Pricing Insurance Risk. Stephen J. Mildenhall

is the sum of losses from its portfolio over one period.

      Ins Co.’s operations are controlled by eight variables: (expected) loss, premium, assets, margin, capital, loss ratio, cost of capital, and leverage. The first five are monetary quantities, and the last three are unitless ratios. They obey five relationships:

      1 premium equals loss plus margin,

      2 assets equal premium plus capital, which we call the asset funding constraint,

      3 loss ratio equals loss divided by premium,

      4 cost of capital equals margin divided by capital, and

      5 asset leverage equals assets divided by premium.

Figure 2.1 lays out these variables and relationships. Monetary quantities are the vertices, the bold diagonal lines correspond to the ratios, and the two shaded triangles signify the asset and premium sum conditions.

      Figure 2.1 The eight variables that control insurance operations and five relationships between them.

      Premium is the amount charged for providing insurance. Premium is net of (i.e., excludes) underwriting expenses but includes an allowance for risk called the margin. Profit, profit load, profit margin, risk margin, and risk load are all synonyms for margin.

      Premium is the critical variable; it is the foundation of the schematic. It is the bridge between investor cash flows on the left and insurance cash flows on the right. At the expected outcome, premium is shared, with margin flowing to investors and expected loss to the insured.

      Policyholders are liable for their expected loss—as Adam Smith pointed out in 1789; by “common loss” he means expected loss. Financing the remaining assets is the shared liability of policyholders and investors. The shared liability equals assets minus expected loss, or equivalently capital plus expected margin. Pricing apportions the shared liability to policyholders and investors.

The loss ratio is the ratio of loss to premium. Because premiums exclude expenses, a 90% loss ratio includes a healthy margin. The premium markup is the inverse expected loss ratio, the ratio of premium to expected loss. Catastrophe bond pricing often quotes markups rather than loss ratios. Premium leverage refers to the ratio of premium to capital.

      The margin is distinct from the contingency provision, which the Actuarial Standards Board (2011) defines as a correction for persistent biases in ratemaking. It says the “contingency provision is not intended to measure the variability of results and, as such, is not expected to be earned as profit.”

      A catastrophe or catastrophe event refers to an single event causing loss to multiple units, such as a hurricane, typhoon, earthquake, winter storm, terrorist attack, or pandemic. A catastrophe loss is the total loss across all units from a catastrophe event. A catastrophe unit means a unit prone to catastrophe losses. A catastrophe risk is a peril likely to result in catastrophe losses. Catastrophe risks tend to attract large margins, making them particularly interesting.

      At various points we mention catastrophe models. These are computer simulation tools used to estimate potential catastrophe losses from an insurance portfolio. Mitchell-Wallace et al. (2017) provides helpful background about the operation and use of catastrophe models.

      Losses in a thick-tailed unit have a high coefficient of variation, are right-skewed and leptokurtic (high kurtosis), and have a significant probability of assuming a very substantial value. Catastrophe losses are usually thick-tailed.

      A long-tailed unit has a slow payout pattern, meaning claims are not settled until many years after they occur.

      Reinsurance is a type of insurance, so we say insurance to cover both, and reinsurance if that is all we mean. Cedents cede business to reinsurers.

      The accounting distinction between capital and equity causes unnecessary confusion.

      Capital refers to funds intended to assure the payment of obligations from insurance contracts, over and above reserves for policyholder liabilities. Capital is also referred to as net assets. The book value of capital depends on accounting conventions. Capital is usually regulated by statute. Surplus is a synonym for capital used in US statutory accounts.

      Equity is the value of the owner’s residual interest. In a stock company, it is called shareholder’s equity. Accounting equity is typically lower than capital since debt can be included in capital but not equity. Equity also has a market value for public stock companies, based on the value of shares outstanding. Equity levels are not regulated. Accounting equity can be negative. The market value of equity is always non-negative because of limited liability.

      2.2 Ins Co.: A One-Period Insurer

      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