Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 23

Insurers are one period, limited liability entities with no existing liabilities. We consider multi-period insurers in Part IV only. The insurer is called Ins Co. It sells insurance policies to insureds or policyholders. Policyholder and insured are treated as synonyms, and both include claimants. Ins Co.’s portfolio is the collection of policies it writes. The length of the period is usually one year. Its length is relevant only because of the time value of money, since interest is a rate per year.

Policyholder liabilities are any amounts Ins Co. owes to policyholders. The two biggest are loss reserves and unearned premium reserves. We incorporate reserves in Part IV only. In property-casualty insurance, loss reserves cover claims that have been incurred but not paid, whether reported or not. In life insurance, liabilities include policy values for long duration contracts.

Assets are the total financial resources owned by Ins Co. that it can use to meet its policyholder liabilities. A regulator usually stipulates required assets, a minimum amount of assets that Ins Co. must hold; see Section 1.2.

Ins Co., like all firms, finances its assets by issuing liabilities. It sells policies, creating policyholder liabilities, in exchange for premiums, and it raises capital from investors by selling them its residual value (equity) or other promises (debt, reinsurance).

Investors can be shareholders when Ins Co. is a stock company or insureds when it is a mutual company or debt holders or reinsurers.

Insurers are intermediaries between insureds and investors. Intermediary always means an insurance company intermediary, and never an agent or broker.

Portfolio components are referred to as units. A unit can be a single policy or a group of policies or be defined by line, geography, branch, business unit, or other characteristics. Unit can also represent the segmentation between reinsurance ceded losses and retained losses.

Ins Co.’s aggregate loss 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.