target="_blank" rel="nofollow" href="#ulink_70176397-da7c-5a0d-849d-224384dad8aa">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 mandatory for a useful risk measure, and they lead us to coherent risk measures. Coherent risk measures have an intuitive representation, providing us with some guidance on forming and comparing them. Spectral risk measures (SRMs)—also known as distortion risk measures—are a subset of coherent measures. They have additional properties and a particularly straightforward representation via a distortion function. Spectral risk measures can be viewed in four equivalent ways:
1 as expected values with varying distorted probabilities,
2 as a weighted sum of TVaRs at different thresholds,
3 as a weighted sum of VaRs at different thresholds, where the weights have specific properties, and
4 as the worst expected value across a set of different probability scenarios.
Spectral risk measures alter or distort the underlying pattern of probabilities and compute expected values based on the new probabilities, analogous to the effect of stochastic discount factors in modern finance. The distorted probability treats large losses as more likely, creating a positive pricing margin. TVaR is the archetypal SRM. It is simple yet powerful and has many desirable properties. We gain analytical insights into the nature of SRMs because they are all weighted averages of TVaRs. For example, we can allocate any SRM-derived quantity by bootstrapping a TVaR allocation.
1.3.2 Part II: Portfolio Pricing
Part II is about portfolio pricing, where the entire portfolio is treated as a single risk. Risk is related to return, suggesting we should apply a risk measure to portfolio losses and use the result to indicate a price. Our principal goal is to determine what price is sufficient for assuming the portfolio risk. Secondary goals include understanding, making inferences about, and calibrating to, market prices.
Insurance is characterized by risk transfer through risk pooling. Figure 1.1 combines all insureds into one portfolio. It shows how the capital and pricing risk measures interact to determine the insurer’s risk pool premium. Part II of the book treats the cash flows on the lower right, between the insurer and the investors.
We are aware that pricing actuaries and underwriters do not set premiums; markets do. However, the aggregate effect of individual company risk-return decisions drives quotes and acceptances in the market. When we talk about setting premiums, we understand it in the framework of evaluating market pricing or offering a quote.
How are the parameters of a pricing model determined? This is a difficult question that must be answered to put theory into practice. We provide examples showing how different parametrization methods perform, link pricing to capital structure, and calibrate an SRM to catastrophe bond pricing.
1.3.3 Part III: Price Allocation
Insurers must allocate a portfolio price and margin to its constituent units to sell policies and run their business. Price allocation is the topic of Part III.
We examine how units contribute to portfolio risk. For example, the models may show several outcomes that lead to insolvency. Which units are the drivers of
