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Abstract
This paper is devoted to asymptotic analysis for a continuous-time risk model with the insurance surplus process and the log-price process of the investment driven by two dependent jump-diffusion processes. We take into account arbitrary dependence between the insurance claims and their corresponding investment return jumps caused by a sequence of systematic factors, whose arrival times constitute a renewal counting process. Under the framework of regular variation, we obtain a simple and unified asymptotic formula for the finite-time ruin probability as the initial wealth becomes large. It turns out that, in the weakly dependent case, the tails of the claims determine the exact decay rate of the finite-time ruin probability while the investment return jumps only contribute to the coefficient of the asymptotic formula; however, in the strongly dependent case, they both produce essential impacts on the finite-time ruin probability which is under-estimated in the weakly dependent case.
Acknowledgments
The authors would like to thank the anonymous referee for his/her suggestive comments and very careful reading of the paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 According to the American Council of Life Insurers, ‘life insurers are a major source of bond financing for American business, holding more than 22% of all U.S. corporate bonds’ and ‘life insurers provide long-term capital to the commercial mortgage market, financing more than 515 billion dollars, or almost one-sixth, of U.S. commercial mortgages’. The Annual Report 2021 of Allianz Group stated that as of 31 December 2021, the overall asset portfolio has reached 808.5 billion euros mainly in the debt investments (672.3 billion decreased by 10.1 billion compared to year end 2020, mainly due to market movements), about 91% of which was invested in investment-grade bonds and loans. See page 90. Available at https://www.allianz.com/en/investorrelations/results-reports/annual-reports.
2 See a report from the International Monetary Fund (2016). The insurance sector: trends and systemic risks implications. Available at https://www.imf.org/External/Pubs/FT/GFSR/2016/01/pdf/c3.pdf.
3 The collapse and near-failure of the insurance giant American International Group was caused largely by its 526 billion dollars portfolio of credit default swaps.
4 The American Property Casualty Insurance Association estimated that the monthly COVID-19 business interruption losses just for businesses with 100 or fewer employees was 255–431 billion dollars per month. Available at http://www.pciaa.net/pciwebsite/cms/content/viewpage?sitePageId=60052.
5 The business segment Asset Management of the Allianz Group was impacted by the severe financial market disruption and related investor uncertainties which led to a negative market valuation of assets under management, and net outflows in the first quarter of 2020. However, in the subsequent quarters, the business segment fully recovered by reaching strong positive market effects and third-party net inflows. Meanwhile, the business segment Property-Casualty was negatively impacted with an amount of 1.1 billion pounds as of 31 December 2020. Losses resulted from unintended business interruption coverage, entertainment, and credit. See page 122 of The Annual Report 2020 of Allianz Group. Available at https://www.allianz.com/en/investor_relations/results-reports/annual-reports/annual-report-archive.html.
6 The Insurance Council of Australia estimated the total annual insured losses due to climate-related extreme events was 3.70 billion dollars, of which the severe bushfires accounted for 2.2 billion. http://www.icadataglobe.com/access-catastrophe-data/ (accessed June 2020).
7 With more than 16 trillion dollars in funds under management, the global assets of the private insurance industry will also have substantial exposure to future climate risk. The physical impacts of climate change will directly impact the investment arms of insurance and reinsurance companies. It should be recognized that risk assessments of an insurer's investment portfolio need to incorporate an analysis of the impact of climate change. See Herweijer et al. (Citation2009) and Giuzio et al. (Citation2019).