Pricing Insurance Risk. Stephen J. Mildenhall

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 losses in those scenarios? Parallel questions can be asked of other, less catastrophic, levels of loss.

      Having computed technical premiums as distorted expected values, we can then apply the same distorted probabilities to unit losses, based on their co-occurrence with the total portfolio losses, to allocate the premiums to units. This technique provides a high degree of consistency and synchronization in calculating technical premiums by unit.

There is a particular approach to handling business unit performance assessment and reinsurance decision making that makes use of a capital measure, typically VaR or TVaR, but appears not to make use of a pricing metric. This approach is rooted in return on risk-adjusted capital style financial logic. It takes two steps: allocate capital and then assign a cost of capital hurdle rate that every unit must meet on its allocated capital. All decisions, such as reinsurance purchases, use the same cost of capital as a benchmark.

      Practitioners recognize that this approach tends to place uncomfortably large burdens on catastrophe exposed units relative to units that do not participate much in solvency threatening events. In addition, when applied to reinsurance purchasing, it tends to favor, almost without exception, deals that operate at high levels of loss with low probabilities. We show that the problem here stems from the implicit use of what we call the Constant Cost of Capital (CCoC) SRM. The overall hurdle rate for the entire portfolio may not be appropriate for every unit. What is needed is a pricing risk measure—different from CCoC—that responds to varying levels of riskiness with different required rates of return. Whereas Part II discusses the construction of such measures, Part III discusses how to deploy them.

      1.3.4 Part IV: Advanced Topics

      The last part of the book touches on five topics that go beyond the coverage of previous chapters: asset risk, reserves, going concern and franchise value, reinsurance optimization, and portfolio optimization.

      1.3.5 Further Structure

      Parts II and III divide portfolio pricing from allocation considerations. Within each part, we further distinguish classical from modern approaches, and theory from practice. The hierarchy, reflected in the sequence of eight core chapters, is:

       Chapters 8 and 9: classical portfolio pricing theory and practice,

       Chapters 10 and 11: modern portfolio pricing theory and practice,

       Chapters 12 and 13: classical price allocation theory and practice, and

       Chapters 14 and 15: modern price allocation theory and practice.

      Our dividing line between classical and modern is 1997, the average publication date of three highly influential papers. Relating to Part I: Artzner et al. (1997) introduced coherent risk measures and revolutionized thinking about measuring risk. Relating to Part II: Wang (1996) introduced the premium density and developed the theory of pricing by layer. And, relating to Part III: Phillips, Cummins, and Allen (1998) rigorously derived financial prices in a multiline company accounting for default. The classical versus modern bifurcation serves a convenient organizational purpose but should not be taken too seriously.

      Classical pricing is predominantly actuarial and risk theoretic. A stability requirement, often linked to the probability of eventual ruin, determines required assets. There is no direct consideration of the cost of capital. On the other hand, modern approaches combine risk theory with financial and mathematical economics and decision science. They relate risk to the investors’ return and the cost of capital and pay attention to uncertainty and risk under pooling. They leverage powerful mathematics to understand intuitively reasonable risk measures.

1.4 What’s in It for the Practitioner?

      This book is intended to be a practitioner-friendly reference as well as providing a theoretical framework. Our methods must have a firm theoretical foundation to justify their real world application. Many topics are inherently technical and require a mathematical background to understand fully. At the same time, the methods we describe can and should be implemented in practice. We have structured the book so readers eager to get their hands dirty can do that more easily. Throughout Parts II and III, we alternate theoretical and practical chapters. The Practice chapters present a range of simple numerical examples and apply all the methods we propose to three realistic Case Studies.

      We also pay more attention to institutional arrangements—the way things get done—than the typical