Academic journal article Journal of Risk and Insurance

# Solvency Measurement for Property-Liability Risk-Based Capital Applications

Academic journal article Journal of Risk and Insurance

# Solvency Measurement for Property-Liability Risk-Based Capital Applications

## Article excerpt

To determine the lognormal expected policyholder deficit at the end of one period with no time value (i = 0), we use the fact that the EPD for risky losses is a call option with exercise price A and current "stock price" L. Since the well-known Black-Scholes (1973) option pricing model assumes that the future stock price is subject to geometric Brownian motion with instantaneous variance [[Sigma].sup.2], at time t the price is lognormally distributed with dispersion parameter [Sigma][square root of t]. The option price is

[Mathematical Expression Omitted]

Introduction

The recent failure of several large life insurers, following the disastrous experience of the savings and loan industry, has pushed solvency oversight to the top of the regulatory agenda. In late 1990, the National Association of Insurance Commissioners began a mission to establish risk-based capital formulas for both life and property-liability insurance, as well as model laws to institute the capital requirements.

Formula-driven capital requirements are not new to insurance. For about 40 years, European authorities have used various formulas to set solvency margins (see Byrnes, 1986, or Daykin et al., 1987, for an extensive overview of international approaches to solvency regulation). In the United States, detailed risk-based capital formulas for other financial institutions (banks and savings and loans) have been adopted and are now undergoing a phase-in period.

This article shows how risk can be quantified in establishing risk-based capital for property-liability insurers. The first section discusses the roles of parties to the insurance contract, establishing why the risk-based capital concept is economically useful. The second section defines capital and risk, leading to the expected policyholder deficit as the relevant risk measure for solvency analysis. A balance sheet model relates capital levels to the size of the expected deficit, providing results for the commonly-used normal and lognormal distributions.

The third section introduces the time dimension with a discussion of bias. Market valuation, for both assets and liabilities, is used to remove bias in risk measurement. This section next describes diffusion processes to show how insurance risk is time-dependent and then shows that a proper time horizon for solvency determination is the period between risk-based capital evaluations. The model is completed by taking the present value of the policyholder deficit and showing that this measure is equivalent to a financial option implicitly given by the policyholders.

The fourth section shows that the degree of correlation between risk elements is a critical factor in properly setting capital levels. Using linear approximation, a simple square root rule incorporates the correlation; its application is illustrated with a hypothetical balance sheet. The final section discusses implications and applications of the results.

Economic Basis for Risk-Based Capital

The purpose of insurance is to spread the costs of unforeseen economic loss over a wide base of policyholders. In turn, the main purpose of solvency regulation is to ensure that the promised insurance protection is available to an acceptable degree of certainty.

The solvency of an insurer is intimately linked to the condition of its balance sheet. Capital, the excess of assets over obligations, represents the owners' stake, or equity in the firm. Under statutory accounting principles (SAP), capital is called surplus; under generally accepted accounting principles (GAAP), capital is called equity.

An insolvency occurs when obligations exceed assets, a condition called technical insolvency. Usually at this point regulators will have intervened to place the company in conservatorship or will have severely curtailed its operations (theoretically, however, an insurer could operate beyond the point of technical insolvency if payment of losses and expenses sufficiently lagged cash inflows). …

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