Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 24

In this section, we introduce the hypothetical insurer called Ins Co., that we use as the base for our theory and examples. Ins Co. is a limited liability company that intermediates between insureds and investors.

Ins Co.’s customers are insureds who are subject to risks they wish to insure, for the three reasons explained in Section 8.1.1. Insureds who use insurance for risk transfer or financing are sensitive to insurer quality and possible default because it correlates with their own misfortune (Merton and Perold 1993; Froot 2007).

Ins Co. is owned by investors who provide risk bearing capital. The investor group overlaps with the insured group in a mutual insurer. Investors may be risk averse. They often have limited capacity to evaluate insurance risk, giving insurers a competitive advantage in risk assessment and pricing (Froot and O’Connell 2008).

Ins Co. exists for one period. It comes into existence at time t = 0 and lasts for one period. Ins Co. has no initial liabilities. At t = 0 it writes one or more single-period insurance contracts and collects premiums from its insureds. At the same time, it raises capital from investors by selling them all or part of its uncertain t = 1 residual value. Ins Co.’s liabilities can be structured as a combination of equity, debt, or reinsurance.

When Ins Co. writes a policy, it collects premium at t = 0 and earns it over the period. We assume all other transactions occur at the end of the period. Therefore all the premium is earned and available to pay claims at t = 1. There is no need to consider an unearned premium reserve because there are no intermediate cash flows or solvency tests between t = 0 and t = 1.

At time t = 1, Ins Co. pays any claims due and gives any residual value to its investors. If Ins Co.’s assets are insufficient to pay the claims, then it defaults. Investors have limited liability: they lose their original investment but owe nothing more.

The length of the policy period is relevant because it determines the investors’ cost to fund the insurer. At t = 0 the investors must pay-in the capital, i.e., cause cash to be transferred into a separate legal entity. They may incur a time-based funding cost. Since funding costs are expressed per year, it is usual to use a one-year time period.

Premiums cover expected losses and loss adjustment expenses, the cost of capital, and frictional capital costs. All other expenses are outside the model. The epigraph to Section 1 shows that Adam Smith was already aware of the cost of capital for insurers in 1789, and wrote about it in a surprisingly modern manner.

Table 2.1 summarizes the aggregate cash flows between Ins Co., investors, and policyholders. At t = 0 all amounts are fixed. At t = 1 the random loss outcome X is revealed. The model can include stochastic asset returns.

Table 2.1 Investor and insured transactions with Ins Co. at t=0,1. In the last row X∧a′=min(X,a′)

View	Total		Loss		Margin		Capital
At issue, t = 0
Insured	Premium	=	Expected loss cost	+	Margin
Investor	Capital	=					Capital
Ins Co. Total	Assets	=	Expected loss cost	+	Margin	+	Capital
expected	a	=	E[X]	+	M	+	Q
At expiration, t = 1
Ins Co. Total	Assets	=	Loss outcome			+	Residual value
random	a′=a(1+r)	=	X∧a′			+	(a′−X)+

In an unregulated insurance market, insured relative risk aversion interacts with investor opportunity costs of capital to determine the amount of assets supporting the risk and the split of those assets into premium and capital. In a regulated market, the amount of assets is determined, or at least heavily influenced by a regulator (or quasi-regulatory rating agency), but the market still determines the split of funding between premium and equity.