Valuation of Probabilities

By Ramesh Akkineni

from 'A Treatise on Probability' by John Maynard Keynes

If we pass from the opinions of theorists to the experience of practical men, it might perhaps be held that a presumption in favour of the numerical valuation of all probabilities can be based on the practice of underwriters and the willingness of Lloyd's to insure against practically any risk. Underwriters are actually willing, it might be urged, to name a numerical measure in every case, and to back their opinion with money. But this practice shows no more than that many probabilities are greater or less than some numerical measure, not that they themselves are numerically definite. It is sufficient for the underwriter if the premium he names exceeds the probable risk. But, apart from this, I doubt whether in extreme cases the process of thought, through which he goes before naming a premium, is wholly rational and determinate; or that two equally intelligent brokers acting on the same evidence would always arrive of the same result. In the case, for instance, of insurances effected before a Budget, the figures quoted must be partly arbitrary. There is in them an element of caprice, and the broker's state of mind, when he quotes a figure, is like a bookmaker's when he names odds. Whilst he may be able to make sure of a profit, on the principle of the bookmaker, yet the individual figures that make up the book are, within certain limits, arbitrary. He may be almost certain, that is to say, that there will not be new taxes on more than one of the articles tea, sugar, and whisky; there may be an opinion abroad, reasonable or unreasonable, that the likelihood is in the order--whisky, tea, sugar; and he may, therefore, be able to effect insurances for equal amounts in each at 30 per cent, 40 per cent, and 45 percent. He has thus made sure of a profit of 15 per cent, however absurd and arbitrary his quotations may be. It is not necessary for the success of underwriting on these lines that the probabilities of these new taxes are really measurable by the figures 3/10, 4/10, and 45/100; it is sufficient that there should be merchants willing to insure at these rates. These merchants, moreover, may be wise to insure even if the quotations are partly arbitrary; for they may run the risk of insolvency unless their possible loss is thus limited. That the transaction is in principle one of bookmaking is shown by the fact that, if there is a specially large demand for insurance against one of the possibilities, the rate rises; -- the probability has not changed, but the "book" is in danger of being upset. A Presidential election for the United States supplies a more precise example. On August 23, 1912, 60 per cent was quoted at Lloyd's to pay a total loss should Dr. Woodrow Wilson be elected, 30 per cent should Mr. Taft be elected, and 20 per cent should Mr. Roosevelt be elected. A broker, who could effect insurances in equal amounts against the election of each candidate, would be certain at these rates of a profit of 10 per cent. Subsequent modifications of these terms would largely depend upon the number of applicants for each kind of policy. Is it possible to maintain that these figures in any way represent reasoned numerical estimates of probability?
In some insurances the arbitrary element seems even greater. Consider, for instance, the reinsurance rates for the Waratah, a vessel which disappeared in South African waters. The lapse of time made rates rise ; the departure of ships in search of her made them fall; some nameless wreckage is found and they rise; it is remembered that in similar circumstances thirty years ago a vessel floated, helpless but not seriously damaged, for two months, and they fall. Can it be pretended that the figures which were quoted from day to day--75 per cent, 83 per cent, 78 percent -- were rationally determinate, or that the actual figure was not within wide limits arbitrary and due to the caprice of individuals? In fact underwriters themselves distinguish between risks which are properly insurable, either because their probability can be estimated between comparatively narrow numerical limits or because it is possible to make a "book" which covers all possibilities, and other risks which cannot be dealt with in this way and which cannot form the basis of a regular business of insurance, --although an occasional gamble may be indulged in. I believe, therefore, that the practice of underwriters weakens rather than supports the contention that probabilities can be measured and estimated numerically.

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