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Brainstorms
1969
by Stephen W. PhilbrickWhen The History Channel runs a retrospective, the year 1969 sticks out in my mind. It was a year of contrasts: Men walking on the moon, boys burning their draft cards, the culmination of a scientific revolution, and the anguish of a social revolution.
In some ways, the CAS reminds me of 1969. CASNET features a serious discussion of our profession's future; for the first time in my memory, it is marked by clouds of doubt. In contrast, the profession is making new achievements on the technical side.
I'll leave the discussion of the social revolution to others. I'd like to comment briefly on some of the scientific gains.
Our profession has a rich history of practicality. When classical statisticians told us that the concept of credibility didn't make sense, we decided we needed it and pressed on. We developed formulas that worked. Later, with the help of Bühlmann and others, we formalized our calculations, and to our delight, found that the ad hoc formulas developed by practitioners matched the theoretically derived formulas.
In a similar way, we have developed algorithms to determine capital adequacy and allocation. However, in these cases, when the theory is laid down, the results are less kind to the ad hoc approaches.
In the case of capital adequacy, we start with a probability of ruin approach. To the extent that capital is required to ensure a company's solvency, it is natural to determine capital such that the likelihood of insolvency is reduced to an acceptable level. (This approach has a counterpart in the banking industry, where it is referred to as Value at Risk, or VaR.) Progress occurred when Butsic introduced the concept of Expected Policyholder Deficit (EPD), essentially noting that we should analyze the cost of insolvency, rather than just the probability of insolvency.
However, when certain plausible axioms (called the axioms of coherence) are selected, both approaches fail to satisfy the axioms. One measure, which in essence combines the VaR and EPD approaches, does satisfy the axioms. This measure is called TCE (Tail Conditional Expectation) or TVaR (Tail Value at Risk), and is the subject of several recent papers.
When allocating capital (or, more properly, the cost of capital), we use risk measures such as the standard deviation or variance of results. Several approaches exist, but a common one analyzes each business segment's marginal contribution to the aggregate risk measure, then allocates the aggregate amount in proportion to the marginal contributions. Jean LeMaire brought game theory to bear on the problem, and showed that, under certain reasonable axioms, the marginal approach fails to satisfy the axioms. An alternative approach, called the Shapley value, does satisfy the axioms.
Of course, when algorithms fail to satisfy axioms, one needs to examine the axioms. Axioms that seem reasonable upon first glance may, in fact, lead to unreasonable restrictions on the resulting formulas. But, as a profession, we must change one or the other. We should not engage in cognitive dissonance: accepting axioms as plausible, then using methods inconsistent with those axioms.
Both of these approaches—TCE as a risk measure, and the Shapley value as an allocation procedure—are relatively new to casualty actuaries. Both depend on sets of axioms that should be accepted or rejected by the actuarial profession. If accepted, we should work as a profession to spread the use of technically superior approaches.
In a few decades we will look back at this period in our history. I hope, and trust we will still be thriving. I also hope we will point to this period as a time when we added scientific underpinnings to some of our well-used tools, and selected new tools where warranted.