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On the Limitation of Risks; being an Essay towards the Determination of the Maximum Amount of Risk to be retained by a Life Insurance Company on a Single Contingency
Published online by Cambridge University Press: 18 August 2016
Extract
It is my object in the present paper to discuss the principles which should guide the actuary in fixing the maximum amount of risk to be retained by a Life Insurance Company upon a single contingency.
Those principles are by no means limited to life insurance, but extend with suitable modifications to other kinds of insurance, as fire, marine, hailstorm, health, and accidental death insurance. They are also applicable to the risks of mercantile transactions, and in particular to the risks incurred by a bank in discounting its customers' bills. There can be little doubt that some recent failures of banks have been caused principally by the neglect of these principles; in other words, the manager has failed to put a proper limit on the risks undertaken, i.e., has trusted individuals to an imprudent extent, and the loss of heavy sums consequent on their failure has proved disastrous to the bank.
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- Copyright © Institute and Faculty of Actuaries 1867
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page 21 note * The mortality being such that out of 1,000 persons alive at the beginning of a year 10 die during the year, it appears correct to say, either that the “annual mortality is one per cent., “or that” the rate of mortality is one per cent, per annum,” or that “the annual rate of mortality is ·01.” In the latter forms of expression, the phrase “rate of mortality” is used in a sense strictly analogous to the common one “rate of interest.” Thus we say that “the rate of interest is five per cent, per annum,” if £1,000 has by the operation of interest become £1,050 at the end of a year.
It must be borne in mind by the student, in reading the Reports of the Registrar-General and similar works, that the term “rate of mortality” is used by Dr. Farr–most injudiciously and improperly in my opinion–to denote a different quantity:–that, namely, which has been called by Mr, Gomp'ertz the “intensity of mortality,” and by Mr. Woolhouse (very happily) the “force of mortality.” It is not easy to give in popular language a strict definition of the “force of mortality;” but it is approximately equal to the ratio which the number dying in a year bears to the number living in the middle of the year.
To enable the reader to understand the distinction more readily, I quote the remarks of Dr. Farr, given in the Blue Book, on the Sanitary State of the Army in India (p 37):·“In the reports upon the Indian Fund, the probability of dying is incorrectly called the rate of mortality, so as to mislead the unwary reader. Thus, if on an average out of 100 men living at the beginning of a year there are 10 deaths in the year following, the probability of dying is expressed by the fraction 
, which is incorrectly called in the reports ‘the rate of mortality’; but the rate of mortality is 
, for the numbers living at the end of the year are 90, and the years of life are 90 + 10 half years, which it may be assumed are lived by the 10 who died in the coarse of the year.”This appears to me much as if we should say, “If £100 has by the operation of interest become £105 at the end of a year, the rate of interest is not 
but 
. In my opinion Dr. Farr is wrong, and the authors he criticizes are right, in the meaning they attach to the phrase, “rate of mortality.”
I have not been able to ascertain that any writer, with the exception of Mr. Edmonds, agrees with Dr. Farr in his views on this point; and I conclude that the attempt to attach a new meaning to the familiar phrase, “rate of mortality,” is an innovation much to be deplored, and in every way to be discountenanced. By this and similar innovations–as his definition of an “Annuity,” and his alteration of the “N Column”–Dr. Farr has greatly diminished the usefulness of his writings.
If Lxdenote the number living at any age x, integral or fractional, then the annual “rate of mortality” at that age x will be 
; and the “force of mortality” at the same age, 
. If the mortality follow Gompertz's law so that 
, then the annual rate of mortality at the age will be found to be 
; and the force of mortality, loge
g. loge
q,q
x.
page 23 note * These values have been calculated as follows:—P0=·99100, and is easily found by logarithms. Then
