1
Introduction

In order to make insurance a trade at all, the common premium must be sufficient to compensate the common losses, to pay the expense of management, and to afford such a profit as might have been drawn from an equal capital employed in any common trade.

Adam Smith, The Wealth of Nations (Book 1, Ch X, Part I, 5th Edition, 1789)

1.1 Our Subject and Why It Matters

Pricing insurance risk is the last mile of underwriting. It determines which risks are accepted onto the balance sheet and makes an insurer's risk appetite operational. It is critical to successful insurance company management.

As the last mile, pricing depends on all that has come before. Actuaries and underwriters have analyzed and classified the risk, trended and developed losses, and on-leveled premiums to pick a best-estimate prospective loss cost. Accountants have allocated fixed and variable expenses. Simulation models place the new risk within the context of the company's existing portfolio. The mechanics of all this work is the subject of much of the actuarial education syllabus: experience and exposure rating, predictive analytics, and advanced statistical methods. That is not the subject of this book! All of that prior effort determines the expected loss, and we take it as a given. Pricing adds the risk margin-to afford capital a reasonable return. The risk margin is our subject.

Since risk margins are often small, how is it they deserve a whole book? Because risk considerations have an outsized market impact. True, personal property may only earn a single-digit margin. But that business often relies on reinsurance priced with margins of 50% or more. When the reinsurance markets fail or become stressed-as seen after Hurricane Andrew and the Northridge earthquake, for example-the tail of high-risk-margin business wags the dog of much larger property lines. Risk margins are critical to the functioning of the insurance market. Even for lines with thin margins, the collective risk and return decisions of firms have profound macro impacts over time such as the secular increases in homeowners pricing over the last twenty years.

We emphasize insurance risk. We do not discuss credit risk nor operational risk. We have only a little to say about asset risk and nothing about interest rate risk. Market risk, underwriting cycles, competitive threats? Sorry, all off-topic. We are focused on the risk of losses arising from insurance contracts. We lean heavily towards a property-casualty perspective and, within that, towards catastrophe risk; however, the principles we lay out apply to any insurance risk. This is not a book about Enterprise Risk Management (ERM) although we do have a few words to say about optimization and portfolio management.

The goal of this book is to demonstrate how to

1. compute a reservation price (technical premium, required premium) for the portfolio, and
2. allocate it to portfolio units (policies, lines of business, etc.) in a defensible manner

starting from a model of the insured risks. These pricing techniques have powerful applications. They allow us to assess the performance of different units, evaluate needed reinsurance, and optimize overall strategy.

1.2 Players, Roles, and Risk Measures

Figure 1.1 shows the participants in the insurance pricing problem. Insureds, left, pay premiums to the insurer and in turn receive loss payments. The regulator, on top, observing the risk that the insurer is taking on, imposes asset requirements. Investors, right, provide capital and in turn receive the residual value (remaining assets) after losses are paid.

Figure 1.1 Players and their roles. The regulator applies a capital risk measure to determine required insurer assets. The pricing risk measure gives the cost of investors' capital. Assets in excess of losses are paid to investors as the residual value of the business.

Insureds buy insurance because of their aversion to risk and because they are required to do so to drive a car, buy a house with a mortgage, etc. Regulators play a social policy role, addressing three principal concerns. First, to ensure mandated third-party insurance provides effective protection. Second, to manage the externality of losses exceeding assets. And third, to prevent insureds being fleeced by excessive premiums. The first concern is present in any tort-based system. We loosely identify the second as European and the third as American. We focus on the second concern, asset adequacy. Our development of technical premiums naturally aligns with the third fairness consideration if we assume that capital markets require fair returns.

Investors indirectly determine premiums because premiums plus capital add up to and fund assets, Figure 1.2. Investors' willingness to provide capital to insurers translates into a pricing risk measure, which the insurer applies to the covered risks. Premium and asset levels are separate problems and need separate tools.

Figure 1.2 The different roles of capital and pricing risk measures.

Two important questions arise from insurance company promises to pay certain sums of money contingent on random events.

1. Are there sufficient assets to honor those promises?
2. Are investors being adequately compensated for taking on those risks?

Crucially, we need to talk about not one but two different risk measures to answer these questions.

Question 1 concerns risk tolerance and is usually answered by an economic capital model. It determines the assets necessary to back an existing or hypothetical portfolio at a given level of confidence. This exercise is also reverse engineered: given existing or hypothetical assets, what are the constraints on business that can be written?

We can imagine a regulator-interpreted broadly as an external authority-considering a portfolio of risks that the insurer proposes to cover. The regulator specifies the amount of assets the insurer must hold to cover the risk. If there is a shortfall after losses are realized, it will be made up by parties external to the insurer, e.g., a guarantee fund or other government entity, or the insureds themselves insofar as they are not reimbursed for claims. The regulator seeks to minimize the nonpayment externality, balanced with a desire for economical insurance.

A capital risk measure is applied to economic capital model output to quantify the level of assets the insurer must hold. Value at Risk (VaR) or Tail Value at Risk (TVaR) at some high confidence level, such as 99.5% or 1 in 200 years, are both popular, but other possible measures exist.

Question 2 concerns risk pricing or risk appetite. We must determine the expected profit insureds need to pay in total to make it worthwhile for investors to bear the portfolio's risk. Regulated insurers are invariably required to hold capital on a regulated balance sheet. We generally assume a funding constraint where premium and investor supplied capital are the only sources of funds. Then, the pricing risk measure determines the split of their asset funding between premium and capital.

The capital and pricing risk measures should not be confused. Historically, capital risk measures have been studied in the context of finance and regulation, e.g., Artzner et al. (1999). In contrast, actuaries have studied pricing risk measures as premium calculation principles (Goovaerts, De Vylder, and Haezendonck 1984). The recent popularity and focus on coherent risk measures has overshadowed actuarial premium calculation principles and led to some confusion about the two distinct purposes of risk measures. Much of the recent literature implicitly or explicitly refers to the capital domain only. However, practitioners dealing with issues such as business unit performance, premium adequacy, and shareholder value are operating in the pricing domain. Taking a risk measure suitable for one use and applying it to the other invites unexpected and confusing results. Instead, we must understand how the capital and pricing measures work together in a complex, nonlinear manner to determine technical prices.

The top-down pricing process we have described may not seem commonplace, although those working in catastrophe reinsurance should find our process familiar. Most individual risk pricing actuaries can legitimately claim to use a bottom-up approach. Nevertheless, deep within almost every company lies a corporate financial model functioning exactly as we describe. It asks: How much capital is needed? What is the cost of that capital? What overall margin is necessary? And, how should it be allocated to each unit?

1.3 Book Contents and Structure

The book has four main parts: one on measuring risk, one about portfolio pricing, one about pricing units within a portfolio, and one addressing advanced topics. The high level overview we provide here supplements the introductory paragraphs in each chapter.

1.3.1 Part I: Measuring Risk

Part I is about risk. What is risk, and how can it be measured and compared? We discuss the mathematical formalism and practical application of representing an insured risk by a random variable. We define a risk measure as a functional taking a random variable to a real number representing the magnitude of its risk. We give numerous examples of risk measures and the different properties they exhibit.

Some properties are more or less...