)-(18), a lecture note (4) and a textbook (5) in Chinese, which I did consult (2). Recently, I found out that my innovative ideas such as the uniform representation of a general life contingency function and its derivative were not even mentioned in (3). Therefore, I feel obliged to write this chapter for the benefit of readers. ...

... (see Peasnell, 1982, p. 108). The above relation coincides with the recursion formula used in financial and actuarial mathematics for computing the balance (residual debt) in a loan contract (Promislow, 2006;Werner and Sotskov, 2006;Kellison, 2009), where b 0 is the amount borrowed, a t b t−1 represents interest and f t is the installment. This fact enables one to interpret f as a loan contract whereby shareholders lend the firm the amount b 0 and receive the installment f t at time t. ...

  • Carlo Alberto Magni Carlo Alberto Magni

This paper presents a theoretical framework for valuation, investment decisions, and performance measurement based on a nonstandard theory of residual income. It is derived from the notion of "unrecovered" capital, which is here named "lost" capital because it represents the capital foregone by the investors. Its theoretical strength and meaningfulness is shown by deriving it from four main perspectives: financial, microeconomic, axiomatic, accounting. Implications for asset valuation, capital budgeting and performance measurement are investigated. In particular: an aggregation property is shown, which makes the simple average residual income play a major role in valuation; a dual relation between the standard theory and the lost-capital theory is proved, clarifying the way periodic performance is computed in the two paradigms and the rationale for measuring performance with either paradigm; the average accounting rate of return is shown to be more reliable than the internal rate of return as a capital budgeting criterion, and maximization of the average residual income is shown to be equivalent to maximization of Net Present Value (NPV). Two metrics are also presented: one enjoys the nice property of robust goal congruence irrespective of the sign of the cash flows; the other one enjoys periodic consistency in the sense of Egginton (1995). The results obtained suggest that this theory might prove useful for real-life applications in firm valuation, capital budgeting decision-making, ex ante and ex post performance measurement, incentive compensation. A numerical example illustrates the implementation of the paradigm to the EVA model and the Edwards-Bell-Ohlson model.

... The expression τ k · w k−1 = w k − w k−1 + a k is a most general framework shared by several economic domains: in economic theory it defines income or profit as the maximum that can be consumed by an individual in a determined period without impairing her wealth or capital (see Fetter, 1937; Hicks, 1946; Lee, 1985. See also Samuelson, 1964); in accounting it gives voice to the so-called clean surplus relation which connects earnings and dividends (see Canning, 1929; Brief and Peasnell, 1995; Penman, 2007); in the theory of financial contracts and in capital budgeting, as well as in actuarial sciences, it is essential in the construction of amortization tables and project balances as a function of cash flows and return rates (see Robichek and Myers, 1965; Teichroew, Robichek and Montalbano, 1965a,b; Francis, 2004; Van de Mieroop, 2005; Fabozzi, 2006; Promislow, 2006; Werner and Sotkov, 2006). We will also make use of the notion of internal return vector r, introduced by Weingartner (1966) , which generalizes the notion of internal rate of return and enables one to overcome the problems related to the existence and uniqueness of internal rate of return. ...

This paper presents an axiomatization of residual income, aka excess profit, and illustrates how it may univocally engenders fixed-income or variable-income assets. In the first part it is shown that, depending on the relations between excess profit and the investor's excess wealth, a well-specified theory of residual income is generated: one is the standard theory, which historically traces back to Hamilton (1777) and Marshall (1890) and is a deep-rooted notion in economic theory, finance, and accounting. Another one is the systemic value added or lost-capital paradigm: introduced in Magni (2000, 2003), the theory is enfolded in Keynes's (1936) notion of user cost and is naturally generated by an arbitrage-theory perspective. In the second part, the paper reverts the usual analysis: instead of computing residual incomes profits from a pattern of cash flows, residual incomes are fixed first to derive vectors of cash flows. It is shown that variable- or fixed-income assets may be constructed on the basis of either theory starting from pre-determined growth rates for excess profit. In particular, zero-coupon bonds and coupon bonds traded in a capital market are shown to be deducted as equilibrium vectors of residual-income-based assets.

... Por outro lado, Jesus (2015) afirma que a desvinculação dos demais benefícios do salário mínimo faz com que os beneficiários que ganham mais do que um salário mínimo recebam reajustes dos seus benefícios em um nível inferior aos daquele e, consequentemente, vejam seus benefícios perderem o poder de compra. iv Para calcular a EVN para a tábua projetada é assumida a tradicional hipótese de distribuição uniforme das mortes ao longo do ano (Promislow, 2011). v Disponível em http://www.atuarios.org.br/noticia/89reforma-previdenciaria, ...

  • Filipe Costa de Souza

Com o objetivo de reformar o sistema previdenciário brasileiro, o Governo Federal enviou para a Câmara dos Deputados a Proposta de Emenda à Constituição nº 287/2016, a qual, dentre outras medidas, propõe instituir uma idade mínima progressiva de aposentadoria. Este trabalho avaliou os aspectos distributivos que esta medida pode gerar aos contribuintes, a partir do cálculo das alíquotas atuarialmente justas frente à heterogeneidade da mortalidade da população brasileira. Como principal resultado, constatou-se que haverá uma redistribuição negativa, em que os indivíduos com menor expectativa de vida, que ingressaram no mercado de trabalho mais jovens e que têm baixo crescimento salarial irão financiar as pensões daqueles mais longevos, que ingressaram no mercado de trabalho mais velhos e com maior perspectiva de crescimento salarial.

... We will not work with cumulative numbers and other items, since it is in our case not needed. Further reading and extension might be found in [10]. The probability of survival in the i th interval is ...

  • Rafael Schwarzenegger Rafael Schwarzenegger

This master's thesis is describing and applying parametric and nonparametric reliability models for censored data. It shows the implementation of reliability in the Six Sigma methodology. The methods are used in survival/reliability of real technical data.

... The stationary state of the phenomenon is then described by the limiting behavior of the Markov chain 295 (Ngoko, Sugihara, and Funaki 2014). After a sufficiently enough time, the resulting probability vector will be the same as the original one whenever it is multiplied by the transition matrix (Promislow 2015): ...

This paper presents a method of building a two-state multi-criteria Markov chain for stochastic solar generators. The design is based on different criteria which are defined using some statistical thresholds such as quartiles and averages. These thresholds have been used for classifying the state of all individual data samples. The proposed model is analyzed via computer simulation on Matlab environment using the historical data recorded over a period of 29 years by three meteorological stations in Algeria (Tamanrasset, Ghardaïa, and Oran). In order to validate and compare the performance of our model, several statistical tests were performed. The root mean square error and mean absolute error range, respectively, from 0.18 to 0.26 and −0.10 to 0.16. The obtained results proved that the model is effective to predict the data. Moreover, this study can be applied in its methodology not only in Algeria sites but also in other countries of the world and can serve as a reference for different climates, which are Mediterranean coastal climate, desert climate, and Sahel climate. Hence, it is useful to the designers of solar energy systems.

... Referensi mengenai teori asuransi jiwa gabungan antara lain Bowers (1991), Promislow (2011), dan Sertdemir (2013). Penelitian lebih lanjut, perhitungan nilai aktuaria berdasarkan hukum mortalita Gompertz dapat menggunakan pendekatan kontinu dari percepatan mortalita (force of mortality) (Yang and Zhou, 2003) sehingga memudahkan dalam perhitungan. ...

  • Khoiroh Alfiana
  • Ade Ima Afifa Himayati

p class="JRPMAbstractBody">Penelitian ini berfokus pada pembahasan mengenai perhitungan premi asuransi jiwa gabungan dwiguna. Asuransi jiwa gabungan dwiguna memberikan dua manfaat sekaligus yaitu manfaat diberikan jika salah satu tertanggung meninggal dunia atau memberikan manfaat jika kedua tertanggung masih hidup hingga akhir masa kontrak asuransi. Perhitungan premi asuransi jiwa gabungan dwiguna ini melibatkan asumsi Gompertz pada probabilitas hidup dan pengaruh nilai tukar pada tingkat bunga. Perhitungan premi dihitung berdasarkan asumsi mortalita Gompertz, yang mempengaruhi probabilitas hidup tertanggung status gabungan. Selain itu, adanya pengaruh fluktuasi dollar terhadap tingkat bunga yang berdampak pada nilai premi asuransi. Berdasarkan hasil yang diperoleh, turunnya nilai tukar rupiah mengakibatkan tingkat bunga meningkat jika dibandingkan dengan tingkat bunga acuhan. Sehingga, dapat disimpulkan nilai premi asuransi dipengaruhi nilai tukar. Hal ini berakibat adanya peningkatan risiko yang dialami perusahaan asuransi. Kata kunci : asuransi jiwa gabungan dwiguna, hukum mortalita Gompertz, nilai tukar. </p

... (The face amount can be allowed to change from year to year.) Type B Universal Life insurance policies have such death benefits [2,8]. ...

  • Elias Shiu Elias Shiu
  • Xiaoyi Xiong

For a general fully continuous life insurance model, the variance of the loss-at-issue random variable is the expectation of the square of the discounted value of the net amount at risk at the moment of death. In 1964 Jim Hickman gave an elementary and elegant derivation of this result by the method of integration by parts. One might expect that the method of summation by parts could be used to treat the fully discrete case. However, there are two difficulties. The summation-by-parts formula involves shifting an index, making it somewhat unwieldy. In the fully discrete case, the variance of the loss-at-issue random variable is more complicated; it is the expectation of the square of the discounted value of the net amount at risk at the end of the year of death times a survival probability factor. The purpose of this note is to show that one can indeed use the method of summation by parts to find the variance of the loss-at-issue random variable for a fully discrete life insurance policy.

The book aims at presenting technical and financial features of life insurance, non-life insurance, pension plans. The book has been planned assuming non-actuarial readers as its "natural" target, namely - advanced undergraduate and graduate students in Economics, Business and Finance; - professionals and technicians operating in Insurance and pension areas, whose job may regard investments, risk analysis, financial reporting, etc, and hence implies a communication with actuarial professionals and managers. Given the assumed target, the book focuses on technical and financial aspects of insurance, however avoiding the use of complex mathematical tools. In this sense, the book can be placed at some "midpoint" of the existing literature, part of which adopts more formal approaches to insurance problems implying the use of non-elementary mathematics, whereas another part addresses practical questions totally avoiding even simple mathematical tools (which, in our opinion, can conversely provide effective tools for presenting technical and financial features of the insurance business). © Springer-Verlag Berlin Heidelberg 2011. All rights are reserved.

This paper analyzes a novel type of mortality contingent-claim called a ruin-contingent life annuity (RCLA). This product fuses together a path-dependent equity put option with a "personal longevity" call option. The annuitant's (i.e. long position) payoff from a generic RCLA is \$1 of income per year for life, akin to a defined benefit pension, but deferred until a pre-specified financial diffusion process hits zero. We derive the PDE and relevant boundary conditions satisfied by the RCLA value (i.e. the hedging cost) assuming a complete market where No Arbitrage is possible. We then describe some efficient numerical techniques and provide estimates of a typical RCLA under a variety of realistic parameters. The motivation for studying the RCLA on a stand-alone basis is two-fold. First, it is implicitly embedded in approximately \$1 trillion worth of U.S. variable annuity (VA) policies; which have recently attracted scrutiny from financial analysts and regulators. Second, the U.S. administration - both Treasury and Department of Labor - have been encouraging Defined Contribution (401k) plans to offer stand-alone longevity insurance to participants, and we believe the RCLA would be an ideal and cost effective candidate for that job.

  • Carlo Alberto Magni Carlo Alberto Magni

This paper analyzes the relations among different concepts such as earnings, profit, interest, rate, consumption, dividend, installment, cash flow, capital. It aims atembracing these notions in a unique conceptual "umbrella" , consisting of five perspectives: (1) accounting, (2) economic theory, (3) theory of finance, (4) loan theory, (5) financial mathematics. These notions and these domains constitute a seeming mishmash: in fact, the hub of the umbrella is given by a unique fundamental relation, shared by all five perspectives and whose ingredients are capital, profit, and cash flow. On the basis of the fundamental relation, market value and book value of a firm are easily obtained.

  • Carlo Alberto Magni Carlo Alberto Magni

Residual income as commonly described in academic papers and in real-life applications may be formally described as a function of three variables: (i) the capital invested, (ii) the rate of return, (iii) the opportunity cost of capital. This paper shows that a different paradigm of residual income is generated if a fourth element is added: (iv) the capital that investors lose if they infuse their funds into the firm (or project). The lost-capital paradigm has various interesting economic, nancial, accounting interpretations and bears intriguing formal and conceptual relations to the standard paradigm. It may be soundly employed in real-life applications as a tool for rewarding managers as well as for appraising firms. Firm value is shown to be a function of total abnormal earnings and independent of time, if the new paradigm is used: what matters is only the book value and the sum of total expected residual incomes, not the periods in which they are generated. This aggregation property is particular important for highlighting the link between accounting values and market values. A numerical example illustrates the practical implementation of the new paradigm to the Economic Value Added and the Edwards-Bell-Ohlson model; also, a model is presented which has the nice property of being aligned in sign with the Net Present Value: this makes it a good candidate for use in value-based management.

The insurer's debt position, which is an obvious implication of the single premium arrangement, must be realized also when other premium arrangements are adopted. This need clearly emerged in Sect. 4.4.1. We recall that an asset accumulation - decumulation process develops, throughout the policy duration, against the insurer's debt position. A technical tool for assessing the insurer's debt is provided by the socalled mathematical reserve.

  • Michael Cheffena Michael Cheffena

A novel physical-statistical channel model for simulating the signal effect by moving human bodies is presented. The human body is modeled as vertically oriented dielectric cylindrical volume. The received signal is assumed to be composed of a direct component which might be subject to shadowing and a multipath component due to reflection and diffuse scattering, i.e., a Ricean channel. The shadowing effect of the direct signal component is calculated using Kirchhoff diffraction equation. The multipath component is parameterized by calculating the reflected fields from the floor, ceiling and walls of the indoor environment as well as scattered fields from moving human bodies. Poisson and Exponential distributions are used to describe the shadowing and inter-shadowing events caused by multiple bodies, respectively. Furthermore, simulation results of the first and second order statistics of the received signal affected by moving human bodies for 3.35 GHz and 60 GHz signals are presented. In addition, initial validation of the developed model are performed using an empirical model for human body shadowing and reported measurement results.

Some insurance firms challenged with a portfolio of high-variance risks face the classic trade-off between risk spreading and risk retaining. Using crop insurance as an example, a new solution to this problem is undertaken to uncover an improved reinsurance design. Joint self-managed reinsurance pooling and private reinsurance are combined in a portfolio approach utilizing combinatorial optimization with a genetic algorithm (Model C), achieving high surplus, high survival probability, and low deficit at ruin. This portfolio model may also be useful for other large natural disaster and weather-related insurance portfolios, and other portfolio applications.

When writing insurance contracts, the insurer takes risks originating from various causes. In life insurance, causes of risk relate to financial aspects (e.g., investment yield, inflation, etc.), demographical aspects (e.g., lifetimes of policyholders, lapses and surrenders, etc.), and expenses. In this chapter, we deal with demographical aspects only, focussing on policyholders' lifetimes, which in turn determine the frequency of death in a portfolio.

  • Charles Sutcliffe Charles Sutcliffe

Annuities are perceived as being illiquid financial instruments, and this has limited their attractiveness to consumers and inclusion in financial models. However, short positions in annuities can be replicated using life insurance and debt, permitting long positions in annuities to be offset, or short annuity positions to be created. The implications of this result for the annuity puzzle, arbitrage between the annuity and life insurance markets, and speculation on expected longevity are investigated. It is argued that annuity replication could help solve the annuity puzzle, improve the price efficiency of annuity markets and promote the inclusion of annuities in household portfolios.

Tontines were once a popular type of mortality-linked investment pool. They promised enormous rewards to the last survivors at the expense of those died early. And, while this design appealed to the gambling instinct, it is a suboptimal way to generate retirement income. Indeed, actuarially-fair life annuities making constant payments–where the insurance company is exposed to longevity risk–induce greater lifetime utility. However, tontines do not have to be structured the historical way, i.e. with a constant cash flow shared amongst a shrinking group of survivors. Moreover, insurance companies do not sell actuarially-fair life annuities, in part due to aggregate longevity risk.

  • David Raymond Christiansen

We present an in-progress domain-specific language for actuaries. Due to the mathematical sophistication of actuaries and the relatively high degree of formalization of the field, we conjecture that a dependently-typed functional language with special support for actuarial models will enable actuaries to develop software that is robust and understandable.

  • Shailaja Deshmukh

In some life insurance policies, benefit to a single life or a group is subject to a type of contingency. For example, the death of an individual may be due to an accident or due to any other cause. In most of the insurance policies the coverage is first given for the base cause, and then there are policy riders for additional benefits. If the death is due to an accident, then the benefit structure is different, and usually the benefit is more than the base coverage. In such cases, the benefit structure and consequently the premium structure depend on time to death as well as on the cause of death. Survivorship models incorporating two random mechanisms, time to termination, and various modes of termination are known as multiple decrement models. Chapter 1 introduces multiple decrement model and the construction of multiple decrement table. Section 1.2 discusses the joint distribution theory of time to decrement and cause of decrement random variables with several illustrative examples. The highlight of the book is its usage of R software for statistical computations. R software is freely available from public domain. A brief introduction of R software is given in the second section, and R code is used for all the computations in subsequent chapters. Sections 1.3 and 1.4 are devoted to the construction of multiple decrement table, using associated single decrement model and central rate bridge. R commands are given for the construction of multiple decrement table.

  • Pascal Winter
  • Frédéric Planchet

In the context of global aging population, improved longevity and ultra-low interest rates, the question of pension plan under-funding and adequate elderly financial planning is gaining awareness worldwide, both among experts, regulatory bodies, and popular media. Additional emergence of societal changes—Peer to Peer business model and Financial Disintermediation—have contributed to the resurgence of the concept of "Tontines" in various papers and the proposal of further models. These generalizations can offer efficient decumulation schemes with high longevity protection which is particularly well adapted for retirement needs—both for its members and carriers. In this paper, we revisit the mechanism proposed by Fullmer and Sabin (Journal of Accounting and Finance, 2019. https://doi.org/10.33423/jaf.v19i8.2615)—which allows the pooling of Modern Tontines through a self-insured community. This "Tontine" generalization retains the flexibility of an individual design: open contribution for a heterogeneous population, individualized asset allocation and predesigned annuitization plan. The actuarial fairness is achieved by allocating the deceased proceedings to survivors using a specific individual pool share which is a function of the prospective expected payouts for the period considered. After a brief introduction, this article provides a formalization of the mathematical framework with prospective analysis, characterizes the inherent bias, generalizes the mechanism to joint lives, and analyses simulated outcomes based on various assumptions. A reverse moral hazard limit is exposed and discussed (the "Term Dilemma"). Some solutions are then proposed to overcome scheme shortcomings and some requirements for practical implementation are discussed.

  • Nadine Gatzert
  • Hato Schmeiser

This chapter provides an overview of new life insurance financial products. After a general market overview, Sect. 36.2 presents different forms of traditional and innovative life insurance financial products and their main characteristics. Since unit-linked and equity- indexed type contracts represent the basis for most innovative products in recent years, Sect. 36.3 presents basic aspects of the modeling, valuation, and risk management of unit- linked life insurance contracts with two forms of investment guarantees (interest rate and lookback guarantees) and different underlying investment strategies. In Sect. 36.4, variable annuities are discussed and focus is laid on challenges for insurers in regard to pricing and risk management of the various embedded options. Section 36.5 finally puts the customer's perspective in the center of the analysis, along with a discussion of current developments regarding product information documents and performance and risk-return profiles, which is of special relevance for new and traditional products.

  • Muhsin Tamturk Muhsin Tamturk
  • Sergey Utev

The finite time ruin probability in the classical surplus process setup with additional capital injections and withdrawals is investigated via the Quantum Mechanics Approach. The results are compared with the Picard–Lefevre Appell Polynomial approach and the traditional Markov Chain approach. In addition, several optimization problems in the insurance market are numerically solved by applying the Quantum Mechanics Approach.

At vehicle insurance companies, the determination of the appropriate pure premium will make the business run well. In this study, we were modeling claims frequency data by considering the characteristics of policyholder such as policyholder's age, marital status, sex, car engine capacity, and age. The data used in this study is a non-motor vehicle and non-truck motor vehicle insurance data, which filed claims during 2013 in a general insurance company. Explaining the significance or value of the research. We are using Generalized Linear Model Multivariate Poisson with Artificial Marginal (GLM-MPAM) to estimate model parameters. The parameter values of this model are estimated using the Maximum Likelihood Estimation method. Furthermore, the estimation result of the parameter can be alternative in the calculation of the pure premium in the next period.

Credit Value Adjustment is the charge applied by financial institutions to the counter-party to cover the risk of losses on a counterpart default event. In this paper we estimate such a premium under the Bates stochastic model (Bates in The Review of Financial Studies 9(1): 69–107, 1996), which considers an underlying affected by both stochastic volatility and random jumps. We propose an efficient method which improves the Finite-Difference Monte Carlo (FDMC) approach introduced by de Graaf et al. (Journal of Computational Finance 21, 2017) In particular, the method we propose consists in replacing the Monte Carlo step of the FDMC approach with a finite difference step and the whole method relies on the efficient solution of two coupled partial integro-differential equations which is done by employing the Hybrid Tree-Finite Difference method developed by Briani et al. (arXiv:1603.07225 2016;IMA Journal of Management Mathematics 28(4): 467–500, 2017;The Journal of Computational Finance 21(3): 1–45, 2017). Moreover, the direct application of the hybrid techniques in the original FDMC approach is also considered for comparison purposes. Several numerical tests prove the effectiveness and the reliability of the proposed approach when both European and American options are considered. Subject classification numbers as needed.

  • Moshe Arye Milevsky Moshe Arye Milevsky

Continuing from where the prior Chap. 10.1007/978-3-030-51434-1_7 left off, this chapter explains how to construct and work with (a.k.a. cook) the remaining lifetime random variable: Tx. The approach to lifetime randomness is based on the underlying mortality hazard rate λx, which is the continuous-time (and probabilistic) analog of the 1-year death rate qx. This chapter models and constructs Tx variables for a variety of given mortality hazard rates λx. This then sets the stage for the main intellectual objective, which is to introduce (and justify) the Benjamin Gompertz law of mortality. That important and famous law is experienced via a number of simulation exercises and experiments in R. The Gompertz model is the computational backbone for many of the subsequent computations (and recipes) in the book.

  • Moshe Arye Milevsky Moshe Arye Milevsky

This chapter develops a methodology for valuing simple cash-flow streams that last a lifetime, which are part of most Defined Benefit (DB) pensions. The focus is on the longevity-contingent building blocks of: (1) immediate, (2) temporary, and (3) deferred income annuities. The chapter begins with a discussion of the value of a longevity-contingent claim and how it differs from the market price versus the manufacturing cost of the product. The algorithms and user-defined R functions are mostly based on the Gompertz law of mortality, although a number of alternative continuous and discrete mortality models are discussed as well. The chapter concludes with a mathematical derivation and implementation of a closed-form expression for the Gompertz Annuity Valuation Model.

  • Doron Kliger Doron Kliger
  • Benny Levikson

This paper presents a new approach for pricing insurance contracts, based both on economic and probabilistic arguments. The novel property of this approach is that it uses the demand for insurance to find the optimal premium an insurer should charge. Our approach stands in contrast to the standard loading factor methods used in actuarial science, where the number of insureds is constant regardless of the charged premium. The insurer maximizes its expected profit, defined as the difference between the expected net revenue from selling insurance contracts and the expected loss due to insolvency. We show how to find the expected-profit maximizing premium, π∗, and its corresponding optimal number of insureds, n∗. The first proposition presented in our paper identifies the premium (and number of insureds that minimize the expected loss due to insolvency). The second proposition gives, for a broad class of demand curves, sufficient conditions for the existence and uniqueness of an internal optimal solution. The third proposition asserts that, due to the suggested expected loss function, the insurer's objective function demonstrates economies to scale. Lastly, we provide a numerical solution for the case of a linear demand curve, giving the optimal premium and number of insureds.

  • Bruce L. Jones

Continuing care retirement communities (CCRCs) offer housing and a variety of services, including long-term care. Typically, the cost of this long-term care is wholly or partially covered by entry and/or periodic fees. Thus, CCRCs provide a long-term-care insurance benefit. For this and other reasons, actuarial involvement in the financial management of CCRCs is desirable. To carry out actuarial analyses of CCRCs, appropriate models are required to describe the status of individual residents and the CCRC population.

  • Daniel Teichroew
  • Alexander A. Robichek
  • Michael Montalbano

The objective of this paper is to investigate the decision-making procedure for accepting or rejecting investment or financing alternatives available to the firm. The properties of the decision rules based on discounted present value and internal rate of return are studied for the class of projects described by a finite sequence of cash flows. The necessary and sufficient conditions under which the decision rules lead to unique solutions are derived. Where the decision rule does not provide a unique solution, it is necessary to define two rates: the project investment rate and the project financing rate. The extension of the project analysis in terms of the two rates permits the derivation of unambiguous decision rules for all projects. The relevance of the results is discussed in the summary.

  • Daniel Teichroew
  • Alexander A. Robichek
  • Michael Montalbano

The purpose of this paper is to prove certain properties of the present and future values of a sequence of cash flows which have applications in the theory of capital budgeting. This is done in Theorems III, IV and V. As an introduction, certain previously available results about the present value function are stated and proved as Theorems I and II. A summary of the relevance of these results in capital budgeting is given in the Summary.

  • Michael R. Powers Michael R. Powers

In recent years, state regulators and state and federal lawmakers in the United States have become increasingly concerned with issues involving both the price and solvency regulation of insurers. In this article, we develop a risk-theoretic framework for studying both rate of return and solvency issues. Employing a certain class of diffusion processes to model insurer net worth, we use the expected discounted cost of insolvency (EDCI), a generalization of the probability of ruin, in constructing the regulator's objective function. This objective function is then used to solve for the optimal rate of return and the optimal loss-to-net worth ratio.