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.
{158}
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?