Chapter 10 Premium Foundations

Chapter Preview. Setting prices for insurance products, i.e., premiums, is an important task for actuaries and other data analysts. This chapter introduces the foundations for pricing non-life products.

The presentation of this chapter follows the premium equation.

  • In Section 10.2, we first present the sources of information that support premium development.
  • We discuss this development of the pure premiums in Section 10.5.
  • In Section 10.6, we discuss fixed and variable non-claim expenses.
  • In Section 10.7.3, we discuss the provision for profit.
  • Section 10.9 summarizes alternative premium principles that incorporate uncertainty into our pricing.

10.1 Introduction to Ratemaking

In this section, you will learn how to:

  • Describe relationship between between exposures, rates, and premiums
  • Describe the components of the rate

This chapter explains how you can determine the appropriate price for an insurance product. As described in Section 1.2, one of the core actuarial functions is ratemakingProcess used by insurers to calculate insurance rates, which drive insurance premiums, where the analyst seeks to determine the right price for a risk.

A priceA quantity, usually of money, that is exchanged for a good or service is the consideration exchanged for a good or service. In insurance, we refer to this consideration as the premiumAmount of money an insurer charges to provide the coverage described in the policy and the service provided by the insurer is protection against contingent events.

The amount of protection will vary by risk being insured. For example, in homeowners insurance, the amount of insurance protection depends on the home value. In life insurance, the amount of protection may depend on a policyholder’s financial status (e.g., income and wealth) and their perceived need for financial security. So, it is common to express insurance prices as a unit of the protection being purchased, for example, a price per thousand dollars of coverage on a home or benefit in the event of death. We refer to the unit of protection as the exposureA measure of the rating units for which rates are applied to determine the premium. for example, exposures may be measured on a per unit basis (e.g. a family with auto insurance under one contract may have an exposure of 2 cars) or per $1,000 of value (e.g. homeowners insurance).. These prices/premiums are known as ratesA rate is the price, or premium, charged per unit of exposure. a rate is a premium expressed in standardized units. because they are expressed in standardized units.

Unlike other products, the costs of insurance protection are not known at the sale of the contract. If the insured contingent eventA condition that results in an insurance claim, such as an automobile accident of the loss of life, does not occur, then the contract costs are only administrative (e.g., to set up the contract) and are relatively minor. If an insured event occurs, then the cost includes not only administrative costs but also claim payment(s) and expenses to settle claims. So, the cost is random when the contract is written, and protection from that randomness is the basis of insurance.

Because costs are unknown at the time of sale, insurance pricing differs from common economic approaches. This chapter introduces traditional actuarial approaches to determine prices as a function of insurance costs. Insurance involves a promise of the insurer to pay a claim when presented by the insured. For this reason, insurance is a regulated business, particularly for personal lines insurance. The role of the regulator is to ensure that the insurer is able to satisfy its promise to its policyholders. In executing this mandate, the regulator often requires the insurer to file support for its rates. The regulator will review that filing to determine whether those rates are reasonable, not excessive, not inadequate, and not unfairly discriminatory.

The actuarial pricing approach we present is sufficient for some insurance markets, such as personal automobile or homeowners, where the insurer has a portfolio of many similar independent risks. However, there are other insurance markets where actuarial prices only provide an input to general market prices. To reinforce this distinction, actuarial cost-based premiums are sometimes known as technical prices.

To develop technical prices, it is helpful to think of a premium as revenue source that provides for

  • Pure Premium - Claim payments are amounts due to the insured under the terms of the insurance contract. Pure premiums include claim payments costs to administer and investigate such claims.
  • Insurer expenses - Non-claim Expenses include insurer costs that vary by premium (such as sales commissions), and those that do not (such as building costs and employee salaries). We include those costs through the Fixed Expenses and Variable Expense Rate of the (10.2).
  • Profit - An insurer requires capital to support operations. The capital provider will reasonably expect to earn a profit from insuring risk. Insurers have two sources of profit: underwriting income and investment income.

We formalize this relationship in our simplified premium equation. \[\begin{equation} \small{ \text{Premium} = \dfrac{\text{Pure Premiums} + \text{Fixed Expenses} }{1 - \text{Variable Expense Rate} - \text{Profit}} } \tag{10.1} \end{equation}\] where \[ \small{ \text{Pure Premiums} = \dfrac{\text{Estimated Claims and Claims Adjustment Expense}}{\text{Exposures}} } \] or \[ \small{ \text{Pure Premiums} = \dfrac{\text{Estimated Claim Counts}}{\text{Exposures}} \times \dfrac{\text{Estimated Claims and Claims Adjustment Expense}}{\text{Estimated Claim Counts}}\\ } \] This simplified premium equation promotes a general understanding of the Relationship of insurance costs and revenue. We refer to this equation as the simplified premium equation because (i) it does not include explicit consideration for investment income, and (ii) we combine consideration of claims and claims adjustment expenses. We will refine the simplified premium equation to consider these items later in this chapter.

We observe that the pure premium in equation (10.1) is a ratio of claims and exposures. We discuss the development of the claims provision in Section 10.3 and the development of exposures in Section 10.4.

10.2 Data Sources

In this section, you will learn how to:

  • Describe the types of data used to develop rates

Insurers consider aggregate information for ratemaking such as exposures, premiums, expenses, claims, and payments. This aggregate information is also useful for managing an insurer’s activities. The information is typically summarized in financial reports which are commonly compiled at least annually and often quarterly. At any given financial reporting date, information about recent policies and claims will be ongoing and necessarily incomplete; this section introduces concepts for projecting risk information so that it is useful for ratemaking purposes.

Insurers generally store information about insured risks, such as exposures, premiums, claim counts, losses, and rating factors, in a relational database that will include:

  • policy database - contains information about the risk being insured, the policyholder, and the contract provisions
  • claims database - contains information about each claim. The claims database is linked to the policy database.
  • payment database - contains information on each claims transaction, typically payments but may also include changes to case reserves. The payment database is linked to the claims database.

Insurers will aggregate the information in these detailed databases to develop the information needed for financial reports. As described in this chapter, insurers’ actuaries will also use this information to develop the premiums.

10.3 Claims

In this section, you will learn how to:

  • Describe the basis for the provision for claims, the numerator of the pure premium
  • Adjust claims to the level of the prospective period

The terms lossThe amount of damages sustained by an individual or corporation, typically as the result of an insurable event. and claimThe amount paid to an individual or corporation for the recovery, under a policy of insurance, for loss that comes within that policy. refer to the amount of compensation paid or payable to the claimant under the terms of the insurance policy. Definitions can vary:

  • Sometimes, claim is used interchangeably with the term loss.
  • In some insurance and actuarial sources, loss is the amount of damage sustained in an insured event. The claim is the amount paid by the insurer. Differences between loss and claim amounts are typically due to the coverage terms such as deductibles and policy limits.
  • In economics, a claim is a demand for payment by an insured or by an injured third party under the terms and conditions of the insurance contract, and the loss is the amount paid by the insurer.

This text will follow the convention of the second bullet.

Also, there are two categories of claim adjustment expenses.

  • Allocated claim adjustment expenses are attributed to a specific claim and are generally comprised of investigation and legal expenses to defend or settle the claim.

    Claims and allocated claim adjustment expenses are sometimes inversely correlated, as additional defense expenses may result in lower claim payments. In this section, references to claims also include allocated claims adjustment expenses.

  • Unallocated claim adjustment expenses cannot be assigned to individual claims (e.g., claim adjuster salaries.) Actuaries often review claims and allocated claim adjustment expenses in aggregate, as the latter is often a function of the former.

    Insurers will include a provision for unallocated claims adjustment expenses either as a percentage of claims and allocated claims expenses or premiums, or a combination thereof. We discuss unallocated claims adjustment expenses separately in Section 10.3.2.1.

10.3.1 Estimated Ultimate Claims

Recall that a claim is the amount paid or payable to claimants under the terms of insurance policies. In more detail, one can consider paid claims, those losses for a particular period that have actually been paid to claimants. When there is an expectation that payment will be made in the future, a claim will have an associated case reserve representing the estimated amount of that payment. Case adjusters establish case reserves separately for each open claim based on the information available. In addition, reported claims, also known as case incurred claims or incurred claims, are the sum of paid claims and case reserves. The ultimate claim is the amount required to close and settle all claims for a defined group of policies. We describe the estimation of ultimate claims and claims adjustment expenses in Section 14.

Alternatively, we can estimate projected claims and claims expenses as the product of projected claim frequency and claims severity:

\[ \text{Claims and Claims Expenses}_{i}^{(t)} = E[X] \times E[N] \]

where:

  • \(E[X]\) is the projected average ultimate severity per claim, and
  • \(E[N]\) is the projected ultimate number of claims.

We note the frequency-severity alternatives to support our discussion of trends in the following section.

10.3.2 Adjustments to Claims and Allocated Claims Adjustment Expenses

In this section, we review adjustments to experience period ultimate claims that are required to support the development of prospective rates. These adjustments include trending, large loss adjustments and provisions for catastrophes. Finally, we discuss approaches to incorporate unallocated claims adjustment expense.

Large Loss and Catastrophe Provisions

Consider, for example, if a five-year experience period included a one-in-20-year event. If we did not adjust the data, we would effectively overestimate claim amounts for that category of claims.

In ratemaking, we remove these unusual large losses and catastrophe losses from the experience period data, and then add a provision consistent with the longer-term average cost of large losses or catastrophes. Although large loss adjustments are commonly based on the insurer’s claims experience, the provision for catastrophes is often based on models developed by specialists. Adjustments for catastrophes are more common in property insurance, while adjustments for large losses are more common in liability insurance.

10.3.2.1 Unallocated Claims Adjustment Expenses

Some insurers include unallocated claims adjustment expenses as a percentage of claims and allocated claims adjustment expenses, while other insurers include unallocated claims adjustment expenses as a percentage of premiums. In our discussion, we use the former approach. For insurers that use the latter approach, the inclusion of a provision for unallocated claims adjustment expenses would follow that described below for other non-claim expenses. Generally, insurers estimate \(UE\) by reviewing historical ratios of those payments to claims and allocated claims adjustment expense payments.

10.4 Exposures

In this section, you will learn how to:

  • Describe the consideration exposures in the developing pure premiums
  • Select an exposure base
  • Adjust historical exposures to the level of the prospective period

The denominator of the pure premium equation is “exposureA type of rating variable that is so important that premiums and losses are often quoted on a ”per exposure” basis. that is, premiums and losses are commonly standardized by exposure variables..” We use exposures to standardize heterogeneous risks. To explain exposures, we can consider scale distributions that we learned about in Chapter 4. To recall a scale distributionSuppose that y = c x, where x comes from a parametric distribution family and c is a positive constant. the distribution is said to be a scale distribution if (i) the distributions of y and x come from the same family and (ii) only a single parameter differs and that by a factor of c., suppose that \(X\) has a parametric distributionModel assumption that the sample data comes from a population that can be modeled by a probability distribution with a fixed set of parameters and define a rescaled version \(R = X/E\), \(E > 0\). If \(R\) is in the same parametric family as \(X\), then the distribution is said to be a scale distribution. As we have seen, the gamma, exponential, and Pareto distributions are examples of scale distributions.

Intuitively, the idea behind exposures is to make risks more comparable to one another. For example, it may be that risks \(X_1, \ldots, X_n\) are from different distributions and yet, with the choice of the right exposures, the rates \(R_1, \ldots, R_n\) are from the same distribution. Here, we interpret the rate \(R_i = C_i/E_i\) as the loss divided by exposure.

Table 10.5.1 provides a few examples.

Table 10.5.1. Commonly used Exposures in Different Types of Insurance

\[ \small{ \begin{matrix} \begin{array}{ll} \textit{Type of Insurance} & \textit{Exposure Basis} \\\hline \text{Personal Automobile} & \text{Earned Car Year, Amount of Insurance Coverage} \\ \text{Homeowners} & \text{Earned House Year, Amount of Insurance Coverage}\\ \text{Workers Compensation} & \text{Payroll}\\ \text{Commercial General Liability} & \text{Sales Revenue, Payroll, Square Footage, Number of Units}\\ \text{Commercial Business Property} & \text{Amount of Insurance Coverage}\\ \text{Physician's Professional Liability} & \text{Number of Physician Years}\\ \text{Professional Liability} & \text{Number of Professionals (e.g., Lawyers or Accountants)}\\ \text{Personal Articles Floater} & \text{Value of Item} \\ \hline \end{array} \end{matrix} } \]

10.4.1 Criteria for Choosing an Exposure

An exposure base should meet the following criteria. It should:

  • be an accurate measure of the quantitative exposure to loss
  • be easy for the insurer to determine (at the time the policy is initiated) and not subject to manipulation by the insured,
  • be easy to understand by the insured and easy to calculate by the insurer,
  • consider any preexisting exposure base established within the industry.

To illustrate, consider personal automobile coverage. Instead of the exposure basis “earned car year,” a more accurate measure of the quantitative exposure to loss might be number of miles driven. Historically, this measure had been difficult to determine at the time the policy is issued and subject to potential manipulation by the insured, so it was not typically used. Modern telematic devices that allow for accurate mileage recording support the use of this exposure base in some marketplaces.

As another example, the exposure measure in commercial business propertyLine of business that insures against damage to their buildings and contents due to a covered cause of loss, e.g., fire insurance, is typically the amount of insurance coverage. As property values grow with inflation, so will the amount of insurance coverage. Thus, rates quoted on a per amount of insurance coverage are less sensitive to inflation.

10.4.2 Written and Earned Exposures

In developing premiums and rates, it’s important that we use claims information and exposure information that is comparable. Most ratemaking uses an accident year approach. In this approach, we relate claims incurred during a specified period to the premium or exposure “earned” during that same period without consideration of the period in which the underlying policy was written. For example, a 12-month policy issued on 1 July 2019 insures claims events in 2019 or 2020, and the claims are assigned to the year of the event. Generally, we earn premiums and exposures on a pro-rata as to time basis as presented in Table 10.5.2, which displays illustrative calculations for a portfolio of four illustrative policies.

Table 10.5.2. Exposures for Four 12-Month Policies

\[ \small{ \begin{matrix} \begin{array}{cl|cc|cc|cc|c} \hline & & & & & & &&\text{In-Force} \\ &\text{Effective} & \text{Written}& \text{Exposure} & \text{Earned} &\text{Exposure}& \text{Unearned} &\text{Exposure}& \text{Exposure} \\ {Policy} &\text{Date} & 2019 & 2020 & 2019 & 2020 & 1/1/2019 & 1/1/2020 & 1/1/2020 \\ \hline \text{A}&\text{1 Jan 2019} & 1.00 & 0.00 & 1.00 & 0.00& 0.00 & 0.00 & 0.00 \\ \text{B}&\text{1 April 2019} & 1.00 & 0.00 & 0.75 & 0.25 & 0.25 & 0.00& 1.00 \\ \text{C}&\text{1 July 2019} & 1.00 & 0.00 & 0.50 & 0.50 & 0.50 & 0.00& 1.00 \\ \text{D}&\text{1 Oct 2019} & 1.00 & 0.00 & 0.25 & 0.75 & 0.75 & 0.00& 1.00 \\ \hline & Total & 4.00 & 0.00 & 2.50 & 1.50 & 1.50 & 0.00 & 3.00 \\ \hline \hline \end{array} \end{matrix} } \]

10.4.3 Adjustments to Exposures

Exposure Trend

Sometimes exposure units are inflation sensitive. For example, payroll is a common exposure base for workers compensation coverage. Even if the insured firm does not grow, it’s payroll may increase due to wage inflation. We refer to the adjustment applied to inflation sensitive exposures as exposure trend.

10.5 Pure Premiums

In this section, you will learn how to:

  • Calculate the expected pure premium

The pure premium in equation (10.1) is a random variable, and so, as a baseline, we use the expected costs to determine rates. To develop our initial understanding, we will consider the insurer that enters into many contracts with risks that are similar except, by pure chance, in some cases, there are losses on some contracts but not on others. The insurer is obligated to pay the total amount of claim payments for all contracts. If the risks are similar, then all policyholders are equally likely to contribute to the total loss. From probability theory, specifically the law of large numbers, we know that the average of iidIndependent and identically distributed risks is close to the expected amount, so we use the expectation as a baseline pricing principle.

In this chapter, we present the development of average premium levels for a portfolio of homogeneous risks. In Chapter 11, we present approaches to develop classification plans which adjust those average premiums to recognize various risk characteristics. In Chapter 15, we present approaches to develop premiums that consider the claim experience of an individual insured.

10.5.1 Experience Period

To develop expected pure premiums, actuaries will typically review claims and exposure experience over a multi-year (typically three to seven years) period. The use of a multi-year period smooths the year-to-year randomness. We refer to this multi-year period as the experience period.

10.5.2 Expected Pure Premium

The expected pure premium is generally calculated as the weighted average of the observations in the experience period. The weights balance responsiveness to more recent experience and the stability of a longer-term average.

\[ \text{Pure Premium}^{(t)} = \text{Exposure}_t \times \sum_{i=1}^n w_i \dfrac{\text{Ultimate Claims and Claims Expenses}_{i}^{(t)}}{\text{Exposure}_i^{(t)}} \]

where:

  • \(w_i\) is the weight for year \(i\) in an \(n\) year experience period.

The superscript (t) indicates that the ultimate claim estimate for accident year \(i\) is adjusted to the level of the prospective program period \(t\). We discussed these adjustments in Section 10.3.2. The following equation demonstrates this process of adjustment. \[ \begin{array}{ll} \text{Claims and Claims Expenses}_{i}^{(t)} &= C_i^{\text{x}LL, \text{x}Cat} \times T_{i}^{(t)} \times LL \times CP \times UE \\ \text{Exposure}_{i}^{(t)}&= \text{Exposure}_i\times E_{i}^{(t)} \end{array} \] where:

  • \(C_i^{\text{x}LL, \text{x}Cat}\) is the estimated ultimate claims for year \(i\), excluding large losses and catastrophes
  • \(T_i\) is a claim trend factor to adjust year \(i\) experience to the cost level of year \(t\)
  • \(E^{(t)}_i\) is an exposure trend factor to adjust year \(i\) experience to the cost level of year \(t\)
  • \(LL\) is a large loss factor
  • \(CP\) is a catastrophe provision
  • \(UE\) is the unallocated claims adjustment expense factor

We discussed these adjustments earlier in this chapter.

10.6 Non-Claim Expenses

In this section, you will learn how to:

  • Describe the consideration of operational expenses in the development of premiums

Non-claim insurer Operating expense costs include commissions, premium taxes, and other expenses such as salaries, rent, and inspections.

  • Some expenses (such as commissions and premium taxes) vary with premiums are “variable” or “premium variable expenses.”
  • Other expenses (such as general administrative and head office costs) are not proportional to the premium.

For non-claim expenses, insurers will typically rely on either historical expense ratios, budgeted amounts, or financial forecasts.

We include fixed expenses in our premium equation on a per-exposure basis and we include variable expenses as a rate per unit premium.

10.7 Investment Income

In this section, you will learn how to:

  • Describe the consideration of the timing of cash flows in the development of the rate
  • Calculate a required provision for underwriting profit

A portion of the required profit is earned from investment income from two sources: policyholder cash flows and investment of the insurer’s surplus. To the extent that investment income is insufficient to provide the required rate of return, the premiums will also need to include an underwriting profitProfit an insurer derives from providing coverage, excluding investment income provision.

As we described, we presented a simplified premium equation in Section 10.5.2 to promote the understanding of the claims and expense provisions in Section 10.3 and exposures in Section 10.6. We now refine the equation to consider investment income. We now consider the other source of an insurer’s profit, investment income.

10.7.1 Investment Income on Policyholder Cash Flows

We first consider policyholder cash flows, i.e., premiums, claims, claim adjustment expenses, and non-claim expenses. We consider investment income on policyholder cash flows by discounting each of the cash flows of each of these components of the premium equation.

  • There may be a delay in the insurer’s receipt of premium, perhaps because the insurer offers payment plans to the insured.
  • Claims and claims adjustment expenses are paid over a period that typically extends beyond the policy term. Generally, property coverages have the shortest payment stream, with all claims being settled and paid over a period that extends between 2 and 5 years, depending on the complexity of the determination of damages. Litigated liability coverages will have intermediate payment streams that range from three to 10 years. Finally, coverages such as workers compensation, offer lifetime benefits that can extend forty years or longer.
  • Non-claim expenses are generally paid over the term of the policy period.

We can rewrite our premium equation to capture the discounting. We replace the unity in the denominator with a premium delay factor and we discount the claims and claims adjustments expenses (in the pure premium) and non-claim expenses. \[ \small{ \text{Premium} = \dfrac{\text{Discounted Pure Premiums} + \text{Discounted Fixed Expenses}} {\text{Premium Delay} - \text{Variable Expense Rate} - \text{Profit}} .\\ } \] We recognize that the discounting effect on the numerator is significantly greater than the effect on the denominator. As a result, consideration of investment income on policyholder cash flows serves to reduce the premium.

The consideration of profit serves in the denominator serves to increase the required premium. We now turn to the determination of that profit provision.

10.7.2 Investment Income on Surplus

The insurer’s surplus is also comprised of invested assets which provide a rate of return. In ratemaking we assume that the investment income on surplus is earned over the policy term, generally 12 months. Investment income of surplus will reduce the required profit provision. For example, if the insurer were able to earn a rate of return on assets of 5% per annum, then the insurer would realize a return of 2.5% of premiums assuming a 2:1 premium: surplus ratio.

10.7.3 The Underwriting Profit Provisions

An insurer requires capital to support operations. The insured pays a premium for the promise of the insurer to pay a claim in the future. Capital serves as protection for the policyholder in the event that premiums are insufficient to pay claims. The capital provider will reasonably expect to earn a profit to insure the risk and subject its capital to loss. Generally, the required profit is expressed as after-tax return on equity.

If the profit provision in the premium equation were 0, then the premium would equal the present value of the present value of cash flows of the insurance policy. However, as we discussed the insurer will require a return on its capital. Generally, coverages that are riskier, i.e., have more variability, will require more supporting capital. Every claim submitted to the insurer has access to all of the capital of the insurer. In insurance, capital is often referred to as surplus. For ratemaking, we notionally allocate capital to coverage using premium to surplus ratios and we state the required rate of return on an after-tax basis. We have to convert that return to a “percent of premium basis” to include in our premium equation. For example, if we assume a 2:1 premium:surplus ratio, a required after-tax rate of return of 12% and a tax rate of 30%, then the profit provision in the premium would be: \[ \begin{array}{ll} \dfrac{12\%~\text{after tax return}}{\text{surplus}} &\times \dfrac{1 \times \text{surplus}}{2 \times \text{premium}} \times \dfrac{1~\text{pre-tax}}{0.7~\text{after tax}} \\~\\ &= \dfrac{8.6\%~\text{pre-tax return}}{\text{premium}} . \end{array} \] We can then reduce the required underwriting profit to consider investment income on surplus. Using the example of Section 10.7.2, the resulting required underwriting profit provision would reduce from 8.6% to 6.1%.

10.8 The Premium Equation

In this section, you learn how to:

  • Calculate the rate for a class of risk
  • Calculate premiums

We can now remove the simplifying assumptions included in Equation (10.1) and provide our final premium equation. The term “pure premium” can be used to refer to rate per exposure unit of provision for claims costs included in the premium for an insured (which may have a quantum of exposure more or less than one exposure unit). In this section, we use the latter definition. \[\begin{equation} \small{ \text{Premium} = \dfrac{\text{Discounted Pure Premium} + \text{Discounted Fixed Expenses}} {\text{Premium Delay} - \text{Variable Expense Rate} - \text{Required Underwriting Profit}} .\\ \\ }\tag{10.2} \end{equation}\]

10.9 Pricing Principles

In this section, you learn how to:

  • Describe common actuarial pricing principles
  • Describe properties of pricing principles
  • Choose a pricing principle based on a desired property

Approaches to pricing vary by the type of contract. For example, personal automobile is a widely available product throughout the world and is known as part of the retail general insurance market in the United Kingdom. Here, one can expect to do pricing based on a large pool of independent contracts, a situation in which expectations of losses provide an excellent starting point. In contrast, an actuary may wish to price an insurance contract issued to a large employer that covers complex health benefits for thousands of employees. In this example, knowledge of the entire distribution of potential losses, not just the expected value, is critical for starting the pricing negotiations. To cover a range of potential applications, this section describes general premium principles and their properties that one can use to decide whether or not a specific principle is applicable in a given situation.

10.9.1 Premium Principles

The prior sections of this chapter introduce traditional actuarial pricing principles that provide a price based only target rates of return and the cost to insure the risk; the price does not depend on the demand for insurance.

Assume that the loss \(X\) has distribution function \(F(\cdot)\) and that there exists some rule (which in mathematics is known as a functional), say \(H\), that takes \(F(\cdot)\) into the positive real line, denoted as \(P = H(F)\). For notation purposes, it is often convenient to substitute the random variable \(X\) for its distribution function and write \(P = H(X)\). Table 10.8.1 provides several examples.

Table 10.8.1. Common Premium Principles

\[ \small{ \begin{array}{ll} \textit{Description } & \textit{Definition } (H(X)) \\\hline \text{Net (pure) premium} & {\rm E}[X] \\ \text{Expected value} & (1+\alpha){\rm E}[X]\\ \text{Standard deviation} & {\rm E}[X]+\alpha ~SD(X)\\ \text{Variance} & {\rm E}[X]+\alpha ~{\rm Var}(X)\\ \text{Zero utility} & \text{solution of }u(w) = {\rm E} [u(w + P - X)]\\ \text{Exponential} & \frac{1}{\alpha} \log {\rm E} [e^{\alpha X}]\\ ~\\ \hline \end{array} } \]

A premium principle is similar to a risk measureA measure that summarizes the riskiness, or uncertainty, of a distribution that is introduced in Section 13.3. Mathematically, both are rules that map the loss rvRandom variable of interest to a numerical value. From a practical viewpoint, a premium principle provides a guide as to how much an insurer will charge for accepting a risk \(X\). In contrast, a risk measure quantifies the level of uncertainty, or riskiness, that an insurer can use to decide on a capital level to be assured of remaining solvent.

The net, or pure, premium essentially assumes no uncertainty. The expected value, standard deviation, and variance principles each add an explicit loading for uncertainty through the risk parameter \(\alpha \ge 0\). For the principle of zero utility, we think of an insurer with utility function \(u(\cdot)\) and wealth w as being indifferent to accepting and not accepting risk \(X\). In this case, \(P\) is known as an indifference price or, in economics, a reservation price. With exponential utility, the principle of zero utility reduces to the exponential premium principle, that is, assuming \(u(x) = (1-e^{-\alpha x})/\alpha\).

For small values of the risk parameters, the variance principle is approximately equal to exponential premium principle, as illustrated in the following special case.

Special Case: Gamma Distribution. Consider a loss that is gamma distributed with parameters \(\eta\) and \(\theta\) (we usually use \(\alpha\) for the location parameter but, to distinguish it from the risk parameter, for this example we call it \(\eta\)). From the Appendix Chapter 20, the mean is \(\eta~ \theta\) and the variance is \(\eta ~\theta^2\). Using \(\alpha_{Var}\) for the risk parameter, the variance premium is \(H_{Var}(X) = \eta~ \theta+\alpha_{Var} ~(\eta ~\theta^2)\). From this appendix, it is straightforward to derive the well-known moment generating function, \(M(t) = {\rm E} [e^{tX}] = (1-t\theta)^{-\eta}\). With this and a risk parameter \(\alpha_{Exp}\), we may express the exponential premium as \[ H_{Exp}(X) = \frac{-\eta}{\alpha_{Exp}} \log\left(1-\alpha_{Exp} \theta\right). \] To see the relationship between \(H_{Exp}(X)\) and \(H_{Var}(X)\), we choose \(\alpha_{Exp} = 2 \alpha_{Var}\). With an approximation from calculus (\(\log(1-x) = -x - x^2/2 - x^3/3 - \cdots\)), we write \[ \begin{array}{ll} H_{Exp}(X) &= \frac{-\eta}{\alpha_{Exp}} \log\left(1-\alpha_{Exp} ~\theta\right) = \frac{-\eta}{\alpha_{Exp}} \left\{ -\alpha_{Exp} ~\theta -(\alpha_{Exp} ~\theta)^2/2 - \cdots\right\} \\ & \approx \eta~ \theta + \frac{\alpha_{Exp}}{2}(\eta ~\theta^2 ) = H_{Var}(X). \end{array} \]

10.9.2 Properties of Premium Principles

Properties of premium principles help guide the selection of a premium principle in applications. Table 10.8.2 provides examples of properties of premium principles.

Table 10.8.2. Common Properties of Premium Principles

\[ \small{ \begin{array}{ll} \textit{Description } & \textit{Definition }\\\hline \text{Nonnegative loading} & H(X) \ge {\rm E}[X] \\ \text{Additivity} & H(X_1+X_2) = H(X_1) + H(X_2), \text{ for independent }X_1, X_2 \\ \text{Scale invariance} & H(cX) = c H(X), \text{ for }c \ge 0 \\ \text{Consistency} & H(c+X) = c + H(X)\\ \text{No rip-off } & H(X) \le \max \{X\}\\ \hline \end{array} } \] This is simply a subset of the many properties quoted in the actuarial literature. For example, the review paper of Young (2014) lists 15 properties. See also the properties described as coherent axioms that we introduce for risk measures in Section 13.3.

Some of the properties listed in Table 10.8.2 are mild in the sense that they will nearly always be satisfied. For example, the no rip-off property indicates that the premium charge will be smaller than the largest or “maximal” value of the loss \(X\) (here, we use the notation \(\max \{X\}\) for this maximal value which is defined as an “essential supremum” in mathematics). Other properties may not be so mild. For example, for a portfolio of independent risks, the actuary may want the additivity property to hold. It is easy to see that this property holds for the expected value, variance, and exponential premium principles but not for the standard deviation principle. Another example is the consistency property that does not hold for the expected value principle when the risk loading parameter \(\alpha\) is positive.

The scale invariance principle is known as homogeneity of degree one in economics. For example, it allows us to work in different currencies (e.g., from dollars to Euros) as well as a host of other applications. Although a generally accepted principle, we note that this principle does not hold for a large value of \(X\) that may border on a surplus constraint of an insurer; if an insurer has a large probability of becoming insolvent, then that insurer may not wish to use linear pricing. It is easy to check that this principle holds for the expected value and standard deviation principles, although not for the variance and exponential principles.

10.10 Reviewing Rate Adequacy

After establishing the initial premiums, insurance company actuaries will perform rate reviews to measure the current adequacy of those rates. For many regulated coverages (typically, personal lines insurance), actuaries file those rate reviews with the insurance regulator. Actuaries review rates regularly as rate levels require updates to keep pace with inflationary pressures. At times, the required rate will have a decreasing trend; for example with improvements in vehicle safety technology or workplace safety. Of course, the primary purpose of the rate is to test whether the experience of the rate program is consistent with loss and expense assumptions underlying the current rates.

10.10.1 The Loss Ratio Method

The “loss ratio method” is a common approach to assess rate adequacy. The loss ratioThe sum of losses divided by the premium. is the ratio of loss to the premium.

\[ \small{ \text{Loss Ratio} = \frac{\text{Loss}}{\text{Premium}} . } \]

When determining premiums, it is a bit counter-intuitive to emphasize this ratio because the premium component is built into the denominator. As we will see, the loss ratio method develops rate changes rather than rates; we can use rate changes to adjust the current rate to the current costs levels.

We calculate rate changes by comparing the loss ratio of the experience period to the target loss ratio. This adjustment factor is then applied to current rates to determine new indicated rates.

10.10.2 Target Loss Ratio

Let us return to equation (10.2). Noting that the “pure premium” is the provision for loss in the rates, we can start with \[ \small{ \begin{array}{ll} \text{Premium} &= \dfrac{\text{Discounted Losses} + \text{Discounted Fixed Expenses} }{\text{Premium Delay} - \text{Variable Expense Rate} - \text{Profit}} \end{array} } \] With this, we have \[ \small{ \begin{array}{ll} \text{Premium Delay} &\text{- Variable Expense Rate - Profit} \\ & = \dfrac{\text{Discounted Losses}}{\text{Premium}} + \dfrac{\text{Discounted Fixed Expenses}}{\text{Premium}}\\ \text{Premium Delay} &\text{ - Variable Expense Rate - Profit} - \dfrac{\text{Discounted Fixed Expenses}}{\text{Premium}} \\ &= \dfrac{\text{Discounted Losses}}{\text{Premium}}\\ &= \text{Target Discounted Loss Ratio} .\\ \end{array} } \] For simplification, we will not repeat that the components of the rate change factor are discounted. In the loss ratio method, we compare the projected loss ratio to the target loss ratio. A projected loss ratio that exceeds the target loss ratio implies the need for a rate increase. A projected loss ratio that is less than the target loss ratio implies the need for a rate decrease.

10.10.3 Experience Period Loss Ratios

Earlier in this section, we described the required adjustments to estimate premiums. We apply those same adjustments to the experience period loss ratios.

10.10.4 Adjustments to Loss

  • As with the development of pure premiums described above, actuaries will typically review claims experience over a multi-year (typically three to seven years) period to smooth the year-to-year randomness. The years in the experience period are similarly weighted to balance responsiveness to more recent experience and the stability of a longer-term period.

  • The numerator of the loss ratio will be ultimate losses.

  • We will consider the presence of catastrophe and large losses in the claims experience.

  • We need to adjust the experience period losses to the cost level of the proposed rate program. We discussed this trend adjustment in Section 10.3.2. We apply the trend factor from the average accident date of the experience period to the average accident date of the proposed rate program. For example, if we are estimating rates that will underlie twelve month policies written in calendar year 2025, the average accident date of the prospective rate program will be 31 December 2025 (sometimes rounded to 1 January 2026). The first policy of the prospective period will be written on 1 January 2025 and expire on 31 December 2025. Assuming even distribution of claim events during the policy, the average accident date (midpoint) of that policy is 1 July 2025. Correspondingly, the last policy of the prospective period will be written on 31 December 2025 and expire on 31 December 2026 with an average accident date (midpoint) of that policy is 1 July 2026. Therefore, the midpoint of all policies written under the proposed rate program is 31 December 2025. To adjust experience for accident year 2022, we apply 3.5 years of trend. The average accident date of accident year 2022 is 1 July 2022 - so 3.5 years is the distance in time to the average accident date of the proposed rate program.

10.10.5 Premium On-Level Adjustment

We also need to adjust premiums for the effect of rate changes. We refer to this adjustment as “on-leveling.” There are two common approaches to on-leveling.

  • The Parallelogram Method: Premium on-level factors use historical rate change calculations. For example, if the company adopted a +10% rate change on 1 July 2022, then the 2022 earned premium would need to be adjusted by +7.5%. - Policies written prior to 1 July 2022 would need to be adjusted by +10%; - For the premium earned after 1 July 2022, half would be earned on policies written under the old rate levels and require the 10% adjustment and half would be written on policies written under the higher rate levels and require no adjustment. The weighted average of these adjustments if +7.5%.
  • Extension of Exposures: The extension of exposures method is a more detailed approach which involves the re-rating of all historical policies at current rates. It is more precise as the parallelogram method relies on rate changes that were calculated as the average rate change given the mix of business at that time. However, the mix of business may change and the rate change effect on the current mix may be different. The extension of exposures does not rely on those average rate changes and instead relies only on current rates.

10.10.6 Premium Trend

Experience period premiums must also be adjusted for for premium trend, and the basis of premium must match the loss trend. For example, insureds may purchase higher limits of coverage to protect against higher inflation. These higher limits would be reflected in the internal claims experience and may underlie the data used to measure loss trend. If we are considering these changes in the loss trend, then we also need to consider the effect of higher limit purchases in premium trend.

10.10.7 Credibility

Oftentimes, the experience being reviewed is not “fully credible.” That is, the predictive value of the data is limited. We, therefore, need to consider an alternative indication of the projected loss ratio to calculate a credibility-weighted loss ratio. We refer to this alternative indicator as the complement of credibility. A common complement is the net loss trend (loss trend/premium trend). The assumption underlying the use of net loss trend as a complement is that in the absence of an alternative indication, we would need to adjust the rate level to consider changes in cost level. Chapter 12 describes credibility in detail.

Example. Loss Ratio Indicated Change Factor. Assume the following information:

  • Experience period loss and LAE ratio = 65%
  • Experience period credibility = 80%
  • Loss Trend = 5%
  • Premium On-Level Adjustment = 1.075
  • Premium Trend = 2%
  • Premium Delay Factor = 0.99
  • Projected fixed expense ratio = 6.5%
  • Variable expense = 25%
  • Target UW profit = 6.1% .

With these assumptions, the indicated change factor can be calculated as

\[ \small{ \begin{array}{ll} \text{Experience Period Loss Ratio} = 65\% \times \dfrac{1.05}{1.075 \times 1.02} = 62.2\%\\ \text{Target Loss Ratio} = 0.99 - 6.5\% - 25\% - 6.1\% = 61.4\%\\ \text{Complement of Credibility} = 0.614 \times \dfrac{1.05}{1.02} = 63.2\%\\ \text{Credibility-weighted loss ratio} = 62.2\% \times 80\% + 63.2\% \times (1 - 80\%) = 62.4\%\\ \text{Indicated loss ratio} = 62.4\% / 61.4\% = 1.016 . \end{array} } \]

This means that overall average rate level should be increased by 1.6%.

10.11 Further Resources and Contributors

This chapter serves as a bridge between the technical introduction of this book and an introduction to pricing and ratemaking for practicing actuaries. For readers interested in learning practical aspects of pricing, we recommend introductions by the Society of Actuaries in Friedland (2013) and by the Casualty Actuarial Society in Werner and Modlin (2016). For a classic risk management introduction to pricing, see Niehaus and Harrington (2003). See also Finger (2006) and Frees (2014).

Bühlmann (1985) was the first in the academic literature to argue that pricing should be done first at the portfolio level (he referred to this as a top down approach) which would be subsequently reconciled with pricing of individual contracts. See also the discussion in Kaas et al. (2008), Chapter 5.

For more background on pricing principles, a classic treatment is by Gerber (1979) with a more modern approach in Kaas et al. (2008). For more discussion of pricing from a financial economics viewpoint, see Bauer, Phillips, and Zanjani (2013).

  • Edward (Jed) Frees, University of Wisconsin-Madison, and José Garrido, Concordia University were the principal authors of the initial version of this chapter.
    • Chapter reviewers included Chun Yong Chew, Curtis Gary Dean, Brian Hartman, and Jeffrey Pai.
  • Rajesh Sahasrabuddhe, Oliver Wyman, is the author of the second edition of this chapter. Email: for chapter comments and suggested improvements.

TS 10.A. Rate Regulation

Insurance regulation helps to ensure the financial stability of insurers and to protect consumers. Insurers receive premiums in return for promises to pay in the event of a contingent (insured) event. Like other financial institutions such as banks, there is a strong public interest in promoting the continuing viability of insurers.

Market Conduct

To help protect consumers, regulators impose administrative rules on the behavior of market participants. These rules, known as market conduct regulationRegulation that ensures consumers obtain fair and reasonable insurance prices and coverage, provide systems of regulatory controls that require insurers to demonstrate that they are providing fair and reliable services, including rating, in accordance with the statutes and regulations of a jurisdiction.

  1. Product regulation serves to protect consumers by ensuring that insurance policy provisions are reasonable and fair, and do not contain major gaps in coverage that might be misunderstood by consumers and leave them unprotected.
  2. The insurance product is the insurance contract (policy) and the coverage it provides. Insurance contracts are regulated for these reasons:
    1. Insurance policies are complex legal documents that are often difficult to interpret and understand.
    2. Insurers write insurance policies and sell them to the public on a “take it or leave it” basis.

Market conduct includes rules for intermediaries such as agents (who sell insurance to individuals) and brokers (who sell insurance to businesses). Market conduct also includes competition policy regulation, designed to ensure an efficient and competitive marketplace that offers low prices to consumers.

Rate Regulation

Rate regulation helps guide the development of premiums and so is the focus of this chapter. As with other aspects of market conduct regulation, the intent of these regulations is to ensure that insurers not take unfair advantage of consumers. Rate (and policy form) regulation is common worldwide.

The amount of regulatory scrutiny varies by insurance product. Rate regulation is uncommon in life insurance. Further, in non-life insurance, most commercial lines and reinsurance are free from regulation. Rate regulation is common in automobile insurance, health insurance, workers compensation, medical malpractice, and homeowners insurance. These are markets in which insurance is mandatory or in which universal coverage is thought to be socially desirable.

There are three principles that guide rate regulation: rates should

  • be adequate (to maintain insurance company solvency),
  • but not excessive (not so high as to lead to exorbitant profits),
  • nor unfairly discriminatory (price differences must reflect expected claim and expense differences).

Recently, in auto and home insurance, the twin issues of availability and affordability, which are not explicitly included in the guiding principles, have been assuming greater importance in regulatory decisions.

Rates are Not Unfairly Discriminatory

Some government regulations of insurance restrict the amount, or level, of premium rates. These are based on the first two of the three guiding rate regulation principles, that rates be adequate but not excessive. This type of regulation is discussed further in the following section on types of rate regulation.

Other government regulations restrict the type of information that can be used in risk classification. These are based on the third guiding principle, that rates not be unfairly discriminatory. “Discrimination” in an insurance context has a different meaning than commonly used; for our purposes, discrimination means the ability to distinguish among things or, in our case, policyholders. The real issue is what is meant by the adjective “fair.”

In life insurance, it has long been held that it is reasonable and fair to charge different premium rates by age. For example, a life insurance premium differs dramatically between an 80 year old and someone aged 20. In contrast, it is unheard of to use rates that differ by:

  • ethnicity or race,
  • political affiliation, or
  • religion.

It is not a matter of whether data can be used to establish statistical significance among the levels of any of these variables. Rather, it is a societal decision as to what constitutes notions of “fairness.”

Different jurisdictions have taken different stances on what constitutes a fair rating variable. For example, in some jurisdictions for some insurance products, gender is no longer a permissible variable. As an illustration, the European Union now prohibits the use of gender for automobile rating. As another example, in the U.S., many discussions have revolved around the use of credit ratings to be used in automobile insurance pricing. Credit ratings are designed to measure consumer financial responsibility. Yet, some argue that credit scores are good proxies for ethnicity and hence should be prohibited.

In an age where more data is being used in imaginative ways, discussions of what constitutes a fair rating variable will only become more important going forward and much of that discussion is beyond the scope of this text. However, it is relevant to the discussion to remark that actuaries and other data analysts can contribute to societal discussions on what constitutes a “fair” rating variable in unique ways by establishing the magnitude of price differences when using variables under discussion.

Types of Rate Regulation

There are several methods, that vary by the level of scrutiny, by which regulators may restrict the rates that insurers offer.

The most restrictive is a government prescribedGovernment sets the entire rating system including coverages regulatory system, where the government regulator determines and promulgates the rates, classifications, forms, and so forth, to which all insurers must adhere. Also restrictive are prior approvalRegulator must approve rates, forms, rules filed by insurers before use systems. Here, the insurer must file rates, rules, and so forth, with government regulators. Depending on the statute, the filing becomes effective when a specified waiting period elapses (if the government regulator does not take specific action on the filing, it is deemed approved automatically) or when the government regulator formally approves the filing.

The least restrictive is a no fileInsurers may use new rates, forms, rules without approval from regulators or record maintenance system where the insurer need not file rates, rules, and so forth, with the government regulator. The regulator may periodically examine the insurer to ensure compliance with the law. Another relatively flexible system is the file onlyInsurers must file rates, forms, rules for record keeping and use immediately system, also known as competitive rating, where the insurer simply keeps files to ensure compliance with the law.

In between these two extremes are the (1) file and use, (2) use and file, (3) modified prior approval, and (4) flex rating systems.

  1. File and Use: The insurer must file rates, rules, and so forth, with the government regulator. The filing becomes effective immediately or on a future date specified by the filer.
  2. Use and File: The filing becomes effective when used. The insurer must file rates, rules, and so forth, with the government regulator within a specified time period after first use.
  3. Modified Prior Approval: This is a hybrid of “prior approval” and “file and use” laws. If the rate revision is based solely on a change in loss experience then “file and use” may apply. However, if the rate revision is based on a change in expense relationships or rate classifications, then “prior approval” may apply.
  4. Flex (or Band) Rating: The insurer may increase or decrease a rate within a “flex band,” or range, without approval of the government regulator. Generally, either “file and use” or “use and file” provisions apply.

For a broad introduction to government insurance regulation from a global perspective, see the website of the International Association of Insurance Supervisors (IAIS).