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27 pages, 974 KiB

Open AccessArticle

Longevity Risk and Annuitisation Decisions in the Absence of Special-Rate Life Annuities

by Jorge de Andrés-Sánchez and Laura González-Vila Puchades

Risks 2025, 13(2), 37; https://doi.org/10.3390/risks13020037 - 19 Feb 2025

Viewed by 258

Longevity risk affecting older adults can be transferred to the insurance market by purchasing a lifetime annuity. Special-rate life annuities, which are priced, among other factors, on the basis of health and lifestyle factors, go beyond traditional considerations of age and sex by [...] Read more.

Longevity risk affecting older adults can be transferred to the insurance market by purchasing a lifetime annuity. Special-rate life annuities, which are priced, among other factors, on the basis of health and lifestyle factors, go beyond traditional considerations of age and sex by using modified mortality tables. However, they are not available in many countries. In regions where life annuities are priced solely via standard mortality tables, retirees with below-average life expectancy may face unfair pricing. This study aims to quantify this actuarial unfairness and proposes an alternative annuitisation strategy for these retirees. The strategy allows them to transfer longevity risk by acquiring a life annuity on the basis of their actual mortality probabilities, thereby mitigating actuarial inequities. Additionally, the paper examines how tax incentives can exacerbate actuarial unfairness and, specifically for Spanish tax regulations, compares different alternatives under two scenarios related to the sources used for purchasing life annuities. Full article

(This article belongs to the Special Issue Applied Financial and Actuarial Risk Analytics)

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26 pages, 572 KiB

Open AccessArticle

Uncertainty in Pricing and Risk Measurement of Survivor Contracts

by Kenrick Raymond So, Stephanie Claire Cruz, Elias Antonio Marcella, Jeric Briones and Len Patrick Dominic Garces

Risks 2025, 13(2), 35; https://doi.org/10.3390/risks13020035 - 18 Feb 2025

Viewed by 304

Abstract

As life expectancy increases, pension plans face growing longevity risk. Standardized longevity-linked securities such as survivor contracts allow pension plans to transfer this risk to capital markets. However, more consensus is needed on the appropriate mortality model and premium principle to price these [...] Read more.

As life expectancy increases, pension plans face growing longevity risk. Standardized longevity-linked securities such as survivor contracts allow pension plans to transfer this risk to capital markets. However, more consensus is needed on the appropriate mortality model and premium principle to price these contracts. This paper investigates the impact of the mortality model and premium principle choice on the pricing, risk measurement, and modeling of survivor contracts. We present a framework for evaluating risk measures associated with survivor contracts, specifically survivor forwards (S-forward) and survivor swaps (S-swaps). We analyze how the mortality model and premium principle assumptions affect pricing and risk measures (value-at-risk and expected shortfall). Four mortality models (Lee–Carter, Renshaw–Haberman, Cairns–Blake–Dowd, and M6) and eight premium principles (Wang, proportional hazard, dual power, Gini, exponential, standard deviation, variance, and median absolute deviation) are considered. Our analysis highlights the need to refine mortality models and premium principles to enhance pricing accuracy and risk management. We also suggest regulators and practitioners incorporate expected shortfall alongside value-at-risk to capture tail risks and improve capital allocation. Full article

(This article belongs to the Special Issue Applied Financial and Actuarial Risk Analytics)

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16 pages, 1521 KiB

Open AccessArticle

Data Mining for the Adjustment of Credit Scoring Models in Solidarity Economy Entities: A Methodology for Addressing Class Imbalances

by Ivan Mauricio Bermudez Vera, Jaime Mosquera Restrepo and Diego Fernando Manotas-Duque

Risks 2025, 13(2), 20; https://doi.org/10.3390/risks13020020 - 22 Jan 2025

Viewed by 853

Abstract

This study addresses the quantification of credit risk in solidarity economy entities, proposing a new methodology to redefine the concept of a “default” in the frequent situations of extreme class imbalances. The objective is to develop and evaluate credit scoring models that enhance [...] Read more.

This study addresses the quantification of credit risk in solidarity economy entities, proposing a new methodology to redefine the concept of a “default” in the frequent situations of extreme class imbalances. The objective is to develop and evaluate credit scoring models that enhance risk management by incorporating internal and external data to assess default risk. Data mining techniques are applied to address class imbalances, redefining the term “default” to include external credit information and increasing the representation of the minority class. The effectiveness of machine learning and statistical models is evaluated using class-balancing methods such as under-sampling, over-sampling, and the Synthetic Minority Over-sampling Technique (SMOTE). The evaluation is based on the Balanced Accuracy metric and the holding power of the performance, ensuring a consistent predictive power of the model while avoiding overfitting. While machine learning methods can improve credit scoring, logistic regression-based models remain effective, especially when combined with class-balancing techniques. It is concluded that a balanced sample in a class size is essential to improve predictive performance. Full article

(This article belongs to the Special Issue Applied Financial and Actuarial Risk Analytics)

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27 pages, 905 KiB

Open AccessArticle

Optimal Benefit Distribution of a Tontine-like Annuity Fund with Age-Structured Models

by Fan Zhang, Ping Chen and Xueyuan Wu

Risks 2025, 13(1), 4; https://doi.org/10.3390/risks13010004 - 30 Dec 2024

Viewed by 674

Abstract

This paper introduces a tontine-like annuity fund designed to provide lifelong income to its participants. Initially, each member contributes a lump-sum payment into a trust fund as a joining premium. Participants then receive benefits over time, based on their survival. As members pass [...] Read more.

This paper introduces a tontine-like annuity fund designed to provide lifelong income to its participants. Initially, each member contributes a lump-sum payment into a trust fund as a joining premium. Participants then receive benefits over time, based on their survival. As members pass away, their share of payouts is redistributed among the survivors, resulting in increased payouts for those remaining. Differing from traditional tontines, which assume a uniform mortality risk, this fund accommodates participants of various ages and allows new members to join during its operation. To accommodate these features, the authors utilize age-structured models (ASMs) to determine fair premiums for new entrants and to analyze the dynamics of benefit distribution. The core objective of this paper is to develop a pension model using ASMs, recognizing its significant potential for adaptation and expansion. The primary mathematical approach employed is the Maximum Principle from optimal control theory, which helps in deriving explicit solutions for the optimal subsidy strategy. Through numerical examples and detailed illustrations, the paper demonstrates that participants who remain in the cohort longer receive greater subsidies. Additionally, the study finds that adverse shocks lead to a smaller population and thus fewer subsidies. Conversely, starting with a larger initial cohort population tends to increase the overall population, resulting in more subsidies. However, higher costs associated with subsidies lead to their reduction. Our analysis reveals the complex interplay of factors influencing the sustainability and effectiveness of the proposed annuity model. Full article

(This article belongs to the Special Issue Applied Financial and Actuarial Risk Analytics)

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12 pages, 732 KiB

Open AccessFeature PaperArticle

A Basic Asymptotic Test for Value-at-Risk Subadditivity

by Marius Hofert

Risks 2024, 12(12), 199; https://doi.org/10.3390/risks12120199 - 10 Dec 2024

Viewed by 671

Abstract

An asymptotic hypothesis test for value-at-risk subadditivity is introduced and studied. The test is derived based on an equivalent formulation of the value-at-risk subadditivity inequality in terms of the distribution of the underlying risks’ sum. Its size is considered mathematically, and its power [...] Read more.

An asymptotic hypothesis test for value-at-risk subadditivity is introduced and studied. The test is derived based on an equivalent formulation of the value-at-risk subadditivity inequality in terms of the distribution of the underlying risks’ sum. Its size is considered mathematically, and its power and p-value are studied empirically for different dependence structures, strength of dependence, marginal distributions, sample sizes, number of risks and value-at-risk confidence levels. Full article

(This article belongs to the Special Issue Applied Financial and Actuarial Risk Analytics)

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15 pages, 446 KiB

Open AccessFeature PaperArticle

Multivariate Spectral Backtests of Forecast Distributions under Unknown Dependencies

by Janine Balter and Alexander J. McNeil

Risks 2024, 12(1), 13; https://doi.org/10.3390/risks12010013 - 17 Jan 2024

Viewed by 1647

Abstract

Under the revised market risk framework of the Basel Committee on Banking Supervision, the model validation regime for internal models now requires that models capture the tail risk in profit-and-loss (P&L) distributions at the trading desk level. We develop multi-desk backtests, which simultaneously [...] Read more.

Under the revised market risk framework of the Basel Committee on Banking Supervision, the model validation regime for internal models now requires that models capture the tail risk in profit-and-loss (P&L) distributions at the trading desk level. We develop multi-desk backtests, which simultaneously test all trading desk models and which exploit all the information available in the presence of an unknown correlation structure between desks. We propose a multi-desk extension of the spectral test of Gordy and McNeil, which allows the evaluation of a model at more than one confidence level and contains a multi-desk value-at-risk (VaR) backtest as a special case. The spectral tests make use of realised probability integral transform values based on estimated P&L distributions for each desk and are more informative and more powerful than simpler tests based on VaR violation indicators. The new backtests are easy to implement with a reasonable running time; in a series of simulation studies, we show that they have good size and power properties. Full article

(This article belongs to the Special Issue Applied Financial and Actuarial Risk Analytics)

19 pages, 5357 KiB

Open AccessArticle

Equity Price Dynamics under Shocks: In Distress or Short Squeeze

by Cho-Hoi Hui, Chi-Fai Lo and Chi-Hei Liu

Risks 2024, 12(1), 1; https://doi.org/10.3390/risks12010001 - 20 Dec 2023

Viewed by 2041

Abstract

This paper proposes a simple bounded stochastic motion to model equity price dynamics under shocks. The stochastic process has a quasi-bounded boundary which can be breached if the probability leakage condition is met. The quasi-boundedness of the process at the boundary can thus [...] Read more.

This paper proposes a simple bounded stochastic motion to model equity price dynamics under shocks. The stochastic process has a quasi-bounded boundary which can be breached if the probability leakage condition is met. The quasi-boundedness of the process at the boundary can thus provide an indicator of the possible risk of equities under price shocks or in distress. Empirical calibration of the model parameters of the proposed process for equities can be performed easily due to the availability of an analytically tractable probability density function which generates fat-tailed distributions consistent with empirical observations. The volatility and mean-reversion of the S&P500 dynamics calibrated by the process are positively and negatively co-integrated, respectively, with the VIX index representing the level of market distress. The process captures the high likelihood of Hertz’s default about two months earlier, using only information until that point, and before the firm filed for Chapter 11 bankruptcy in May 2020 as a result of the COVID-19 pandemic. Empirical calibration of the process for GameStop’s stock price shows that the short squeeze in the stock occurred when the condition for breaching the upper boundary was met on 14 January 2021, i.e., about two weeks before major short-sellers closed out their positions with significant losses. The trading volume of the stock was positively co-integrated with the probability leakage ratio. Full article

(This article belongs to the Special Issue Applied Financial and Actuarial Risk Analytics)

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37 pages, 582 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Rank-Based Multivariate Sarmanov for Modeling Dependence between Loss Reserves

by Anas Abdallah and Lan Wang

Risks 2023, 11(11), 187; https://doi.org/10.3390/risks11110187 - 26 Oct 2023

Cited by 1 | Viewed by 2336

Abstract

The interdependence between multiple lines of business has an important impact on determining loss reserves and risk capital, which are crucial for the solvency of a property and casualty (P&C) insurance company. In this work, we introduce the two-stage inference method using the [...] Read more.

The interdependence between multiple lines of business has an important impact on determining loss reserves and risk capital, which are crucial for the solvency of a property and casualty (P&C) insurance company. In this work, we introduce the two-stage inference method using the Sarmanov family of multivariate distributions to the actuarial literature. In fact, we study rank-based methods using the Sarmanov distribution to adequately estimate the loss reserves and properly capture the dependence between lines of business. An inadequate choice of the dependence structure may negatively impact the estimation of the marginals and, hence, the reserve. Thus, we propose a two-stage inference strategy in this research to address this, while taking advantage of the flexibility of the Sarmanov distribution. We show that this strategy leads to a more robust estimation, and better captures the dependence between the risks. We also show that it generates smaller risk capital and a better diversification benefit. We extend the model to the multivariate case with more than two lines of business. To illustrate and validate our methods, we use three different sets of real data from both a major US property–casualty insurer and a large Canadian insurance company. Full article

(This article belongs to the Special Issue Applied Financial and Actuarial Risk Analytics)

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