{"title":"构建具有内生和正终极远期利率的 Smith-Wilson 无风险利率曲线","authors":"Chaoyi Zhao , Zijian Jia , Lan Wu","doi":"10.1016/j.insmatheco.2023.11.003","DOIUrl":null,"url":null,"abstract":"<div><p><span>We propose several methods for obtaining endogenous and positive ultimate forward rates (UFRs) for risk-free interest rate curves based on the Smith-Wilson method. The Smith-Wilson method, which is adopted by Solvency II, can both interpolate the market price data and extrapolate to the UFR. However, the method requires an exogenously-chosen UFR. To obtain an endogenous UFR, </span><span>de Kort and Vellekoop (2016)</span> proposed an optimization framework based on the Smith-Wilson method. In this paper, we prove the existence of an optimal endogenous UFR to their optimization problem under the condition that the cash flow matrix is square and invertible. In addition, to ensure the positivity of the optimal endogenous UFR during extreme time periods such as the COVID-19 pandemic, we extend their optimization framework by including non-negative constraints. Furthermore, we also propose a new optimization framework that can not only generate endogenous and positive UFRs but also incorporate practitioners' prior knowledge. We prove the feasibility of our frameworks, and conduct empirical studies for both the Chinese government bonds and the EURIBOR swaps to illustrate the capabilities of our methods.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":null,"pages":null},"PeriodicalIF":1.9000,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Construct Smith-Wilson risk-free interest rate curves with endogenous and positive ultimate forward rates\",\"authors\":\"Chaoyi Zhao , Zijian Jia , Lan Wu\",\"doi\":\"10.1016/j.insmatheco.2023.11.003\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p><span>We propose several methods for obtaining endogenous and positive ultimate forward rates (UFRs) for risk-free interest rate curves based on the Smith-Wilson method. The Smith-Wilson method, which is adopted by Solvency II, can both interpolate the market price data and extrapolate to the UFR. However, the method requires an exogenously-chosen UFR. To obtain an endogenous UFR, </span><span>de Kort and Vellekoop (2016)</span> proposed an optimization framework based on the Smith-Wilson method. In this paper, we prove the existence of an optimal endogenous UFR to their optimization problem under the condition that the cash flow matrix is square and invertible. In addition, to ensure the positivity of the optimal endogenous UFR during extreme time periods such as the COVID-19 pandemic, we extend their optimization framework by including non-negative constraints. Furthermore, we also propose a new optimization framework that can not only generate endogenous and positive UFRs but also incorporate practitioners' prior knowledge. We prove the feasibility of our frameworks, and conduct empirical studies for both the Chinese government bonds and the EURIBOR swaps to illustrate the capabilities of our methods.</p></div>\",\"PeriodicalId\":54974,\"journal\":{\"name\":\"Insurance Mathematics & Economics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.9000,\"publicationDate\":\"2023-11-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Insurance Mathematics & Economics\",\"FirstCategoryId\":\"96\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0167668723000963\",\"RegionNum\":2,\"RegionCategory\":\"经济学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"ECONOMICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Insurance Mathematics & Economics","FirstCategoryId":"96","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167668723000963","RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ECONOMICS","Score":null,"Total":0}
Construct Smith-Wilson risk-free interest rate curves with endogenous and positive ultimate forward rates
We propose several methods for obtaining endogenous and positive ultimate forward rates (UFRs) for risk-free interest rate curves based on the Smith-Wilson method. The Smith-Wilson method, which is adopted by Solvency II, can both interpolate the market price data and extrapolate to the UFR. However, the method requires an exogenously-chosen UFR. To obtain an endogenous UFR, de Kort and Vellekoop (2016) proposed an optimization framework based on the Smith-Wilson method. In this paper, we prove the existence of an optimal endogenous UFR to their optimization problem under the condition that the cash flow matrix is square and invertible. In addition, to ensure the positivity of the optimal endogenous UFR during extreme time periods such as the COVID-19 pandemic, we extend their optimization framework by including non-negative constraints. Furthermore, we also propose a new optimization framework that can not only generate endogenous and positive UFRs but also incorporate practitioners' prior knowledge. We prove the feasibility of our frameworks, and conduct empirical studies for both the Chinese government bonds and the EURIBOR swaps to illustrate the capabilities of our methods.
期刊介绍:
Insurance: Mathematics and Economics publishes leading research spanning all fields of actuarial science research. It appears six times per year and is the largest journal in actuarial science research around the world.
Insurance: Mathematics and Economics is an international academic journal that aims to strengthen the communication between individuals and groups who develop and apply research results in actuarial science. The journal feels a particular obligation to facilitate closer cooperation between those who conduct research in insurance mathematics and quantitative insurance economics, and practicing actuaries who are interested in the implementation of the results. To this purpose, Insurance: Mathematics and Economics publishes high-quality articles of broad international interest, concerned with either the theory of insurance mathematics and quantitative insurance economics or the inventive application of it, including empirical or experimental results. Articles that combine several of these aspects are particularly considered.