Ruin probabilities under capital constraints

Abstract

In this paper, we generalise the classic compound Poisson risk model, by the introduction of ordered capital levels, to model the solvency of an insurance firm. A breach of the higher capital level, the magnitude of which does not cause further breaches of either the lower level or the so-called intermediate confidence level (of the shareholders), requires a capital injection to restore the surplus to a solvent position. On the other hand, if the confidence level is breached capital injections are no longer a viable method of recapitalisation. Instead, the company can borrow money from a third party, subject to a constant interest rate, which is paid back until the surplus returns to the confidence level and subsequently can be restored to a fully solvent position by a capital injection. If at any point the surplus breaches the lower capital level, the company is considered 'insolvent' and is forced to cease trading. For the aforementioned risk model, we derive an explicit expression for the 'probability of insolvency' in terms of the ruin quantities of the classical risk model. Under the assumption of exponentially distributed claim sizes, we show that the probability of insolvency is in fact directly proportional to the classical ruin function. It is shown that this result also holds for the asymptotic behaviour of the insolvency probability, with a general claim size distribution. Explicit expressions are also derived for the moment generating function of the accumulated capital injections up to the time of insolvency and finally, in order to better capture the reality, dividend payments to the companies shareholders are considered, along with the capital constraint levels, and explicit expressions for the probability of insolvency, under this modification, are obtained.

On the edge: The impacts of cash flow at risk on the shareholders’ equity of public companies in Brazil

“…These requirements are derived from international agreements known as "Basel III" for financial institutions (Andrieş et al, 2022;Oliveira & Ferreira, 2019) and "Solvency II" for insurance companies (Carvalho & Cardoso, 2021;Chen & Yuan, 2017;Macohon et al, 2017). In essence, regulations based on these agreements require institutions to assess their risks and, based on this, to maintain a minimum amount of capital to minimize the risk of insolvency (Euphasio Junior & Carvalho, 2022;Gupta & Liang, 2005;Ramsden & Papaioannou, 2019).…”

“…It is no coincidence that the rules imposed by banking and insurance regulators require a minimum amount of capital for entities to be able to operate in the market, addressing the different risks to which they are exposed and thus minimizing the likelihood of insolvency (Areias & Carvalho, 2021;Ramsden & Papaioannou, 2019), which is damaging both to shareholders and to the health of the financial system (Harrington, 2009). In Brazil, for example, banks must follow the determinations of the Central Bank, which are based on the "Basel III" pillars (Oliveira & Ferreira, 2019).…”

No limite: impactos do fluxo de caixa em risco sobre o patrimônio líquido de empresas de capital aberto no Brasil

“…Tais exigências derivam de acordos internacionais, conhecidos como "Basileia III" nas instituições financeiras (Andrieş et al, 2022;Oliveira & Ferreira, 2019), e como "Solvência II" nas instituições securitárias (Carvalho & Cardoso, 2021;Chen & Yuan, 2017;Macohon et al, 2017). Em síntese, a regulação baseada nesses acordos faz com que a instituição avalie seus riscos e, com base nisso, seja obrigada a manter um valor mínimo de capital, para minimizar o risco de insolvência (Euphasio Junior & Carvalho, 2022;Gupta & Liang, 2005;Ramsden & Papaioannou, 2019).…”

“…Não por acaso, as regras impostas pelos órgãos reguladores de bancos e seguradoras exigem um capital mínimo para que a entidade possa operar no mercado, fazendo frente aos diversos riscos a que ela está exposta e, assim, minimizando a probabilidade de insolvência (Areias & Carvalho, 2021;Ramsden & Papaioannou, 2019), danosa tanto para os acionistas quanto para a saúde do sistema financeiro (Harrington, 2009). No Brasil, por exemplo, os bancos devem seguir as determinações do Banco Central, que se baseiam nos pilares da "Basileia III" (Oliveira & Ferreira, 2019).…”

Reinsurance and Solvency Capital: Mitigating Insurance Companies’ Ruin Probability

“…But it is not just claims ordinances that matter: Dickson and Qazvini (2016) assess the role of reinsurers to minimize abrupt falls in the insurer's adjusted equity. Ramsden and Papaioannou (2019) introduce order on the capital level, arguing that the firm could take out loans from third parties (not just insurance companies), in case of insufficient own equity and difficulties in recapitalize its financial position, as a result of the shareholder confidence loss. In this case, the authors derive explicit expressions for the moments generating function of accumulated capital injections until the moment of bankruptcy, making it possible to estimate the ruin probability and dividend payments.…”