Pub Date : 2022-09-21DOI: 10.1080/10920277.2022.2106576
Jingshu Luo, Martin Grace
Medical liability insurance covers physicians’ liability, and its price could affect physicians’ practice. In this article, we use a unique county-level dataset to study how medical liability insurance prices of three specialties, internal medicine, general surgery, and obstetrics–gynecology (OB-GYN), changed after the Affordable Care Act (ACA) elective Medicaid expansion provision. The Medicaid expansion has largely increased the demand for health care services and potentially exposed physicians to higher medical liability risks. With higher expected losses, insurers could react by increasing medical malpractice insurance prices. We first study all counties in states that elected to expand Medicaid and compare them to counties in nonexpansion states. Then we narrow our analysis to consider differential effects in bordering counties with different Medicaid expansion statuses over the period 2010–2018. In both samples, we find significantly higher medical liability insurance prices 2 years after the expansion (on average) in expansion states in comparison to nonexpansion states, and the difference is larger for physicians practicing internal medicine (6–8% at 2 years after expansion) and general surgery (12–16% at 2 years after expansion) but less so for OB-GYN. Our OB-GYN results are likely because significant numbers of births were already covered under Medicaid and were not affected by the expansion. Our finding suggests that the expansion of health insurance increases liability costs to medical practitioners.
{"title":"Does Public Health Insurance Expansion Influence Medical Liability Insurance Prices? The Case of the ACA’s Optional Medicaid Expansion","authors":"Jingshu Luo, Martin Grace","doi":"10.1080/10920277.2022.2106576","DOIUrl":"https://doi.org/10.1080/10920277.2022.2106576","url":null,"abstract":"Medical liability insurance covers physicians’ liability, and its price could affect physicians’ practice. In this article, we use a unique county-level dataset to study how medical liability insurance prices of three specialties, internal medicine, general surgery, and obstetrics–gynecology (OB-GYN), changed after the Affordable Care Act (ACA) elective Medicaid expansion provision. The Medicaid expansion has largely increased the demand for health care services and potentially exposed physicians to higher medical liability risks. With higher expected losses, insurers could react by increasing medical malpractice insurance prices. We first study all counties in states that elected to expand Medicaid and compare them to counties in nonexpansion states. Then we narrow our analysis to consider differential effects in bordering counties with different Medicaid expansion statuses over the period 2010–2018. In both samples, we find significantly higher medical liability insurance prices 2 years after the expansion (on average) in expansion states in comparison to nonexpansion states, and the difference is larger for physicians practicing internal medicine (6–8% at 2 years after expansion) and general surgery (12–16% at 2 years after expansion) but less so for OB-GYN. Our OB-GYN results are likely because significant numbers of births were already covered under Medicaid and were not affected by the expansion. Our finding suggests that the expansion of health insurance increases liability costs to medical practitioners.","PeriodicalId":46812,"journal":{"name":"North American Actuarial Journal","volume":"27 1","pages":"508 - 529"},"PeriodicalIF":1.4,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46835267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-13DOI: 10.1080/10920277.2022.2102040
T. Signorelli, C. Campani, C. Neves
This article proposes a method to build term structures that are consistent with market data and that provide interest rates for which the volatility, on average, decreases as maturities increase. The method is designed for continuous repetitive use and is consistent with work by Diebold and Li, providing reasonable extrapolated rates, with an appropriate level of volatility over time. The Svensson model is adopted, and its parameters are estimated by the combination of a genetic algorithm and a quasi-Newton nonlinear optimization method. We innovate with a new objective function that focuses on both parts of the estimated curves (interpolated and extrapolated). For this purpose, a stability component is added. The new objective function aims to solve the problem of estimating long-term rates not observable in the market, for which the estimates are usually artificially stable or excessively volatile. The results show that the estimation method is able to bring the volatility of extrapolated rates to levels consistent with those observed for the longest liquid rate. Estimation errors are small enough and there is no statistical evidence that they are biased. The method is useful for the insurance market, since it provides interest rates that do not lead to artificially stable or excessively volatile technical provisions.
{"title":"Extrapolating Long-Run Yield Curves: An Innovative and Consistent Approach","authors":"T. Signorelli, C. Campani, C. Neves","doi":"10.1080/10920277.2022.2102040","DOIUrl":"https://doi.org/10.1080/10920277.2022.2102040","url":null,"abstract":"This article proposes a method to build term structures that are consistent with market data and that provide interest rates for which the volatility, on average, decreases as maturities increase. The method is designed for continuous repetitive use and is consistent with work by Diebold and Li, providing reasonable extrapolated rates, with an appropriate level of volatility over time. The Svensson model is adopted, and its parameters are estimated by the combination of a genetic algorithm and a quasi-Newton nonlinear optimization method. We innovate with a new objective function that focuses on both parts of the estimated curves (interpolated and extrapolated). For this purpose, a stability component is added. The new objective function aims to solve the problem of estimating long-term rates not observable in the market, for which the estimates are usually artificially stable or excessively volatile. The results show that the estimation method is able to bring the volatility of extrapolated rates to levels consistent with those observed for the longest liquid rate. Estimation errors are small enough and there is no statistical evidence that they are biased. The method is useful for the insurance market, since it provides interest rates that do not lead to artificially stable or excessively volatile technical provisions.","PeriodicalId":46812,"journal":{"name":"North American Actuarial Journal","volume":"27 1","pages":"472 - 492"},"PeriodicalIF":1.4,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48136148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-26DOI: 10.1080/10920277.2022.2100425
Jean‐François Bégin
Over the last 40 years, various frameworks have been proposed to model economic and financial variables relevant to actuaries. These models are helpful, but searching for a unique model that gives optimal forecasting performance can be frustrating and ultimately futile. This study therefore investigates whether we can create better, more reliable economic scenario generators by combining them. We first consider eight prominent economic scenario generators and apply Bayesian estimation techniques to them, thus allowing us to account for parameter uncertainty. We then rely on predictive distribution stacking to obtain optimal model weights that prescribe how the models should be averaged. The weights are constructed in a leave-future-out fashion to build truly out-of-sample forecasts. An extensive empirical study based on three economies—the United States, Canada, and the United Kingdom—and data from 1992 to 2021 is performed. We find that the optimal weights change over time and differ from one economy to another. The out-of-sample behavior of the ensemble model compares favorably to the other eight models: the ensemble model’s performance is substantially better than that of the worse models and comparable to that of the better models. Creating ensembles is thus beneficial from an out-of-sample perspective because it allows for robust and reasonable forecasts.
{"title":"Ensemble Economic Scenario Generators: Unity Makes Strength","authors":"Jean‐François Bégin","doi":"10.1080/10920277.2022.2100425","DOIUrl":"https://doi.org/10.1080/10920277.2022.2100425","url":null,"abstract":"Over the last 40 years, various frameworks have been proposed to model economic and financial variables relevant to actuaries. These models are helpful, but searching for a unique model that gives optimal forecasting performance can be frustrating and ultimately futile. This study therefore investigates whether we can create better, more reliable economic scenario generators by combining them. We first consider eight prominent economic scenario generators and apply Bayesian estimation techniques to them, thus allowing us to account for parameter uncertainty. We then rely on predictive distribution stacking to obtain optimal model weights that prescribe how the models should be averaged. The weights are constructed in a leave-future-out fashion to build truly out-of-sample forecasts. An extensive empirical study based on three economies—the United States, Canada, and the United Kingdom—and data from 1992 to 2021 is performed. We find that the optimal weights change over time and differ from one economy to another. The out-of-sample behavior of the ensemble model compares favorably to the other eight models: the ensemble model’s performance is substantially better than that of the worse models and comparable to that of the better models. Creating ensembles is thus beneficial from an out-of-sample perspective because it allows for robust and reasonable forecasts.","PeriodicalId":46812,"journal":{"name":"North American Actuarial Journal","volume":"27 1","pages":"444 - 471"},"PeriodicalIF":1.4,"publicationDate":"2022-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49155718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-18DOI: 10.1080/10920277.2022.2099425
P. Brockett, L. Golden, Pengyu Wei, Charles C. Yang
Within the context of Medicare’s enunciated triple aims of better health, better care, and lower costs, we examine the effectiveness of medical loss ratio (MLR) on health outcomes of Medicare Advantage insurers. We simultaneously examine the effect of an efficiency measure for the insurer performance: medical service utilization efficiency (an assessment of how efficiently an insurer provides medical services). This research is based upon collection and integration of several data sources: health outcome data, financial data, and medical service utilization data. The assessment procedure employs a two-stage analytical approach: efficiency analysis followed by regressions. We quantify insurer efficiency using data envelopment analysis (DEA), which determines the relative efficiency of an insurer when the inputs and outputs can both be multivariate. We then run regressions with the dependent variables being functional health outcomes (“improving or maintaining mental health,” “improving or maintaining physical health,” and “improving or maintaining physical and mental health”) and health improvement efficiency (how cost-efficient the insurer is in improving functional health outcomes). Independent variables include MLR, medical service utilization efficiency, and a rich set of control variables. We find that neither MLR nor medical service utilization efficiency provides a good regulatory and evaluation indicator for stimulating/producing functional health outcomes. On the other hand, they do both significantly relate to health improvement efficiency, and hence are both reasonable regulatory and monitoring indicators for efficiently producing positive health outcomes. Our results suggest that to enhance health improvement efficiency, medical service utilization efficiency should be incorporated as a cost-efficient regulatory and monitoring indicator when evaluating Medical Advantage insurers.
{"title":"Medicare Advantage, Medical Loss Ratio, Service Efficiency, and Efficiently Positive Health Outcomes","authors":"P. Brockett, L. Golden, Pengyu Wei, Charles C. Yang","doi":"10.1080/10920277.2022.2099425","DOIUrl":"https://doi.org/10.1080/10920277.2022.2099425","url":null,"abstract":"Within the context of Medicare’s enunciated triple aims of better health, better care, and lower costs, we examine the effectiveness of medical loss ratio (MLR) on health outcomes of Medicare Advantage insurers. We simultaneously examine the effect of an efficiency measure for the insurer performance: medical service utilization efficiency (an assessment of how efficiently an insurer provides medical services). This research is based upon collection and integration of several data sources: health outcome data, financial data, and medical service utilization data. The assessment procedure employs a two-stage analytical approach: efficiency analysis followed by regressions. We quantify insurer efficiency using data envelopment analysis (DEA), which determines the relative efficiency of an insurer when the inputs and outputs can both be multivariate. We then run regressions with the dependent variables being functional health outcomes (“improving or maintaining mental health,” “improving or maintaining physical health,” and “improving or maintaining physical and mental health”) and health improvement efficiency (how cost-efficient the insurer is in improving functional health outcomes). Independent variables include MLR, medical service utilization efficiency, and a rich set of control variables. We find that neither MLR nor medical service utilization efficiency provides a good regulatory and evaluation indicator for stimulating/producing functional health outcomes. On the other hand, they do both significantly relate to health improvement efficiency, and hence are both reasonable regulatory and monitoring indicators for efficiently producing positive health outcomes. Our results suggest that to enhance health improvement efficiency, medical service utilization efficiency should be incorporated as a cost-efficient regulatory and monitoring indicator when evaluating Medical Advantage insurers.","PeriodicalId":46812,"journal":{"name":"North American Actuarial Journal","volume":"27 1","pages":"493 - 507"},"PeriodicalIF":1.4,"publicationDate":"2022-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43769988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-11DOI: 10.1080/10920277.2022.2077220
Jiang Cheng, Frank Y. Feng, Xudong Zeng
Pay-as-you-drive (PAYD) insurance is an exciting innovation. We develop a dynamic model to study PAYD insurance from the policyholder’s utility maximization perspective. We demonstrate that PAYD insurance does benefit the policyholder by reducing premium paid and increasing the total utility derived from auto usage and wealth. PAYD insurance may also improve overall social welfare by incentivizing customers to drive less. We illustrate that PAYD insurance is more efficient than fuel tax in reducing mileage due to the concavity relation of premium and driving distance. Finally, we derive a cut-off value of mileage below which policyholders who drive with traditional insurance should switch to a PAYD policy. Our research proposes a reliable theoretical framework, and confirms that PAYD insurance benefits both individual customers and society as a whole.
{"title":"Pay-As-You-Drive Insurance: Modeling and Implications","authors":"Jiang Cheng, Frank Y. Feng, Xudong Zeng","doi":"10.1080/10920277.2022.2077220","DOIUrl":"https://doi.org/10.1080/10920277.2022.2077220","url":null,"abstract":"Pay-as-you-drive (PAYD) insurance is an exciting innovation. We develop a dynamic model to study PAYD insurance from the policyholder’s utility maximization perspective. We demonstrate that PAYD insurance does benefit the policyholder by reducing premium paid and increasing the total utility derived from auto usage and wealth. PAYD insurance may also improve overall social welfare by incentivizing customers to drive less. We illustrate that PAYD insurance is more efficient than fuel tax in reducing mileage due to the concavity relation of premium and driving distance. Finally, we derive a cut-off value of mileage below which policyholders who drive with traditional insurance should switch to a PAYD policy. Our research proposes a reliable theoretical framework, and confirms that PAYD insurance benefits both individual customers and society as a whole.","PeriodicalId":46812,"journal":{"name":"North American Actuarial Journal","volume":"27 1","pages":"303 - 321"},"PeriodicalIF":1.4,"publicationDate":"2022-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46488491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-05DOI: 10.1080/10920277.2022.2078374
Maximilian Bär, Nadine Gatzert
The question how individuals should (optimally) annuitize their wealth remains of high relevance in light of longevity risk and volatile capital markets. In this article, we first present traditional and innovative products and strategies for the decumulation of wealth during retirement, based on a review of 72 selected academic articles in peer-reviewed journals. We further identify relevant factors that generally influence the conception of these products from the retirees’ perspectives, and derive implications for product developers, before concluding with avenues of future research. Our results indicate that innovative suggestions often comprise tontine-like structures, exploit actuarial and accounting smoothing in various ways, defer annuitization to higher ages, or combine it with long-term care options, for instance. Key areas of future research in this field include the consideration of both insurer and retiree perspectives in the analysis of products, using behavioral considerations when evaluating the retirees’ perspective, and taking into account the impact of costs or expenses. While recent articles increasingly consider these aspects, manifold opportunities for future research remain.
{"title":"Products and Strategies for the Decumulation of Wealth during Retirement: Insights from the Literature","authors":"Maximilian Bär, Nadine Gatzert","doi":"10.1080/10920277.2022.2078374","DOIUrl":"https://doi.org/10.1080/10920277.2022.2078374","url":null,"abstract":"The question how individuals should (optimally) annuitize their wealth remains of high relevance in light of longevity risk and volatile capital markets. In this article, we first present traditional and innovative products and strategies for the decumulation of wealth during retirement, based on a review of 72 selected academic articles in peer-reviewed journals. We further identify relevant factors that generally influence the conception of these products from the retirees’ perspectives, and derive implications for product developers, before concluding with avenues of future research. Our results indicate that innovative suggestions often comprise tontine-like structures, exploit actuarial and accounting smoothing in various ways, defer annuitization to higher ages, or combine it with long-term care options, for instance. Key areas of future research in this field include the consideration of both insurer and retiree perspectives in the analysis of products, using behavioral considerations when evaluating the retirees’ perspective, and taking into account the impact of costs or expenses. While recent articles increasingly consider these aspects, manifold opportunities for future research remain.","PeriodicalId":46812,"journal":{"name":"North American Actuarial Journal","volume":"27 1","pages":"322 - 340"},"PeriodicalIF":1.4,"publicationDate":"2022-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44292146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-05DOI: 10.1080/10920277.2022.2087679
Fabio Baione, D. Biancalana, Paolo De Angelis
Beta regression is a flexible tool in modeling proportions and rates, but is rarely applied in th actuarial field. In this article, we propose its application in the context of policyholder behavior and particularly to model surrenders and withdrawals. Surrender implies the expiration of the contract and denotes the payment of the surrender value, which is contractually defined. Withdrawal does not imply the termination of the contract and denotes the payment of a cash amount, left to the discretion of the policyholder, within the limits of the surrender value. Moreover, the Actuarial Standard of Practice 52 states that, for surrender and withdrawal estimation, the actuary should take into account several risk factors that could influence the phenomenon. To this aim, we introduce a two-part Beta regression model, where the first part consists in the estimate of the number of surrenders and withdrawals by means of a multinomial regression, as an extension of the logistic regression model frequently used in the empirical literature just to estimate surrender. Then, considering the uncertainty on the amount withdrawn, we express it as a proportion of surrender value; in this way, it assumes values continuously in the interval and it is compliant with a Beta distribution. Therefore, in the second part, we propose the adoption of a Beta regression approach to model the proportion withdrawn of the surrender value. Our final goal is to apply our model on a real-life insurance portfolio providing the estimates of the number of surrenders and withdrawals as well as the corresponding cash amount for each risk class considered.
{"title":"A Two-Part Beta Regression Approach for Modeling Surrenders and Withdrawals in a Life Insurance Portfolio","authors":"Fabio Baione, D. Biancalana, Paolo De Angelis","doi":"10.1080/10920277.2022.2087679","DOIUrl":"https://doi.org/10.1080/10920277.2022.2087679","url":null,"abstract":"Beta regression is a flexible tool in modeling proportions and rates, but is rarely applied in th actuarial field. In this article, we propose its application in the context of policyholder behavior and particularly to model surrenders and withdrawals. Surrender implies the expiration of the contract and denotes the payment of the surrender value, which is contractually defined. Withdrawal does not imply the termination of the contract and denotes the payment of a cash amount, left to the discretion of the policyholder, within the limits of the surrender value. Moreover, the Actuarial Standard of Practice 52 states that, for surrender and withdrawal estimation, the actuary should take into account several risk factors that could influence the phenomenon. To this aim, we introduce a two-part Beta regression model, where the first part consists in the estimate of the number of surrenders and withdrawals by means of a multinomial regression, as an extension of the logistic regression model frequently used in the empirical literature just to estimate surrender. Then, considering the uncertainty on the amount withdrawn, we express it as a proportion of surrender value; in this way, it assumes values continuously in the interval and it is compliant with a Beta distribution. Therefore, in the second part, we propose the adoption of a Beta regression approach to model the proportion withdrawn of the surrender value. Our final goal is to apply our model on a real-life insurance portfolio providing the estimates of the number of surrenders and withdrawals as well as the corresponding cash amount for each risk class considered.","PeriodicalId":46812,"journal":{"name":"North American Actuarial Journal","volume":"27 1","pages":"380 - 395"},"PeriodicalIF":1.4,"publicationDate":"2022-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47065086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-27DOI: 10.1080/10920277.2022.2078373
R. Thomas
I congratulate the authors on this enjoyable and timely article, which touches on several of my interests. I would like to offer some comments on nonrisk price discrimination, that is, individual price variations that do not reflect expected costs (sometimes described as “ price optimization ” )
{"title":"Discussion on “The Discriminating (Pricing) Actuary,” by Edward W. (Jed) Frees and Fei Huang","authors":"R. Thomas","doi":"10.1080/10920277.2022.2078373","DOIUrl":"https://doi.org/10.1080/10920277.2022.2078373","url":null,"abstract":"I congratulate the authors on this enjoyable and timely article, which touches on several of my interests. I would like to offer some comments on nonrisk price discrimination, that is, individual price variations that do not reflect expected costs (sometimes described as “ price optimization ” )","PeriodicalId":46812,"journal":{"name":"North American Actuarial Journal","volume":"27 1","pages":"206 - 207"},"PeriodicalIF":1.4,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43339698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-15DOI: 10.1080/10920277.2022.2071741
V. Brazauskas, Ponmalar Ratnam
Many risk measures can be defined through the quantile function of the underlying loss variable (e.g., a class of distortion risk measures). When the loss variable is discrete or mixed, however, the definition of risk measures has to be broadened, which makes statistical inference trickier. To facilitate a straightforward transition from the risk measurement literature of continuous loss variables to that of discrete, in this article we study smoothing of quantiles for discrete variables. Smoothed quantiles are defined using the theory of fractional or imaginary order statistics, which was originated by Stigler (1977). To prove consistency and asymptotic normality of sample estimators of smoothed quantiles, we utilize the results of Wang and Hutson (2011) and generalize them to vectors of smoothed quantiles. Further, we thoroughly investigate extensions of this methodology to discrete populations with infinite support (e.g., Poisson and zero-inflated Poisson distributions). Furthermore, large- and small-sample properties of the newly designed estimators are investigated theoretically and through Monte Carlo simulations. Finally, applications of smoothed quantiles to risk measurement (e.g., estimation of distortion risk measures such as Value at Risk, conditional tail expectation, and proportional hazards transform) are discussed and illustrated using automobile accident data. Comparisons between the classical (and linearly interpolated) quantiles and smoothed quantiles are performed as well.
{"title":"Smoothed Quantiles for Measuring Discrete Risks","authors":"V. Brazauskas, Ponmalar Ratnam","doi":"10.1080/10920277.2022.2071741","DOIUrl":"https://doi.org/10.1080/10920277.2022.2071741","url":null,"abstract":"Many risk measures can be defined through the quantile function of the underlying loss variable (e.g., a class of distortion risk measures). When the loss variable is discrete or mixed, however, the definition of risk measures has to be broadened, which makes statistical inference trickier. To facilitate a straightforward transition from the risk measurement literature of continuous loss variables to that of discrete, in this article we study smoothing of quantiles for discrete variables. Smoothed quantiles are defined using the theory of fractional or imaginary order statistics, which was originated by Stigler (1977). To prove consistency and asymptotic normality of sample estimators of smoothed quantiles, we utilize the results of Wang and Hutson (2011) and generalize them to vectors of smoothed quantiles. Further, we thoroughly investigate extensions of this methodology to discrete populations with infinite support (e.g., Poisson and zero-inflated Poisson distributions). Furthermore, large- and small-sample properties of the newly designed estimators are investigated theoretically and through Monte Carlo simulations. Finally, applications of smoothed quantiles to risk measurement (e.g., estimation of distortion risk measures such as Value at Risk, conditional tail expectation, and proportional hazards transform) are discussed and illustrated using automobile accident data. Comparisons between the classical (and linearly interpolated) quantiles and smoothed quantiles are performed as well.","PeriodicalId":46812,"journal":{"name":"North American Actuarial Journal","volume":"27 1","pages":"253 - 277"},"PeriodicalIF":1.4,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43253060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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