For an insurer whose liabilities and corresponding assets extend over a single time period, Butsic [2013] determined the optimal capital level by maximizing the value of the insurance to the policyholder, while providing a fair return to the insurer’s owners. This paper extends those results to determine optimal capital when liabilities last for several time periods until settlement. Given the optimal capital for one period, the analysis applies backward induction to find optimal capital for successively longer time frames.
A key element in this approach is the stochastic process for loss development; another is the choice of capital funding strategy, which must respond to the evolving loss estimate. In addition to the variables that affect the optimal one-period capital amount (such as the loss volatility, frictional cost of capital and the policyholder risk preferences), in this paper I show that the horizon length, the capitalization interval (time span between potential capital flows), and the policy term will influence the optimal capital for multiple time periods. Institutional and market factors, such as the conservatorship process for insolvent insurers and the cost of raising external capital, also play a major role and are incorporated into the model.
Results show that the optimal capital depends on both the annual and the ultimate loss volatility. Consequently, more total capital (ownership plus policyholder-supplied capital) is required as the time horizon increases; however, optimal ownership capital may decrease as the time horizon lengthens due to the policyholder-supplied capital, which includes premium components for risk margins and income taxes. Also, less capital is needed if capital flows can occur frequently and/or if the policy term is shorter. Insurers that are able to more readily raise capital externally will need to carry less of it.
The model is extended to develop asset risk capital and incorporate features, such as present value and risk margins, that are necessary for practical applications. Although the primary focus is property-casualty insurance, the method can be extended to life and health insurance. In particular, the approach used to determine capital required for multi-period asset risk will apply to these firms.
The resulting optimal capital for insurers can form the basis for pricing, corporate governance and regulatory applications.
Keywords: Backward induction, capital strategy, capitalization interval, certainty-equivalent loss, conservatorship, exponential utility, fair-value accounting, policy term, risk margin, stochastic loss process, technical insolvency, time horizon
