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3 pages, 267 KiB

Open AccessEditorial

Special Issue “Actuarial Mathematics and Risk Management”

by Annamaria Olivieri

Risks 2023, 11(7), 134; https://doi.org/10.3390/risks11070134 - 20 Jul 2023

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Abstract

Among the most important implementations of the principles of enterprise risk management (ERM), the risk management process (RMP) involves various quantitative phases, usually encompassed under the label of quantitative risk management (QRM) [...] Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

Research

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8 pages, 351 KiB

Open AccessArticle

A Comparison of Macaulay Approximations

by Stefanos C. Orfanos

Risks 2022, 10(8), 153; https://doi.org/10.3390/risks10080153 - 29 Jul 2022

Cited by 1 | Viewed by 1553

Abstract

We discuss several known formulas that use the Macaulay duration and convexity of commonly used cash flow streams to approximate their net present value, and compare them with a new approximation formula that involves hyperbolic functions. Our objective is to assess the reliability [...] Read more.

We discuss several known formulas that use the Macaulay duration and convexity of commonly used cash flow streams to approximate their net present value, and compare them with a new approximation formula that involves hyperbolic functions. Our objective is to assess the reliability of each approximation formula under different scenarios. The results in this note should be of interest to actuarial candidates and educators as well as analysts working in all areas of actuarial practice. Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

19 pages, 730 KiB

Open AccessArticle

Equivalent Risk Indicators: VaR, TCE, and Beyond

by Silvia Faroni, Olivier Le Courtois and Krzysztof Ostaszewski

Risks 2022, 10(8), 142; https://doi.org/10.3390/risks10080142 - 22 Jul 2022

Cited by 3 | Viewed by 1910

Abstract

While a lot of research concentrates on the respective merits of VaR and TCE, which are the two most classic risk indicators used by financial institutions, little has been written on the equivalence between such indicators. Further, TCE, despite its merits, may not [...] Read more.

While a lot of research concentrates on the respective merits of VaR and TCE, which are the two most classic risk indicators used by financial institutions, little has been written on the equivalence between such indicators. Further, TCE, despite its merits, may not be the most accurate indicator to take into account the nature of probability distribution tails. In this paper, we introduce a new risk indicator that extends TCE to take into account higher-order risks. We compare the quantiles of this indicator to the quantiles of VaR in a simple Pareto framework, and then in a generalized Pareto framework. We also examine equivalence results between the quantiles of high-order TCEs. Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

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23 pages, 937 KiB

Open AccessArticle

Reverse Sensitivity Analysis for Risk Modelling

by Silvana M. Pesenti

Risks 2022, 10(7), 141; https://doi.org/10.3390/risks10070141 - 18 Jul 2022

Cited by 7 | Viewed by 1760

Abstract

We consider the problem where a modeller conducts sensitivity analysis of a model consisting of random input factors, a corresponding random output of interest, and a baseline probability measure. The modeller seeks to understand how the model (the distribution of the input factors [...] Read more.

We consider the problem where a modeller conducts sensitivity analysis of a model consisting of random input factors, a corresponding random output of interest, and a baseline probability measure. The modeller seeks to understand how the model (the distribution of the input factors as well as the output) changes under a stress on the output’s distribution. Specifically, for a stress on the output random variable, we derive the unique stressed distribution of the output that is closest in the Wasserstein distance to the baseline output’s distribution and satisfies the stress. We further derive the stressed model, including the stressed distribution of the inputs, which can be calculated in a numerically efficient way from a set of baseline Monte Carlo samples and which is implemented in the R package SWIM on CRAN. The proposed reverse sensitivity analysis framework is model-free and allows for stresses on the output such as (a) the mean and variance, (b) any distortion risk measure including the Value-at-Risk and Expected-Shortfall, and (c) expected utility type constraints, thus making the reverse sensitivity analysis framework suitable for risk models. Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

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15 pages, 1133 KiB

Open AccessArticle

Marriage and Individual Equity Release Contracts with Dread Disease Insurance as a Tool for Managing the Pensioners’ Budget

by Agnieszka Marciniuk and Beata Zmyślona

Risks 2022, 10(7), 140; https://doi.org/10.3390/risks10070140 - 12 Jul 2022

Cited by 1 | Viewed by 1214

Abstract

In many countries around the world, demographic and civilization changes have brought about the phenomenon of aging societies. This phenomenon affects the economy, especially the pension and health care systems, causing difficulties in their financing. The implementation of a policy that would effectively [...] Read more.

In many countries around the world, demographic and civilization changes have brought about the phenomenon of aging societies. This phenomenon affects the economy, especially the pension and health care systems, causing difficulties in their financing. The implementation of a policy that would effectively manage the problem of the longevity risk is thus required. Using housing resources and private health insurance to improve retirees’ living standards may serve this purpose. The instruments we propose comprise two variants of contracts: the first for a marriage, the second for an individual client. We analysed the cash flow in both the cases. The results suggest that the amount of cash flows related to reverse equity and dread disease insurance benefits depends on the spouse’s economic status, age, and health conditions. The benefits of the two variants of the contract vary. This paper examines numerous strategies for selecting the type of the contract, taking into consideration the abovementioned factors. Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

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10 pages, 549 KiB

Open AccessArticle

The Copula Derived from the SAHARA Utility Function

by Jaap Spreeuw

Risks 2022, 10(7), 133; https://doi.org/10.3390/risks10070133 - 28 Jun 2022

Cited by 1 | Viewed by 1262

Abstract

A new Archimedean copula family is presented that was derived from the SAHARA utility function introduced in the economic literature in 2011. Its properties are discussed, and its flexibility and versatility are demonstrated. It is left tail decreasing or right tail increasing, but [...] Read more.

A new Archimedean copula family is presented that was derived from the SAHARA utility function introduced in the economic literature in 2011. Its properties are discussed, and its flexibility and versatility are demonstrated. It is left tail decreasing or right tail increasing, but unlike mainstream Archimedean families, not necessarily stochastically increasing at the same time. It is shown that the family fits very well to a dataset of previously studied coupled lives in the literature. Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

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17 pages, 1531 KiB

Open AccessArticle

A New Class of Counting Distributions Embedded in the Lee–Carter Model for Mortality Projections: A Bayesian Approach

by Yaser Awad, Shaul K. Bar-Lev and Udi Makov

Risks 2022, 10(6), 111; https://doi.org/10.3390/risks10060111 - 27 May 2022

Cited by 4 | Viewed by 1632

Abstract

The Lee–Carter model, the dominant mortality projection modeling in the literature, was criticized for its homoscedastic error assumption. This was corrected in extensions to the model based on the assumption that the number of deaths follows Poisson or negative binomial distributions. We propose [...] Read more.

The Lee–Carter model, the dominant mortality projection modeling in the literature, was criticized for its homoscedastic error assumption. This was corrected in extensions to the model based on the assumption that the number of deaths follows Poisson or negative binomial distributions. We propose a new class of families of counting distributions, namely, the ABM class, which belongs to a wider class of natural exponential families. This class is characterized by its variance functions and contains the Poisson and the negative binomial distributions as special cases, offering an infinite class of additional counting distributions to be considered. We are guided by the principle that the choice of distribution should be made from a pool of distributions as large as possible. To this end, and following a data mining approach, a training set of historical mortality data of the population could be modeled using the ABM’s rich choice of distributions, and the chosen distribution should be the one that proved to offer superior projection results on a test set of mortality data. As an alternative to parameter estimation via the singular value decomposition used in the classical Lee–Carter model, we adopted Bayesian estimation, harnessing the Markov Chain Monte Carlo methodology. A numerical study demonstrates that when fitting mortality data using this new class of distributions, while traditional distributions may provide desirable projections for some populations, for others, alternative distributions within the ABM class can potentially produce superior results for the entire population or particular age groups, such as the oldest-old. Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

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21 pages, 439 KiB

Open AccessArticle

A Bridge Life Insurance for Households—Diagnosis and Motives

by Anna Jędrzychowska

Risks 2022, 10(4), 81; https://doi.org/10.3390/risks10040081 - 08 Apr 2022

Cited by 1 | Viewed by 2500

Abstract

Purpose: The purpose of this article is to describe the initial concept of household bridging insurance. Design/methodology/approach: In the first part of the article, an extensive literature review is made. This is made to show the research gap of insufficient protection of households [...] Read more.

Purpose: The purpose of this article is to describe the initial concept of household bridging insurance. Design/methodology/approach: In the first part of the article, an extensive literature review is made. This is made to show the research gap of insufficient protection of households against destabilization resulting from the lost personal contribution. Data shown in the text present the scale of the loss of lost unpaid work (based on household time budgets). The existing methods of managing this loss, based on social insurance, are also shown. Findings: This paper discusses the possibility of creating a new insurance. Its need is indicated (research gap, the scale of the problem, and insufficient protection by the social insurance system) and a preliminary outline of its structure is indicated (annuities character, dynamic sum insured related to the lifecycle of the household). The article contains the theoretical background of the new product, and introduces further research on the use of multistate models in the construction and calculation of insurance premiums. Originality/value: So far, studies concerning, inter alia, personal damage indicate the lost personal contribution (unpaid work for household members) and even try to evaluate it. However, no private insurance has been proposed to mitigate the destabilization resulting from the death of an adult household member. The article therefore proposes a new life insurance (a separated policy or as an extension option) that would help the household to return to normal operation after the death of one of the household members. Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

27 pages, 912 KiB

Open AccessArticle

Unit-Linked Tontine: Utility-Based Design, Pricing and Performance

by An Chen, Thai Nguyen and Thorsten Sehner

Risks 2022, 10(4), 78; https://doi.org/10.3390/risks10040078 - 07 Apr 2022

Cited by 3 | Viewed by 2523

Abstract

Due to the low demand for conventional annuities, alternative retirement products are sought. Quite recently, tontines have been frequently brought up as a promising option in this respect. Inspired by unit-linked life insurance and retirement products, we introduce unit-linked tontines in this article, [...] Read more.

Due to the low demand for conventional annuities, alternative retirement products are sought. Quite recently, tontines have been frequently brought up as a promising option in this respect. Inspired by unit-linked life insurance and retirement products, we introduce unit-linked tontines in this article, where the tontine payoffs are directly linked to the development of the underlying financial market. More specifically, we consider two different tontine payoff structures differing in the (non-)inclusion of guaranteed payments. We first price the unit-linked tontines by using the risk-neutral pricing approach. Consequently, we study the attractiveness of these products for a utility-maximizing policyholder and compare them with non-unit-linked tontines. Our numerical analysis sheds light on the design challenges and gives explanations why similar products might not be widely adopted already. Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

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22 pages, 662 KiB

Open AccessArticle

Special-Rate Life Annuities: Analysis of Portfolio Risk Profiles

by Ermanno Pitacco and Daniela Y. Tabakova

Risks 2022, 10(3), 65; https://doi.org/10.3390/risks10030065 - 13 Mar 2022

Cited by 3 | Viewed by 2529

Abstract

Special-rate life annuities are life annuity products whose single premium is based on a mortality assumption driven (at least to some extent) by the health status of the applicant. The health status is ascertained via an appropriate underwriting step (which explains the alternative [...] Read more.

Special-rate life annuities are life annuity products whose single premium is based on a mortality assumption driven (at least to some extent) by the health status of the applicant. The health status is ascertained via an appropriate underwriting step (which explains the alternative expression “underwritten life annuities”). Better annuity rates are then applied in presence of poor health conditions. The worse the health conditions, the smaller the modal age at death (as well as the expected lifetime), but the higher the variance of the lifetime distribution. The latter aspect is due to significant data scarcity as well as to the mix of possible pathologies leading to each specific rating class. A higher degree of (partially unobservable) heterogeneity inside each sub-portfolio of special-rate annuities follows, and this results in a higher variability of the total portfolio payout. The present research aims at analyzing the impact of extending the life annuity portfolio by selling special-rate life annuities. Numerical evaluations have been performed by adopting a deterministic approach as well as a stochastic one, according to diverse assumptions concerning both lifetime distributions and portfolio structure and size. Our achievements witness the possibility of extending the annuity business without taking huge amounts of risk. Hence, the risk management objective “enhancing the company market share” can be pursued without significant worsening of the annuity portfolio risk profile. Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

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41 pages, 1139 KiB

Open AccessArticle

Equalization Reserves for Reinsurance and Non-Life Undertakings in Switzerland

by Anja Breuer and Yves Staudt

Risks 2022, 10(3), 55; https://doi.org/10.3390/risks10030055 - 03 Mar 2022

Cited by 2 | Viewed by 3876

Abstract

Equalization reserves is an insurance liability with features of own capital. By law, Swiss reinsurance and non-life undertakings must hold equalization reserves within their statutory accounts. Regarding Swiss solvency modeling, the equalization reserves are set to zero. Swiss reinsurance and non-life undertakings define [...] Read more.

Equalization reserves is an insurance liability with features of own capital. By law, Swiss reinsurance and non-life undertakings must hold equalization reserves within their statutory accounts. Regarding Swiss solvency modeling, the equalization reserves are set to zero. Swiss reinsurance and non-life undertakings define the upper limit and the corresponding transfer rule to the equalization reserves; however, this information is not disclosed. The goal of the study is to find a relationship between the equalization reserves and the publicly available technical account items, applying a generalized additive model (GAM). Thereafter, we transform the continuous variables into discrete ones, and we apply a generalized linear model (GLM). The study is based on published data from 1997 to 2018, whereby we restate the implicitly published equalization reserves. For reinsurance undertakings, the GAM model captures the relationship better than the GLM one; for non-life undertakings, the GLM model performs better. For reinsurance undertakings, the equalization reserves depend on the equalization reserves of the previous year, on the calendar year, on the legal form, on the technical result, on the administration and commission costs and on other costs. For non-life undertakings, the equalization reserves depend on the net claims payments, on the equalization reserves of the previous year, on the net change in claims reserves without change in equalization reserves, on the calendar year and on the net earned premium. Furthermore, we look at the need for equalization reserves: do the undertakings accumulate and release the equalization reserves? Further, the impact of taxes on the equalization reserves is looked at. The concept of equalization reserves avoids the misuse of tax optimization. We conclude that the discussion about disclosure of equalization reserves will restart. In addition, the definition of the upper limit of the equalization reserves could be widened by linking the equalization reserves to the insurance/reserving risk from the capital modeling. Full article

(This article belongs to the Special Issue Actuarial Mathematics and Risk Management)

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