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Brainstorms
Zero-Sum Contracts
by Stephen W. Philbrick
An article in the local paper, reporting a bonus payment to University of Maryland Coach Ralph Friedgen for taking our team to the Orange Bowl, caught my attention for two reasons. As a Marylander I am happy to see the football team step up into the ranks of the elite. As an actuary, I was intrigued to see that the $300,000 bonus cost the school $15,000. (The taxpayer in me was also happy.)
The school purchased an insurance policy "against" the unfortunate occurrence of a bowl invitation. The premium in this case is probably reasonable, given losing seasons for Maryland in prior years. The price for, say, University of Florida, would be much higher.
The actuarial pricing of performance bonuses raises some intriguing issues. Some schools probably work on the assumption that they are almost certain to go to a bowl game, and wouldn't even offer a contingent bonus. Effectively, it would be priced into the salary.
While a few elite programs expect to go to bowl games on a regular basis, the likelihood for most of the 117 Division I football programs is fairly low. Consequently, the risk associated with any payoff is fairly high. The contract will either pay the total amount, or zero.
As is the case with other insurance contracts, a company cannot justify writing a single contract. A company wants to write a number of contracts with little correlation between the contracts. The law of large numbers will operate, and while the total risk will grow as contracts are added, the risk grows at an ever-slower rate. With a sufficient number of contracts, the risk margin required for each one can be modest.
However, these performance contracts have a potentially interesting feature. The number of teams that will go to a bowl game is fixed. One team managing to win a few extra games and go to a bowl invariably means another team will be crowded out. If a company could write such a contract for every single Division I school, it would know precisely how many contracts will have a claim. The aggregate risk actually drops to zero, even though each individual contract has substantial risk.
In practice, the ideal cannot be met. Not all schools will offer performance bonuses, the terms may differ, and the amounts are likely to differ. This probably turns out to be a benefit. While each contract contains risk transfer, if someone could literally write a contract on every school, someone might step in and argue that the set of contracts, taken as a whole, do not constitute risk transfer. Alternatively, some organization other than an insurance company might decide to offer such a product. So it may be good news that the best achievable market penetration would still have underwriting risk.
Another interesting attribute is the perceived chance of loss as compared to the true chance of loss. For many coverages, such as auto liability, the policyholder believes he or she is less likely to have a claim than is actually the case. The price seems high, based upon this unrealistic belief of a low frequency of loss. Sports fans, on the other hand, are more apt to overestimate their chances of winning. Last year's winners expect to repeat. Last year's losers assume that it was an off-year, and this year will be better. While few Maryland fans "expected" that Maryland would make it into a bowl, most probably would peg their odds at better than the 20-1 (or worse) implied by the premium. The head of an athletic department waxing eloquently about the next year's prospects will have to at least feel guilty to complain about the price of the contract.
At the extreme, this is an example of a zero-sum game, where there is still risk at the individual participant level. Sports offers the clearest examples where contingent bonuses for achieving playoffs have the desired attributes: risk at the individual contract level, but little risk when aggregated. Contests for scientific breakthroughs could be structured this way. There may be other examples, or perhaps a clever insurance company could create situations, in which contracts could be structured with risk at the contract level, but minimal risk in aggregate.