the risk margin must be positive for the portfolio. However, negative margins are appropriate for some units within the portfolio. They occur for units that are hedges, with losses arising more in situations where the portfolio has lower losses and less when the portfolio has more significant losses. The common practice of paying a positive margin for ceded reinsurance proves the point: the outward cash flow (premium) is greater in expectation than the inward cash flow (recovery). Looking at expectations makes reinsurance seem inappropriate, but the key to the value of reinsurance (or any hedge) is when those cash flows occur.
How do I use a risk measure to determine reservation prices? Chapter 10 and Chapter 14 show how pricing and capital risk measures combine to determine premiums. Chapter 20 offers some more advanced considerations.
The reader will recognize a gap between our simplified models of insurance operations and the complexity of the real world. The practitioner who has mastered Parts I, II, and III and is starting to think seriously about implementing risk measures will likely come up with numerous “What about…?” questions. The following more advanced questions commonly arise for insurers with functioning and integrated risk pricing systems. They are addressed in Part IV.
How do I handle asset risk? How do I incorporate risky assets in the model? How much capital does asset risk consume? Should I treat asset risk in a fundamentally different way from insurance risk? We conclude that an additional degree of freedom emerges, but not to any good use. Chapter 8.8 discusses the impact of asset risk on pricing and the market value of equity in an option pricing model. Chapter 16 shows that investing in a risky asset typically lowers the fair price (and quality) of insurance being sold.
How do I price for reserve risk? I write business that takes years to settle. It is unrealistic to assume all losses are paid in one year. How do I incorporate reserves into the model? Reserve volatility consumes underwriting capacity. However, our model shows that the allocated margins are small when reserves are stable. In a sense, reserves can provide ballast for the prospective portfolio. IFRS and other accounting conventions have begun to require a risk margin for reserves for better earnings recognition. We discuss the Solvency II Cost of Capital Risk Margin and a real option approach to reserves in Chapter 17.
How do I manage a going concern? I don’t manage for just one year and then dissolve the business; I manage a going concern with brand recognition and franchise value. How does that change the model? Chapter 18 outlines the theory of optimal dividends and a simple model of franchise value.
How can I optimize ceded reinsurance purchases? I can see how assumed reinsurance can be treated as selling another line of business, but how do I think about ceded reinsurance? More specifically, how should I go about optimizing it? Chapter 19 discusses how to evaluate and optimize a ceded reinsurance program.
How can I optimize my insurance portfolio? I used to think about optimizing my capital usage against capital constraints. Now I think I should be optimizing my cost of capital, but that doesn’t seem to be what you are recommending. Is there a disconnect here? Chapter 20 explores the complex interaction of cost allocation, benefit allocation, and premium regulation. It uncovers some unavoidable market distortions.
1.5 Where to Start
If you have read this far, you likely have a pricing problem. It may be embedded in a broader effort—business unit assessment or portfolio optimization or strategic planning—but it comes down to a pricing problem at its core. At a high level, our recommendations sound simple:
1 Establish your asset requirement.
2 Establish your portfolio cost of capital.
3 Select and calibrate a consistent spectral risk measure.
4 Use what we call the natural allocation to allocate the margin to each unit.
These recommendations presume a lot of work has already been done: gathering and organizing relevant data, developing a mathematical model or numerical tabulation (simulated sample) of the portfolio risks, establishing loss cost estimates for the units, etc. As we said, pricing is the last mile.
The asset requirement should be easy to determine since an external authority usually promulgates it. However, it may require some work to compute, using a standard (e.g., regulatory) capital risk measure. If you find no obvious binding capital constraint, remember that management’s risk tolerance is irrelevant; only the owner’s risk tolerance matters. Try to divine it. This step can be incredibly challenging for mutual companies. If you are engaged in an optimization project, then a capital risk measure is necessary because you will have to what-if the capital requirement. If the problem involves the current portfolio only, say a business unit profitability assessment or reinsurance purchase decision, you need only calculate current required assets.
The portfolio cost of capital may similarly be handed down from on high. It can be expressed 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
