An Essay to ascertain the value of leases, and annuities for years and lives. By W[eyman] L[ee]. London, 1737, 8vo.
A valuation of Annuities and Leases certain, for a single life. By Weyman Lee, Esq. of the Inner Temple. London, 1751, 8vo. Third edition, 1773.
Every branch of exact science has its paradoxer. The world at large cannot tell with certainty who is right in such questions as squaring the circle, etc. Mr. Weyman Lee[344] was the assailant of what all who had studied called demonstration in the question of annuities. He can be exposed to the world: for his error arose out of his not being able to see that the whole is the sum of all its parts.
By an annuity, say of L100, now bought, is meant that the buyer is to have for his money L100 in a year, if he be then alive, L100 at the end of two years, if then alive, and so on. It is clear that he would buy a life annuity if he should buy the first L100 in one office, the second in another, and so on. All the difference between buying the whole from one office and buying all the separate contingent payments at different offices, is immaterial to calculation. Mr. Lee would have agreed with the rest of the world about the payments to be made to the several different offices, in consideration of their several contracts: but he differed from every one else about the sum to be paid to _one_ office. He contended that the way to value an annuity is to find out the term of years which the individual has an even chance of surviving, and to charge for the life annuity the value of an annuity certain for that term.
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It is very common to say that Lee took the average life, or expectation, as it is wrongly called, for his term: and this I have done myself, taking the common story. Having exposed the absurdity of this second supposition, taking it for Lee's, in my _Formal Logic_,[345] I will now do the same with the first.
A mathematical truth is true in its extreme cases. Lee's principle is that an annuity on a life is the annuity made certain for the term within which it is an even chance the life drops. If, then, of a thousand persons, 500 be sure to die within a year, and the other 500 be immortal, Lee's price of an annuity to any one of these persons is the present value of one payment: for one year is the term which each one has an even chance of surviving and not surviving. But the true value is obviously half that of a perpetual annuity: so that at 5 percent Lee's rule would give less than the tenth of the true value. It must be said for the poor circle-squarers, that they never err so much as this.
Lee would have said, if alive, that I have put an _extreme case_: but any _universal_ truth is true in its extreme cases. It is not fair to bring forward an extreme case against a person who is speaking as of usual occurrences: but it is quite fair when, as frequently happens, the proposer insists upon a perfectly general acceptance of his assertion. And yet many who go the whole hog protest against being tickled with the tail. Counsel in court are good instances: they are paradoxers by trade. June 13, 1849, at Hertford, there was an action about a ship, insured against a _total_ loss: some planks were saved, and the underwriters refused to pay. Mr. Z. (for deft.) "There can be no degrees of totality; and some timbers were saved."--L. C. B. "Then if the vessel were burned to the water's edge, and some rope saved in the boat, there would be no total loss."--Mr. Z. "This is putting a very extreme case."--L. C. B. "The argument {159} would go that length." What would _Judge_ Z.--as he now is--say to the extreme case beginning somewhere between six planks and a bit of rope?
