Enhancing insurer value through reinsurance optimization

https://doi.org/10.1016/j.insmatheco.2005.11.004Get rights and content

Abstract

The paper investigates the demand for reinsurance in insurer risk management. The insurer’s objective is to maximize shareholder value under a solvency constraint imposed by a regulatory authority. In a one period model of a regulated market where the required solvency level is fixed, an insurer is assumed to maintain solvency using two control variables: reinsurance and risk capital supplied by shareholders. Two alternative regulatory constraints are considered in the paper. In the first one the required risk capital is determined at the beginning of the period independently of the reinsurance decision. In the second model insurers can reduce the required minimal level of risk capital taking into account reinsurance. The first model does not create a demand for reinsurance in a frictionless market, however, there is demand for reinsurance in this model under the presence of corporate tax and financial distress costs. An optimal tradeoff between the required minimal level of the risk capital and purchase of reinsurance occurs in the second model. The impact of taxation and financial distress costs on the demand for reinsurance is quantified and assessed using the models.

Introduction

Many studies of reinsurance optimization in the classical actuarial literature assume that the insurer objective is to minimize its ruin probability. This assumption has limitations from the point of view of the modern theory of integrated risk management for an insurance company, since it focuses on risk minimization only, without any explicit regard to the company’s economic value. Other more recent studies (e.g. Taksar, 2000 and references therein) that maximize the expectation of the discounted future dividends (company’s value) paid by an insurer to its shareholders allowing for reinsurance do not take into consideration frictional costs such as corporate tax and costs of financial distress.

In this article, we study the demand for reinsurance in the presence of corporate tax and costs of financial distress. In order to construct an objective function of an insurer we should first understand the nature and economic aspects of an insurance business. In contrast to industrial companies, insurers do not generally leverage themselves via capital markets. They collect insurance premiums (borrow money) by issuing debt in the form of insurance policies that pay the policyholders (lenders) compensation (financial benefits) if pre-specified events occur. To create and then issue insurance contracts, insurers rely on diversification and financial markets. By pooling contracts that are not perfectly correlated, aggregate losses become more predictable. While pooling reduces uncertainty, unexpected losses may still arise, potentially jeopardizing the insurer’s ability to meet its obligations. On the other hand, unlike bondholders who can effectively reduce their credit risk exposure by holding a well-diversified portfolio of bonds with different issuers, policyholders generally cannot mitigate insurer default risk in any cost-efficient way. Therefore, policyholders usually accumulate their “ credit” exposure with insurers, the financial strength of which is assessed by rating agencies and/or regulators. Insurers satisfy regulatory requirements on solvency/security by holding risk capital in addition to operating capital including a component of premium income.

We define the premium, net of administrative expenses, as consisting of the expected value of claim loss and risk loading (or risk premium). According to Daykin et al. (1996 (pp. 156–157)), Nakada et al. (1999) and SOA Economic Capital Calculation and Allocation Subgroup (2003) the sum of the risk premium and risk capital determines the value of economic capital. Like industrial companies, insurers are financed by their principals (shareholders) (see Brealey and Myers, 2002 and Culp, 2002). Shareholders of insurance companies supply risk capital (“equity capital” or “ surplus”) that is invested on their behalf in financial assets. In so doing, shareholders expect to earn a fair return on invested risk capital. Insurers create shareholders’ value through investment in assets and borrowing in the insurance market, rather than in capital markets. In the presence of frictional costs such as corporate and individual taxes (double taxation)1 it is more costly for insurers to create value from investment in financial markets compared to direct investment funds. However, insurers have a competitive advantage in creating value by borrowing in the insurance market since they can directly manage the moral hazard and adverse selection costs of insurance risks. Self risk-pooling arrangements are costly and insurance contracts provide an efficient means of lowering these costs.

We assume that the main insurer’s objective is to maximize shareholder value under solvency constraints imposed by a Regulatory Authority. An insurer may traditionally improve its solvency level or reduce insolvency risk, which captures both undesirable large fluctuations and ruin probability (see Gerber, 1980 and Hürlimann, 1993), by buying reinsurance to reduce the unexpected fluctuations in the insurance losses of the insurer. In a regulated insurance market when the solvency level, required by a Regulatory Authority, is fixed we assume that the insurer can maintain this level by two control variables: reinsurance and risk capital. Different regulatory requirements determine the extent to which reinsurance can be used to reduce capital requirements for solvency. In practice there are two main alternatives: (1) (model M1) where risk capital is independent of the reinsurance strategy, and is a required minimum value of risk capital determined at the beginning of the period ignoring future reinsurance strategies; (2) (model M2) where the required minimum value of the risk capital is determined taking into account the reinsurance strategy.

While the first possibility allows reinsurance to influence the future solvency it does not allow it to influence the initial capital so that there are two independent control variables, initial capital and reinsurance. The second model effectively has only reinsurance as a control variable, since the required capital is explicitly dependent on the ceded amount of insurance, and is determined by the reinsurance control variable. Under the second model, purchasing reinsurance will normally decrease the required risk capital, and decreasing risk capital will increase the demand for reinsurance. By considering shareholders of the insurance company as residual claimants, it is natural to consider a measure of performance based on return on risk capital defined as the ratio of the expected payoff to shareholders, allowing for limited liability, to the invested risk capital. We assume that an insurer’s objective is to maximize this ratio and hence maximise return to shareholders consistent with modern financial theory. The proposed “return on risk capital” (RRC) is different from the “risk adjusted return on capital” (RAROC, e.g. see Nakada et al., 1999), which is defined as a ratio: (risk premium plus investment return) divided by (economic capital). RAROC is a measure of capital performance that aims to adjust the returns of an insurer (or usually bank) for risk and expresses this in relation to economic capital (risk premium plus risk capital) employed. We explicitly include the limited liability of shareholders, insolvency constraints as well as frictional costs.

In this article, we study the demand for change-loss reinsurance contracts in single period models M1 and M2 in the presence of corporate tax and costs of financial distress and quantify their impact.

Section snippets

Demand for reinsurance in shareholder’s value creation: one-period frictionless model

We consider a single period model. Let X denote the random aggregate claims of an insurer portfolio PX and let X have the distribution function F(x),x0 defined in the probability space (Ω,P). It is assumed that in perfectly competitive insurance and reinsurance markets, insurers are subject to the risk of insolvency, however we assume the risk of reinsurer solvency is negligible and can be ignored. In practice this is not the case although reinsurers who are not highly rated and credit worthy

Demand for reinsurance in shareholder’s value creation: one-period model under the presence of corporate tax

We now consider the problem of the demand for change-loss reinsurance in a single-period model under the presence of corporate tax. Tax is one of the factors that is known to generate a demand for risk management including reinsurance.

The decision to reinsure can be an important means of altering a company’s capital requirements, which in turn gives an opportunity to create (enhance) shareholders value. Garven (1987) suggests that in order for insurer capital-structure decisions (including

Demand for reinsurance in shareholder’s value creation: one-period model under the presence of financial distress costs

In this section we consider the model M1 allowing for costs of financial distress. The main assumption underlying our inclusion of financial distress costs into the model M1 is the distinction between the economic states “financial distress” and “insolvency”. The notion that financial distress and insolvency are different states has been introduced in the finance literature (Jarrow and Purnanandam, 2004 and references therein). We consider three economic states of the insurance company at the

Conclusion

In this article, we have investigated the demand for (change-loss) reinsurance in two single-period models. We assume that the objective of the insurer is to maximise shareholders’ value by maximising their expected return allowing for the financial effect of insolvency. In both models the gross insurer premium is determined using an expected value premium principle that does not reflect the effect of insolvency on policy payoff consistent with standard actuarial premium principles. The models

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This work was partially supported by Australian Research Council Discovery Grant DP0345036.

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