Showing papers in "The North American Actuarial Journal in 2003"
TL;DR: In this article, the authors derived explicit formulas for computing tail conditional expectations for elliptical distributions, a family of symmetric distributions that includes the more familiar normal and student-t distributions.
Abstract: Significant changes in the insurance and financial markets are giving increasing attention to the need for developing a standard framework for risk measurement. Recently, there has been growing interest among insurance and investment experts to focus on the use of a tail conditional expectation because it shares properties that are considered desirable and applicable in a variety of situations. In particular, it satisfies requirements of a “coherent” risk measure in the spirit developed by Artzner et al. (1999). This paper derives explicit formulas for computing tail conditional expectations for elliptical distributions, a family of symmetric distributions that includes the more familiar normal and student-t distributions. The authors extend this investigation to multivariate elliptical distributions allowing them to model combinations of correlated risks. They are able to exploit properties of these distributions, naturally permitting them to decompose the conditional expectation, and allocate t...
374 citations
TL;DR: In this paper, the authors examine properties of risk measures that can be considered to be in line with some "best practice" rules in insurance, based on solvency margins, and demonstrate that imposing properties that are generally valid for risk measures, in all possible dependency structures, which violate best practice rules, though providing opportunities to derive nice mathematical results, leads to problems.
Abstract: We examine properties of risk measures that can be considered to be in line with some “best practice” rules in insurance, based on solvency margins. We give ample motivation that all economic aspects related to an insurance portfolio should be considered in the definition of a risk measure. As a consequence, conditions arise for comparison as well as for addition of risk measures. We demonstrate that imposing properties that are generally valid for risk measures, in all possible dependency structures, based on the difference of the risk and the solvency margin, though providing opportunities to derive nice mathematical results, violates best practice rules. We show that so-called coherent risk measures lead to problems. In particular we consider an exponential risk measure related to a discrete ruin model, depending on the initial surplus, the desired ruin probability, and the risk distribution.
168 citations
TL;DR: In this paper, the authors present an alternative representation of risk measures originally defined in terms of expectations with respect to distorted probabilities, and show that the right-tail, left-tail and two-sided deviations/indices suggested by Wang (1998) can be represented in this alternative form.
Abstract: The authors present an alternative representation of risk measures originally defined in terms of expectations with respect to distorted probabilities. They also show that the right-tail, left-tail, and two-sided deviations/indices suggested by Wang (1998) can be represented in this alternative form. Empirical estimators for these quantities are proposed and their properties explored.
126 citations
TL;DR: In this paper, the authors analyzed a life insurance endowment policy, paid by annual premiums, in which the benefit is annually adjusted according to the performance of a special investment portfolio and a minimum return is guaranteed to the policyholder.
Abstract: This paper analyzes a life insurance endowment policy, paid by annual premiums, in which the benefit is annually adjusted according to the performance of a special investment portfolio and a minimum return is guaranteed to the policyholder. In particular, the author considers both the case in which the annual premium is constant and the case in which the premium also is adjusted according to the performance of the reference portfolio. Moreover, the policy under scrutiny is characterized by the presence of a surrender option, that is, of an American-style put option that enables the policyholder to give up the contract and receive the surrender value. The aim of the paper is to give sufficient conditions under which there exists a (unique) fair premium. This premium is implicitly defined by an equation (or, alternatively, can be viewed as a fixed point of a suitable function) based on a recursive binomial tree ala Cox, Ross, and Rubinstein (1979). An iterative algorithm is then implemented in orde...
115 citations
TL;DR: In this article, a statistical modeling strategy based on extreme value theory was proposed to describe the behavior of an insurance portfolio, with particular emphasis on large claims, using the 1991-92 group medical claims database maintained by the Society of Actuaries.
Abstract: This paper discusses a statistical modeling strategy based on extreme value theory to describe the behavior of an insurance portfolio, with particular emphasis on large claims. The strategy is illustrated using the 1991–92 group medical claims database maintained by the Society of Actuaries. Using extreme value theory, the modeling strategy focuses on the “excesses over threshold” approach to fit generalized Pareto distributions. The proposed strategy is compared to standard parametric modeling based on gamma, lognormal, and log-gamma distributions. Extreme value theory outperforms classical parametric fits and allows the actuary to easily estimate high quantiles and the probable maximum loss from the data.
103 citations
TL;DR: In this paper, the authors proposed an economic model that has the flexibility of modeling the underlying index fund as well as the interest rates, and illustrated some popular EIAs to assess the implication of the proposed model.
Abstract: This paper considers the pricing of equity-indexed annuities (EIAs). Traditionally, the values of the guarantees embedded in these contracts are priced by modeling the underlying index fund while keeping the interest rates constant. The assumption of constant interest rates becomes unrealistic in pricing and hedging the EIAs since the embedded guarantees are often of much longer maturity. To solve this problem, the authors propose an economic model that has the flexibility of modeling the underlying index fund as well as the interest rates. Some popular EIAs are illustrated to assess the implication of the proposed model.
99 citations
TL;DR: In this article, the moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process were investigated. And they were shown to be asymptotic when the claim size is exponentially and subexponentially distributed.
Abstract: This paper investigates the moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process. Using the integro-differential equation that we establish, we obtain some explicit expressions for the moments. Furthermore, when the claim size is exponentially and subexponentially distributed, asymptotic relationships for the moments are derived as the initial capital tends to infinity. Also, we show the joint probability density function of the surplus before ruin and the deficit at ruin.
70 citations
TL;DR: In this article, a closed-form expression for B(u, T), the Laplace-Stieltjes transform of the expected excess of the running maximum of a Wiener process above a positive constant u in a finite time interval of length T, is presented.
Abstract: Pricing exotic options or guarantees in equity-indexed annuities can be problematic. The authors present closed-form formulas for pricing lookback options and dynamic guarantees that facilitate the hedging and reserving for such products. The principal tool used is a closed-form expression for B(u, T), the Laplace-Stieltjes transform of the expected excess of the running maximum of a Wiener process above a positive constant u in a finite time interval of length T. If the aggregate net income of a company is modeled with a Wiener process, then the excess of the running maximum above u can be interpreted as aggregate dividend payments, and the quantity B(u, T) is the expectation of the discounted value of the dividend payments up to time T. The formula for B(u, T) is used to price European lookback options (call and put, fixed and floating strike). It is also used to price dynamic fund protection, which is a guarantee on an investment fund: The number of units of the investment fund is increased wh...
67 citations
TL;DR: In this paper, the expected present value of the payments to be made by the sponsor as well as that of the refunds to the sponsor is calculated for situations where these two expected values are equal, and the refunds at the upper barrier are interpreted as the dividends paid to the shareholders of a company according to a barrier strategy.
Abstract: In this paper asset and liability values are modeled by geometric Brownian motions. In the first part of the paper we consider a pension plan sponsor with the funding objective that the pension asset value is to be within a band that is proportional to the pension liability value. Whenever the asset value is about to fall below the lower barrier or boundary of the band, the sponsor will provide sufficient funds to prevent this from happening. If, on the other hand, the asset value is about to exceed the upper barrier of the band, the assets are reduced by the potential overflow and returned to the sponsor. This paper calculates the expected present value of the payments to be made by the sponsor as well as that of the refunds to the sponsor. In particular we are interested in situations where these two expected values are equal. In the second part of the paper the refunds at the upper barrier are interpreted as the dividends paid to the shareholders of a company according to a barrier strategy. H...
63 citations
TL;DR: In this article, the authors reviewed 45 recent research papers that look at factors that might lead to increased risk in the individual annuity market and the offering of impaired life annuities.
Abstract: As many countries consider mandatory individual retirement accounts as their answer to a secure social security system, the question arises as to whether all workers can get true “market value” annuities when they retire. It is clear today that private-sector life annuities are priced assuming that the applicant is healthy—very healthy. Very little underwriting or risk classification now exists in the individual annuity marketplace. However, if a large percentage of the population were looking to annuitize their social security accounts upon retirement, there would be strong pressure for more risk classes in the annuity-pricing structure. Even without the advent of individual accounts for social security, the authors of this paper feel there may be real market opportunities for more risk classification in the individual annuity market and the offering of “impaired life annuities.” Given that this pressure does or might soon exist, this paper reviews 45 recent research papers that look at factors ...
59 citations
TL;DR: In this article, the authors apply the principle of equivalent utility to price and reserve equity-indexed life insurance, as introduced by Gerber (1976) and extended by Moore and Young (2002a, b).
Abstract: The author applies the principle of equivalent utility to price and reserve equity-indexed life insurance. Young and Zariphopoulou (2002a, b) extended this principle to price insurance products in a dynamic framework. However, in those papers, the insurance risks were independent of the risky asset in the financial market. By contrast, the death benefit for equity-indexed life insurance is a function of a risky asset; therefore, this paper further extends the principle of equivalent utility. In a second extension, the author applies the principle of equivalent utility to calculate reserves, as introduced by Gerber (1976). In a related paper, Moore and Young (2002) price equity-indexed pure endowments, the building blocks of equity-indexed life annuities.
TL;DR: In this paper, the dynamic fund protection option (DFP) is defined as an American option that provides the amount if it is exercised at time t, t ≥ 0, and the option payoff is guaranteed not to fall below the price of stock 1 and is indexed by stock 2 in the sense that the instantaneous growth rate of F(t) is that of S2(t).
Abstract: Consider an American option that provides the amount if it is exercised at time t, t ≥0. For simplicity of language, we interpret S1(t) and S2(t) as the prices of two stocks. The option payoff is guaranteed not to fall below the price of stock 1 and is indexed by the price of stock 2 in the sense that, if F(t) > S1(t), the instantaneous growth rate of F(t) is that of S2(t). We call this option the dynamic fund protection option. For the two stock prices, the bivariate Black-Scholes model with constant dividend-yield rates is assumed. In the case of a perpetual option, closed-form expressions for the optimal exercise strategy and the price of the option are given. Furthermore, this price is compared with the price of the perpetual maximum option, and it is shown that the optimal exercise of the maximum option occurs before that of the dynamic fund protection option. Two general concepts in the theory of option pricing are illustrated: the smooth pasting condition and the construction of the replic...
TL;DR: A model for critical illness insurance is proposed and rates of onset of ESRD from APKD are estimated and ignoring known genetic risks in underwriting sets a precedent that could have unintended consequences for the underwriting of nongenetic risks of similar magnitude.
Abstract: Adult polycystic kidney disease (APKD) is a single-gene autosomal dominant genetic disorder leading to end-stage renal disease (ESRD, meaning kidney failure). It is associated with mutations in at least two genes, APKD1 and APKD2, but diagnosis is mostly by ultrasonography. We propose a model for critical illness (CI) insurance and estimate rates of onset of ESRD from APKD using two studies. Other events leading to claims under CI policies are included in the model, which we use to study (a) extra premiums under CI policies if the presence of an APKD mutation is known, and (b) the possible costs arising from adverse selection if this information is unavailable to insurers. The extra premiums are typically very high, but because APKD is rare, the possible cost of adverse selection is low. However, APKD is just one of a significant number of single-gene disorders, and this benign conclusion cannot be assumed to apply to all genetic disorders taken together. Moreover, ignoring known genetic risks in...
TL;DR: In this article, the authors extended traditional credibility formulas in two aspects: the first part considers data from both the claims number and claims amount processes, and the second part considers the data from multiple lines of business.
Abstract: Credibility is a form of insurance pricing that is widely used, particularly in North America. It is a special type of experience rating that employs a weighted average of claims experience and a previously established price to determine a new price for each risk class under consideration. This article extends traditional credibility formulas in two aspects. The new procedures are called “multivariate credibility” because both aspects make use of additional sources of data when compared to traditional formulas. Specifically, the first portion of the paper considers data from both the claims number and claims amount processes. Assuming an aggregate loss model for total claims, optimal insurance pricing formulas are derived. The insurance prices turn out to be an intuitively appealing weighted average of the overall mean claim, the claims number experience, and the claims amount experience. The second portion of the paper considers data from claims number and amount processes from multiple lines of...
TL;DR: In this article, the authors proposed a general individual catastrophe risk model that allows damage ratios to be random functions of the catastrophe intensity, and derived some distributional properties of the insured risks and of the aggregate catastrophic loss under this model.
Abstract: The authors propose a general individual catastrophe risk model that allows damage ratios to be random functions of the catastrophe intensity. They derive some distributional properties of the insured risks and of the aggregate catastrophic loss under this model. Through the model and ruin probability calculations, they formally illustrate the well-known fact that the catastrophe risk cannot be diversified through premium collection alone, as is the case with the usual “day-to-day” risk, even for an arbitrary large portfolio. They also derive some risk orderings between different catastrophe portfolios and show that the risk level of a realistic portfolio falls between that of a portfolio of comonotonic risks and that of a portfolio of independent risks. Finally, the authors illustrate their findings with a numerical example inspired from earthquake insurance.
TL;DR: In this article, the authors investigated the asymptotic tail behavior of maxima of a random walk with negative mean and heavy-tailed increment distribution and gave a simple proof to improve the related result in Ng et al. (2002).
Abstract: This paper investigates the asymptotic tail behavior of maxima of a random walk with negative mean and heavy-tailed increment distribution. A simple proof is given to improve the related result in Ng et al. (2002).
TL;DR: In this paper, the role of gradual (or phased) retirement in introducing flexibility into the range of choices between work and retirement is discussed, including legal barriers and barriers relating to pension plan objectives.
Abstract: Aging of the population raises many questions and issues for individuals, families, and society: economic, political, social, psychological, medical, ethical, moral, religious, and legal, all of which bear on the quality of life in its many dimensions. Economic security in old age is one rubric under which many of the problems and their possible solutions may be discussed. How a society arranges for its members to work and retire is an important facet in the provision for old-age economic security. This article is concerned with the implications of demographic and labor force changes for work and retirement. It discusses the role of gradual (or phased) retirement in introducing flexibility into the range of choices between work and retirement. Section 1 explains the rationale for gradual retirement. Section 2 spells out the barriers to implementing gradual retirement programs, including legal barriers and barriers relating to pension plan objectives. Section 3 discusses some possible solutions fo...
TL;DR: In this article, the authors investigate the pricing of discretely monitored dynamic fund protections when the fund price follows a lognormal process or a constant elasticity of variance (CEV) process.
Abstract: The authors investigate the pricing of discretely monitored dynamic fund protections when the fund price follows a lognormal process or a constant elasticity of variance (CEV) process. A backward recursive pricing formula is derived. By employing a numerical technique that combines function approximation and numerical quadrature, the authors demonstrate how to complete each recursion level efficiently. Numerical experiments show that the results compare favorably with those obtained by other pricing methods.
TL;DR: In this article, an integration-by-parts proof of the Hattendorff theorem in the general fully continuous insurance model is presented, motivated by a derivation of the theorem in a general fully discrete insurance model.
Abstract: This paper presents an integration-by-parts proof of the Hattendorff theorem in the general fully continuous insurance model. The proof motivates a derivation of the theorem in the general fully discrete insurance model. Increments of a martingale over disjoint time intervals are uncorrelated random variables; the paper explains that the Hattendorff theorem can be viewed as an application of this result. A notable feature of the paper is the extensive use of the indicator function.
TL;DR: This analysis leads to the development of a framework for improvements in today’s mix of health care financing and to high-level principles for a better-coordinated relationship between public and private programs.
Abstract: Public health systems are finding it increasingly difficult to fund all the health care that their citizens want and need. Fiscal constraint is causing many nations to reconsider the respective roles of their public programs and private health insurance. The author examines the potential for and performance of health systems around the world and the advantages and disadvantages of public and private health financing. This analysis leads to the development of a framework for improvements in today’s mix of health care financing and to high-level principles for a better-coordinated relationship between public and private programs. The role of health actuaries in moving toward more effective health systems is then explored.
TL;DR: In this paper, a new family of contaminated exponential dispersion loss models is defined and some properties of its properties are examined, and a wider family of loss distributions are proposed, allowing the modeling of extreme claims.
Abstract: A new family of contaminated exponential dispersion loss models is defined and some of its properties are examined. These models offer a wider family of loss distributions, allowing the modeling of extreme claims. Their usefulness is illustrated with real data.
TL;DR: J is a general-purpose computer programming language built upon the shoulders of the premier array programming language APL, created by Kenneth Iverson, and an especially novel feature is the “train” notation of J, which encourages “bident’ and “trident” function composition called “hooks” and” forks, respectively.
Abstract: Language for Analytic Computing, Research Studies Press, England, 277 pages, $96.00. J is a general-purpose computer programming language. It is built upon the shoulders of the premier array programming language APL, created by Kenneth Iverson, who is also the architect of J. Since the invention of APL, actuaries have demonstrated its practicality; as of yet, few actuaries have discovered its descendent, J. Thomson’s assessment states that J is “the natural language for analytic computing,” and his complete enthusiasm for J is consistent with mine, especially for actuarial professionals. As Thomson portrays it, a singular feature of J is its succinctness relative to alternative programming languages and its cogency in expressing analytical notions. An especially novel feature is the “train” notation of J, which encourages “bident” and “trident” function composition called “hooks” and “forks,” respectively. Forks are characterized by functional triads: the mean can be computed as a fork as follows; the first of the two expressions below is conceptual code and the second is literal J.
Journal Article•
TL;DR: In this article, Jones and Mereu examined families of fractional age assumptions including the three traditional ones analyzed by Frostig and presented an optimality criterion based on the length of the probability density function over the range of the mortality table.
Abstract: BRUCE L. JONES* AND JOHN A. MEREU We found this paper to be of interest because the topic is one we recently did some work on (Jones and Mereu 2000, 2002). In our papers, we examined families of fractional age assumptions including the three traditional ones analyzed by Frostig. To help in choosing specific fractional age assumptions, we presented an optimality criterion based on the length of the probability density function over the range of the mortality table. Our findings are consistent with those obtained by Frostig. They show that UDD results are better than those found with the constant-force and the hyperbolic (Balducci) assumptions. We postulated assumptions that were better than UDD, but not as simple. These could be used if the error with UDD was not acceptable.
TL;DR: In this paper, the MCVM estimator of the scale parameter behaves like a compromise between the GM estimators with k 1 and k 2 and puts much more emphasis on robustness than any of the other estimators considered by Serfling.
Abstract: which are the values most often found in practice. The bad news is that, as we can see in Figure 3, the efficiency drops rapidly as the value of increases. In summary, the MCVM estimator of the parameter behaves like a compromise between the GM estimators with k 1 and k 2, whereas the MCVM estimator of the parameter puts much more emphasis on robustness than any of the GM estimators considered by Serfling, thereby being much less efficient. In terms of inference about the lognormal mean , this means that efficiency is traded off for more robustness as the true value of the scale parameter increases.