North American Actuarial Journal Research Articles

A Markov multiple state model for epidemic and insurance modelling

AbstractWith recent epidemics such as COVID-19, H1N1 and SARS causing devastating financial loss to the economy, it is important that insurance companies plan for financial costs of epidemics. This article proposes a new methodology for epidemic and insurance modelling by combining the existing deterministic compartmental models and the Markov multiple state models to facilitate actuarial computations to design new health insurance plans that cover epidemics. Our method is inspired by the seminal paper (Feng and Garrido (2011) North American Actuarial Journal, 15, 112–136.) of Feng and Garrido and complements the work of Hillairet and Lopez et al. in Hillairet and Lopez ((2021) Scandinavian Actuarial Journal, 2021(8), 671–694.) and Hillairet et al. ((2022) Insurance: Mathematics and Economics, 107, 88–101.) In this work, we use the deterministic SIR model and the Eyam epidemic data set to provide numerical illustrations for our method.

Hedging Dynamic Fund Protection: A Static Versus Dynamic Comparison

The static nature of the downside protection offered by a standard Put option can motivate investors to use exotic option contracts like dynamic fund protection (DFP), which protects the underlying asset value throughout its life. DFP has a path-dependent payoff structure with its terminal payoff depending on the terminal asset price, the strike price and the minimum price level attained by the asset over the contract tenor. This makes DFP a contract strikingly similar to a Lookback Call option. In this article, we try to borrow the respective strengths from a dynamic and a static hedging strategy, formulate a semi-static hedging procedure for DFP and compare the hedging effectiveness of this procedure with a standard delta hedging strategy. Given the payoff resemblance between a DFP and a Lookback Call, we also seek to verify Tompkins (2002, Journal of Risk Finance, 3(4), 6–34) conclusion, which objectively shows that a dynamic procedure performs better in hedging a Lookback Call option both with and without transaction cost. Using simulated underlying price paths, we perform the delta hedging on DFP based on the closed-form pricing formula given by Gerber and Pafumi (2000, North American Actuarial Journal, 4(2), 28–37) and compare the hedge performance results with its static hedge counterpart.

Historical pricing variability in immediate and deferred income annuities

Risk Management and Insurance ReviewEarly View DATA INSIGHT Historical pricing variability in immediate and deferred income annuities David Blanchett, Corresponding Author David Blanchett [email protected] orcid.org/0000-0002-8288-0122 Retirement Research for PGIM DC Solutions, Newark, New Jersey Correspondence David Blanchett, Retirement Research for PGIM DC Solutions, Newark, NJ. Email: [email protected]Search for more papers by this authorBranislav Nikolic, Branislav Nikolic Insurance at the Index Standard, Toronto, CanadaSearch for more papers by this author David Blanchett, Corresponding Author David Blanchett [email protected] orcid.org/0000-0002-8288-0122 Retirement Research for PGIM DC Solutions, Newark, New Jersey Correspondence David Blanchett, Retirement Research for PGIM DC Solutions, Newark, NJ. Email: [email protected]Search for more papers by this authorBranislav Nikolic, Branislav Nikolic Insurance at the Index Standard, Toronto, CanadaSearch for more papers by this author First published: 22 June 2023 https://doi.org/10.1111/rmir.12244Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL REFERENCES Blanchett, D., Finke, M., & Nikolic, B. (2021). How competitive are income annuity providers over time? Risk Management and Insurance Review, 24(no. 2), 207– 214. CANNEX. 2022. “CANNEX Survey Experience: January 1, 2021 to December 31, 2021.” White Paper. Charupat, N., Kamstra, M. J., & Milevsky, M. A. (2016). The sluggish and asymmetric reaction of life annuity prices to changes in interest rates. Journal of Risk and Insurance, 83(no. 3), 519– 555. Friedman, B. M., & Warshawsky, M. J. (1990). The cost of annuities: Implications for saving behavior and bequests. The Quarterly Journal of Economics, 105, 135– 154. Habib, F., Huang, H., Mauskopf, A., Nikolic, B., & Salisbury, T. S. (2020). Optimal allocation to deferred income annuities. Insurance: Mathematics and Economics, 90, 94– 104. James, M. (1947). The Metropolitan Life: A Study in Business Growth. Viking. Milevsky, M. A. (2006). The Calculus of Retirement Income: Financial Models for Pension Annuities and Life Insurance. Cambridge University Press. Milevsky, M. A. (2005). Real longevity insurance with a deductible: Introduction to advanced-life delayed annuities (ALDA). North American Actuarial Journal, 9(no. 4), 109– 122. Poterba, J., & Solomon, A. (2021). Discount Rates, Mortality Projections, and Money's Worth Calculations for US Individual Annuities. No. w28557 National Bureau of Economic Research. Poterba, J. M., 1997, “The History of Annuities in the US.” Working paper 6001, National Bureau of Economic Research. Early ViewOnline Version of Record before inclusion in an issue ReferencesRelatedInformation

On the surplus management of funds with assets and liabilities in presence of solvency requirements

In this paper, we consider a company whose assets and liabilities evolve according to a correlated bivariate geometric Brownian motion, such as in Gerber and Shiu [(2003). Geometric Brownian motion models for assets and liabilities: From pension funding to optimal dividends. North American Actuarial Journal 7(3), 37–56]. We determine what dividend strategy maximises the expected present value of dividends until ruin in two cases: (i) when shareholders won't cover surplus shortfalls and a solvency constraint [as in Paulsen (2003). Optimal dividend payouts for diffusions with solvency constraints. Finance and Stochastics 7(4), 457–473] is consequently imposed and (ii) when shareholders are always to fund any capital deficiency with capital (asset) injections. In the latter case, ruin will never occur and the objective is to maximise the difference between dividends and capital injections. Developing and using appropriate verification lemmas, we show that the optimal dividend strategy is, in both cases, of barrier type. Both value functions are derived in closed form. Furthermore, the barrier is defined on the ratio of assets to liabilities, which mimics some of the dividend strategies that can be observed in practice by insurance companies. The existence and uniqueness of the optimal strategies are shown. Results are illustrated.

Joint moments of discounted claims and discounted perturbation until ruin in the compound Poisson risk model with diffusion

This paper studies a generalization of the Gerber-Shiu expected discounted penalty function [Gerber and Shiu (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72] in the context of the perturbed compound Poisson insurance risk model, where the moments of the total discounted claims and the discounted small fluctuations (arising from the Brownian motion) until ruin are also included. In particular, the latter quantity is represented by a stochastic integral and has never been analyzed in the literature to the best of our knowledge. Recursive integro-differential equations satisfied by our generalized Gerber-Shiu function are derived, and these are transformed to defective renewal equations where the components are identified. Explicit solutions are given when the individual claim amounts are distributed as a combination of exponentials. Numerical illustrations are provided, including the computation of the covariance between discounted claims and discounted perturbation until ruin.

Tail Moments of Compound Distributions

In this article, we study the moment transform of both univariate and multivariate compound sums. We first derive simple explicit formulas for the first and second moment transforms when the (loss) frequency distribution is in the so-called class. Then we show that the derived formulas can be used to efficiently compute risk measures such as the tail conditional expectation (TCE), the tail variance (TV), and higher tail moments. The results generalize those in Denuit (North American Actuarial Journal, 24 (4):512–32, 2020).

Longevity Risk and Capital Markets: The 2016–2017 Update

This special issue of the North American Actuarial Journal contains 14 contributions to the academic literature, all dealing with longevity risk and capital markets. Draft versions of the papers we...

Longevity Risk and Capital Markets: The 2017–2018 Update

This special issue of the North American Actuarial Journal contains 13 contributions to the academic literature, all dealing with longevity risk and capital markets. Draft versions of the papers we...

CBDX: a workhorse mortality model from the Cairns–Blake–Dowd family

AbstractThe purpose of this paper is to identify a workhorse mortality model for the adult age range (i.e., excluding the accident hump and younger ages). It applies the “general procedure” (GP) of Hunt & Blake [(2014), North American Actuarial Journal, 18, 116–138] to identify an age-period model that fits the data well before adding in a cohort effect that captures the residual year-of-birth effects arising in the original age-period model. The resulting model is intended to be suitable for a variety of populations, but economises on the number of period effects in comparison with a full implementation of the GP. We estimate the model using two different iterative maximum likelihood (ML) approaches – one Partial ML and the other Full ML – that avoid the need to specify identifiability constraints.

Advances in Predictive Analytics

This Special Issue of the North American Actuarial Journal (NAAJ) contains nine articles that contribute to the advances in predictive analytics with a wide range of applications in insurance and r...

BIAS-CORRECTED INFERENCE FOR A MODIFIED LEE–CARTER MORTALITY MODEL

AbstractAs a benchmark mortality model in forecasting future mortality rates and hedging longevity risk, the widely employed Lee–Carter model (Lee, R.D. and Carter, L.R. (1992) Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87, 659–671.) suffers from a restrictive constraint on the unobserved mortality index for ensuring model’s identification and a possible inconsistent inference. Recently, a modified Lee–Carter model (Liu, Q., Ling, C. and Peng, L. (2018) Statistical inference for Lee–Carter mortality model and corresponding forecasts. North American Actuarial Journal, to appear.) removes this constraint and a simple least squares estimation is consistent with a normal limit when the mortality index follows from a unit root or near unit root AR(1) model with a nonzero intercept. This paper proposes a bias-corrected estimator for this modified Lee–Carter model, which is consistent and has a normal limit regardless of the mortality index being a stationary or near unit root or unit root AR(1) process with a nonzero intercept. Applications to the US mortality rates and a simulation study are provided as well.

ON SARMANOV MIXED ERLANG RISKS IN INSURANCE APPLICATIONS

AbstractIn this paper we consider an extension to the aggregation of the FGM mixed Erlang risks, proposed by Cossette et al. (2013 Insurance: Mathematics and Economics, 52, 560–572), in which we introduce the Sarmanov distribution to model the dependence structure. For our framework, we demonstrate that the aggregated risk belongs to the class of Erlang mixtures. Following results from S. C. K. Lee and X. S. Lin (2010 North American Actuarial Journal, 14(1) 107–130), G. E. Willmot and X. S. Lin (2011 Applied Stochastic Models in Business and Industry, 27(1) 8–22), analytical expressions of the contribution of each individual risk to the economic capital for the entire portfolio are derived under both the TVaR and the covariance capital allocation principle. By analysing the commonly used dependence measures, we also show that the dependence structure is wide and flexible. Numerical examples and simulation studies illustrate the tractability of our approach.

Longevity Risk and Capital Markets: The 2012–2013 Update

This Special Issue of the North American Actuarial Journal contains 15 contributions to the academic literature all dealing with longevity risk and capital markets. Draft versions of the articles w...

A Digital Picture of the Actuarial Research Community

The evolution of publication patterns in actuarial research is described through a survey of the contents of four peer-reviewed journals identified in several studies as the most influential in the field, including the North American Actuarial Journal. The research output of countries and institutions is compared over the 30-year period 1982–2011 on the basis of the number of articles and pages published, adjusted for journal page size. While simple counts such as these are only a cursory measure of productivity, and certainly not a measure of quality, they lead to rankings that broadly reflect the level of activity in actuarial research worldwide. Countries and institutions that are most active in the field are easily identified from these rankings. Such information is valuable to governmental funding agencies and administrators in academia or industry responsible for the quantitative assessment of research performance. Young researchers and prospective graduate students may also find it useful to acquaint themselves with the breadth of the actuarial research community.

Longevity Risk and Capital Markets

This Special Issue of the North American Actuarial Journal contains ten contributions to the academic literature all dealing with longevity risk and capital markets. Draft versions of the papers were presented at Longevity Five: the Fifth International Longevity Risk and Capital Markets Solutions Conference that was held in New York on 25-26 September 2009. It was hosted by J. P. Morgan and St John’s University and organized by the Pensions Institute at Cass Business School, London, and the Edmondson-Miller Chair at Illinois State University.

A Direct Approach to the Discounted Penalty Function

This paper provides a new and accessible approach to establishing certain results concerning the discounted penalty function. The direct approach consists of two steps. In the first step, closed-form expressions are obtained in the special case in which the claim amount distribution is a combination of exponential distributions. A rational function is useful in this context. For the second step, one observes that the family of combinations of exponential distributions is dense. Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus process has downward and upward jumps, modeled by two independent compound Poisson processes. If the distribution of the upward jumps is exponential, a series of new results can be obtained with ease. Subsequently, certain results of Gerber and Shiu [H. U. Gerber and E. S. W. Shiu, North American Actuarial Journal 2(1): 48–78 (1998)] can be reproduced. The two-step approach is also applied when an independent Wiener process is added to the surplus process. Certain results are related to Zhang et al. [Z. Zhang, H. Yang, and S. Li, Journal of Computational and Applied Mathematics 233: 1773–1 784 (2010)], which uses different methods.

“Pricing Asian Options and Equity-Indexed Annuities with Regime Switching by the Trinomial Tree Method”, Fei Lung Yuen and Hailiang Yang, April, 2010

(2010). “Pricing Asian Options and Equity-Indexed Annuities with Regime Switching by the Trinomial Tree Method”, Fei Lung Yuen and Hailiang Yang, April, 2010. North American Actuarial Journal: Vol. 14, No. 2, pp. 272-277.

“Computation of Multivariate Barrier Crossing Probability and Its Applications in Credit Risk Models,” Joonghee Huh and Adam Kolkiewicz, July 2008

(2010). “Computation of Multivariate Barrier Crossing Probability and Its Applications in Credit Risk Models,” Joonghee Huh and Adam Kolkiewicz, July 2008. North American Actuarial Journal: Vol. 14, No. 1, pp. 150-156.

“Valuation of Discrete Dynamic Fund Protection under Lévy Processes,” Hoi Ying Wong and Ka Wai Lam, April 2009

(2009). “Valuation of Discrete Dynamic Fund Protection under Levy Processes,” Hoi Ying Wong and Ka Wai Lam, April 2009. North American Actuarial Journal: Vol. 13, No. 4, pp. 520-524.

A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model

In this paper, a risk model where claims arrive according to a Markovian arrival process (MAP) is considered. A generalization of the well-known Gerber–Shiu function is proposed by incorporating the maximum surplus level before ruin into the penalty function. For this wider class of penalty functions, we show that the generalized Gerber–Shiu function can be expressed in terms of the original Gerber–Shiu function (see e.g. [Gerber, Hans U., Shiu, Elias, S.W., 1998. On the time value of ruin. North American Actuarial Journal 2(1), 48–72]) and the Laplace transform of a first passage time which are both readily available. The generalized Gerber–Shiu function is also shown to be closely related to the original Gerber–Shiu function in the same MAP risk model subject to a dividend barrier strategy. The simplest case of a MAP risk model, namely the classical compound Poisson risk model, will be studied in more detail. In particular, the discounted joint density of the surplus prior to ruin, the deficit at ruin and the maximum surplus before ruin is obtained through analytic Laplace transform inversion of a specific generalized Gerber–Shiu function. Numerical illustrations are then examined.