Risk modelling on liquidations with Lévy processes

Highlights

• Our paper represents the first attempt in studying the Liquidations under the Lévy setup.

• We compactly express the joint distribution of the time of liquidation, the surplus at liquidation and the historical high of the surplus until liquidation by the scale functions and Lévy triplets associated with two underlying Lévy processes. When the two Lévy processes are different from each other by a deterministic drift or coincide with each other, even more compact results expressed only by the scale functions are presented.

• Our results are very general and are consistent with the existing literatures on Parisian ruin with (or without) a lower barrier when our model and parameters degenerate.

• Numerical examples are provided to illustrate the underlying features of liquidation ruin.

Abstract

In classical ruin theory, the time of ruin is defined as the time when the surplus of an insurance portfolio falls below zero. This simplification of a single barrier, however, needs careful adaptations to imitate the real-world liquidation process. Inspired by [7] and [24], this paper adopts a three-barrier model to describe the financial stress leading to bankruptcy of an insurer. The financial status of the insurer is classified into three states, namely, the solvent, the insolvent, and the liquidated. The insurer’s surplus processes at the states of solvent and insolvent are modeled by two spectrally negative Lévy processes, which have been taken as good candidates to model insurance risks in the recent literature. Accordingly, the time of liquidation is defined in this three-barrier model. By adopting the techniques of excursions in fluctuation theory, we obtain the joint distribution of the time of liquidation, the surplus at liquidation, and the historical high of the surplus until liquidation, which generalizes the known results on the classical expected discounted penalty function from [16]. The results have semi-explicit expressions in terms of the scale functions and the Lévy triplets associated with the two underlying Lévy processes. The special case when the two underlying Lévy processes coincide with each other or differ from each other by a constant drift term is also studied, and our results are expressed compactly via only the scale functions. The corresponding results are consistent with the classic works of literature on Parisian ruin with (or without) a lower barrier in [4,22], and [14]. Numerical examples are provided to illustrate the underlying features of liquidation ruin.

Introduction

In classical risk theory, the ultimate ruin occurs when an insurance company does not have sufficient assets to meet its liabilities. That is, whenever the surplus falls to zero or below, the insurance company is defined as insolvent. In this somewhat vague statement, the state of solvency is described by a single barrier zero of the surplus process. Under this definition, ruin theory has been a popular research topic for many decades. However, this simplification can describe only a small proportion of bankruptcies in the real world.

In the United States (US), bankruptcy is regulated by the United States Bankruptcy Code. When a business is in financial hardship such that its liability is greater than its assets, two options can be taken when filing for bankruptcy: Chapter 7 or Chapter 11. A business can seek Chapter 7 bankruptcy only if it immediately ceases all operations, and hence, Chapter 7 is usually referred to as “liquidation”; this type of bankruptcy is a popular option for small business owners. Chapter 11 bankruptcy, on the other hand, is often referred to as “reorganization”or “rehabilitation”bankruptcy because the process provides the business with an opportunity to reorganize its financial structure, such as debts, and try to recover as a healthy organization while continuing to operate. Chapter 11 is more expensive than Chapter 7 and is typically intended for medium to large businesses. Refer to the 2020 edition of United States Bankruptcy Code for more details. According to Corbae and D’Erasmo [11], 80% of bankruptcies of publicly traded companies are filed under Chapter 11, whereas only 20% are Chapter 7 liquidations.

From an academic point of view, it is natural and straightforward to describe Chapter 7 bankruptcy by a single-barrier model for the surplus process, as adopted by Gerber and Shiu [16] and Leland [23], in which bankruptcy is triggered when the firm’s asset value falls to the debt’s principal value. While Chapter 11 is a long and costly process, the granting of the opportunity to reorganize debts has been an attractive feature for many large corporations. Therefore, more and more researchers in finance have attempted to imitate the process of Chapter 11 bankruptcy in their models. Broadie et al. [7] proposed a three-barrier model to characterize the financial states of a bank: distress, Chapter 11 bankruptcy, and Chapter 7 liquidation. Antill and Grenadier [2] formulated a model in which shareholders can choose both their timing of default and the chapter of bankruptcy. They also examined how this flexibility alters the capital structure decisions of the firm.

Given the significance of Chapter 11 filing, regulators of the insurance industry also keep up to date with the trend, aiming to provide crucial safeguards for policyholders and for the economy. A review on the changes in the US insurance regulatory system can be found in [24]. We give only a brief introduction on the current regulatory system in the US and European Union (EU) to provide context for this study.

As reviewed by Li et al. [24], insurance companies in the US are not subject to the federal bankruptcy code, but their financial health is monitored by the National Association of Insurance Commissioners and varies from state to state. Their primary goal is to ensure there is sufficient capital for insurers to operate and meet their obligations to policyholders and other claimants. The US method of measuring whether capital is adequate is called risk-based capital. On the basis of the amount of insurance the company writes, the lines of business it writes, the assets it invests in, and other measures, an absolute least amount of capital an insurer needs is calculated. If an insurer capital dwindles, the regulators have the opportunity to intercede. The closer the actual capital is to the risk-based minimum, the more powerful the intervention can be.

Analogous to the regulatory spirit in the US, the current supervisory regime for insurance companies in the EU is also a risk-based capital regime. Since January 2016, EU insurers have been governed by the Solvency II regulatory regime. A three-level capital requirement system is applied: (a) the technical provision fulfils the obligations toward policyholders and other beneficiaries; (b) the minimum capital requirement is a safety net and indicates the level of capital below which ultimate supervisory action would be triggered; and (c) the solvency capital requirement enables an insurance company to absorb significant unforeseen losses and gives reasonable assurance to policyholders and beneficiaries. An overview of this system can be found in [17].

To imitate this real-world process of bankruptcy, the classical definition of ruin describes Chapter 7 liquidation reasonably well using a single barrier, but fails to describe the complex Chapter 11 reorganization. In view of the recent regulatory development described above, and following the work by Li et al. [24] on the insurance sector, we use a three-barrier model to describe the financial stress toward bankruptcy of an insurance company. This paper distinguishes between insolvency and bankruptcy. A company can be insolvent without being bankrupt, but it cannot be bankrupt without being insolvent. Insolvency is the inability to pay debts when they are due, whereas bankruptcy refers to a final alternative following the failure of all other attempts to clear debt. In the context of the Cramér-CLundberg model, a similar triple solvency/insolvency barrier system was considered in [32] and [33].

To better reflect the meaning of the three barriers, this paper adopts a different naming system of the three barriers, which was used by Li et al. [24]. The highest barrier is called the “safety”barrier (denoted by $c$). A surplus process that remains above this barrier indicates that the insurer has enough buffer to settle all claims in extreme situations which means the insurer is a healthy financial institution. If below the barrier, an investigation of the insurer’s business is carried out by the regulator on the insurer’s financial capacity to meet both its short-term and its long-term liabilities. If the surplus process keeps shrinking, the intermediary barrier is called a “reorganization”or “rehabilitation”barrier (denoted by $b$), which triggers the regulator’s intervention in the insurer’s business operation. The regulator may give a broad range of directions to the insurer with respect to carrying on its business, including prohibiting the insurer from issuing further policies, prohibiting it from disposing of assets, requiring it to make provisions in its accounts, and requiring it to increase its paid-up capital. The regulator’s rehabilitation intervention continues until the surplus reaches the safety barrier, or transfers to the wind-up procedure when the lowest barrier is breached. We call the lowest barrier the “liquidation”barrier (denoted by $a$). Once an insurance company has been liquidated, it is completely dissolved and permanently ceases its operations. The three-barrier system ($a<b<c$) classifies the state of an insurer into solvent, insolvent, and liquidated (bankrupt) states. Details of the transition between states are presented in Section 2.

Another feature of this paper is that the insurer’s surplus process is modeled by spectrally negative Lévy processes, which are stochastic processes with stationary independent increments and with sample paths of no positive jumps. Lévy processes have been taken as good candidates to model insurance risks; see [35] for a review of these features of Lévy processes. The applications of spectrally negative Lévy processes in risk theory can be seen in [1], [6], [8], [9], [12], [13], [14], [15], [18], [27], [36], [42], [44], and [39], and references therein. Using the time of liquidation, this paper explores an extended definition of the expected discounted penalty function expressed in terms of the scale functions and the Lévy triplets associated with the Lévy processes. The joint distribution of the time of liquidation, the surplus at liquidation, and the historical high of the surplus until liquidation is derived.

From a technical point of view, our mathematical argument is based on a heuristic idea presented in [24] that consists of distinguishing the excursions away from the three barriers (i.e., $a$, $b$, and $c$) of the underlying diffusion surplus process. In this paper, we provide an analogous definition of the time of liquidation ruin in the context of Lévy processes. With the help of the fluctuation theory for spectrally negative Lévy processes (particularly in [28] and [36], and references therein) and delicate path analysis arguments, this paper derives a semi-explicit and compact expression of the extended Gerber-Shiu function at liquidation ruin, in terms of the so-called scale functions and the Lévy triplets associated with the underlying Lévy processes. From our results, we can easily deduce the Gerber-Shiu distribution and the two-sided exit identities at Parisian ruin, which was originally obtained by Baurdoux et al. [4], Landriault et al. [22], and Frostig and Keren-Pinhasik [14]. In addition, compared with the fluctuation identities obtained in [22] and [14], where the spectrally negative Lévy process is assumed to have bounded path variation, this paper unifies the results for bounded and unbounded variation.

The rest of this paper is organized as follows. In Section 2, we review the basics of the spectrally negative Lévy processes, construct the three-barrier system, and define the liquidation risk with an independent exponentially distributed grace period. Section 3 presents a semi-explicit expression of the extended GerberCShiu function at the time of liquidation ruin. The application of our main results in the case of Parisian ruin is provided in Section 4. In Section 5, several numerical examples are discussed to illustrate the features of our results. All proofs are presented in the appendices.