Pricing Insurance Risk. Stephen J. Mildenhall
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forms of capital, capital compared to equity, accounting, and the mechanism of default, especially equal priority, all play essential roles. Furthermore, we address certain standard practices in the industry and subsume them within our analytical framework so the reader can better appreciate their properties and behavior.
Over the years we have found that putting these tools and techniques into practice raises the following questions.
Which risk measure should I use? A common followup question asks if the risk measure should be sensitive to the tail of the distribution or volatility in the body, which translates into a concern about solvency vs. quarterly earnings. As always in modeling, the measure must be appropriate to the intended purpose. Our framework separates the amount of capital from its cost: the capital risk measure is necessarily tail focused, whereas the pricing risk measure captures investor return expectations. Additionally, the connection between capital structure and the pricing risk measure is a fundamental insight.
How do I reconcile and manage different economic and regulatory views of risk? Often followed by, “Who cares? We manage to [Rating Agency’s] capital model”. We agree: you don’t care. The rating agency model is a binding constraint on the amount of capital for many insurers. You control the form and influence the cost of capital. Again, two risk measures. It isn’t a question of reconciliation; it is a question of understanding each measure’s distinct purposes.
Should pricing target a return on all capital, or can there be excess capital? Genuinely excess capital is exceptionally rare. Our model produces a cost of capital specific to each company, which varies with the amount of capital. A better capitalized company has a lower percentage cost of capital, other things being equal, because higher layer capital is less stressed and exposed to risk. As a result, the problem of applying a uniformly high cost of capital, producing uneconomic premiums, should not occur. The frictional costs of capital are, however, constant for all layers of capital. Indeed, they could be increasing if the management of overcapitalized companies has an incentive to engage in frivolous, self-aggrandizing activities.
How do I determine the cost of capital? Does it vary by unit? The risk cost of capital is the weighted average cost over the actual capital used. Debt and reinsurance have known costs. The cost of equity capital is normally estimated using a peer study. The cost of capital varies by unit according to which capital layers each unit consumes. The frictional cost of capital typically does not vary by unit.
Can risk margins ever be negative? Classical and modern approaches to pricing are unanimous that 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