The marginal cost of risk and capital allocation in a property and casualty insurance company

The marginal cost of risk and capital allocation in a property and casualty insurance company

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Speaker(s): Qiheng Guo (Georgia State University), Daniel Bauer (University of Alabama)

Allocating capital to business lines is a key actuarial task in an insurance company, since the risk associated with different lines has to be supported by capital. In a world with financial frictions, it is costly to carry over the internal surplus and raise capital from external sources. So, allocating capital and its associated cost to lines of business is essential for pricing and performance measurement in multiline settings. A large number of papers on the subject in the actuarial and broader insurance literature focus primarily on technical aspects of how to divide risk capital given by an arbitrary risk measure. Such approaches often yield results that are murky in economic objectives such as profit and value maximization. We develop a multi-period profit maximization model for a property and casualty (P&C) insurance company and use it for determining the marginal cost of risk and resulting economic capital allocations. In contrast to previous literature, our model features a loss structure that matches the characteristics of a P&C company, comprising short-tailed and long-tailed business lines. In particular, we take into account loss history and loss development years. As an example application, we implement the model using two P&C insurance business lines and two development years on the long-tailed line, assuming that the losses develop according to a Chain-Ladder model with jointly normal innovations. We use NAIC data for calibration of loss and premium parameters. We solve our model numerically by dynamic programming on a discretized state space. Our numerical results demonstrate how loss history affects the marginal cost and capital allocations. We find that long-tailed lines are employed as short-term sources of financing, an insight that considerably changes the characteristics and optimal policies relative to short-tailed lines.

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