Pricing Insurance Risk. Stephen J. Mildenhall
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as a rate of return or a monetary margin amount; these are interchangeable representations. In the unlikely case you get to set your portfolio profitability target, you need to examine your firm’s balance sheet—fortunately, this is required in the next task.
Selecting and calibrating a pricing risk measure—specifically a spectral risk measure—is the biggest challenge. We have evolved away from our early fondness for particular parametric SRMs (especially the ones we invented). We now recommend using bespoke nonparametric or semiparametric distortion functions to more closely mirror actual funding costs. It may be that you are not modeling the entire firm’s portfolio but only a part of it. If you do not have access to the whole company risk profile, fear not. You should treat the task as if the parent company is the investor and the portfolio is the company—even though this is a case of suboptimizing. The point here is that the SRM gives shape to how the overall required margin is distributed across layers of assets at risk. More specific advice on selecting a distortion function is given in Chapter 11.5.
With these inputs in hand, allocating margin via the natural allocation is almost a trivial numerical exercise.
Of course, we hope you will read the whole book eventually, but we are not so naїve as to assume you have the time to sit down and read it cover to cover. It takes a lot more to explain and understand why than how. Why spectral risk measures make sense and do not violate the received wisdom of finance theory, and why the natural allocation is justified in being treated as canonical and not merely one of many equally acceptable alternatives—these issues take many more pages than explaining the mechanics of computation. We hope you will appreciate the why and read the whole book. But if you want to jump ahead to a quick grasp of the how, we recommend the following. Make sure to do enough of the exercises as you go along to feel secure that you “get it.”
Read about the insurance market and Ins Co., our model company, in Chapter 2.
Review the introductory material on risk measures in Chapter 3. This should be material you already know. But do pay special attention to the Lee diagram in Chapter 3.5.
Some of the material on VaR and TVaR in Chapter 4 may be new to you, so make sure you are comfortable with the basics.
Chapter 5.1 lays out the big picture of how Ins Co. approaches the task of analyzing its capital and pricing needs.
Read the practical applications in Chapter 6 and Guide to the Practice Chapters, Chapter 7.
Read about classical risk theory in Chapter 8.4 and the DCF model in Chapter 8.7. This will help tie the later material back to material you likely have already seen.
See how classical premium calculation principles work out on our case studies in Chapter 9.
Read the sections in Chapter 10, Modern Portfolio Pricing Theory, down through Chapter 10.8. This is core theory about SRMs.
Read and work examples in Chapter 11, Modern Portfolio Pricing Practice, down through Selecting a Distortion in Chapter 11.5. Read this last section twice and bookmark it for the day you need to select a distortion for your own purposes.
Browse Classical Price Allocation Theory, Chapter 12, down through Loss Payments in Default, Chapter 12.3. This is material that should be more or less familiar to actuaries.
See how classical price allocation works out on our Case Studies in Chapter 13.
Read the first two sections in Chapter 14, Modern Price Allocation Theory. This covers the natural allocation of a coherent risk measure, properties and characterization of allocations, computational algorithms, and comments on selecting an allocation. This is the core theory about allocating SRMs. If you are looking for ways to visualize multidimensional risk, read Chapter 14.3, especially Chapter 14.3.7, as well.
Read Modern Price Allocation Practice, Chapter 15. This is essential “how-to” material.
If reserves feature prominently in your project, you may want to read Chapter 17 in Part IV. This also covers the Solvency II risk margin in Chapter 17.3.
If reinsurance purchasing features prominently in your project, you may want to read Chapter 19.
If you are working with portfolio optimization, you may want to read Chapter 20.
2 The Insurance Market and Our Case Studies
In this chapter, we outline the operation of the insurance market as we model it and describe the hypothetical Ins Co. used in our examples. We then introduce a Simple Discrete Example and three more realistic Case Studies which are used throughout the book to illustrate the methods presented in the theory chapters.
2.1 The Insurance Market
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.
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