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Bonuses allotted to Policies in force for upwards of Ten Years.

Policies in force for

Age at

15 Years. 20 Years. 25 Years. 30 Years. 35 Years. 40 Years. 45 Years.

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The amounts apportioned under the various modes in which the Bonus might be received were as follows:

£57,113 in Reversionary Bonus.

1,958 in Reduction of Premiums.
830 in converting Whole Life Assurances into Endowment

Assurances, payable at a fixed age, or at death.

£59,901 Consolidated Revenue Account of the Crown Life Assurance Company

for Five Years, commencing 25th March, 1865, and ending
25th March, 1870.
£ 8. d.

8. d. Amount of Funds on 25th

Claims under Policies (after March, 1865, the begin

Deduction of Sums Rening of the period 1,008,590 16 6 assured)

459,168 85 Premiums (after Deduction


24,775 14 7 of Re-Assurance Pre


10,209 19 8 miums). 527,334 14 9 Commission

27,073 3 0 Consideration for Annuities

Expenses of Management.. 59,344 4 2 granted.

1,672 9 4 Dividends and Bonuses to Interest and Dividends 219,111 17 4 Shareholders..

67,237 16 2 Other Receipts

Other PaymentsFines from Proprietors


4,589 00 for Non-Assurance

860 11 6 Claims Admitted, but Fines from Assured for

not paid

32,478 10 0 Revival of Policies

53 4 2 Dividends due, but not Registration Fees ..

119 0

1,144 16 3 Balance of Profit and Loss 1,985 0 11 Amount of Funds on 25th Scottish Friendly Com

March, 1870, the end of pany

5,350 6 1 the period, as per First Premiums, 1870, Out


1,126,281 18 5 standing

7,082 5 1 Interest, 1870, Outstanding

16,004 12 10 Bonuses to Shareholders,

24,138 13 2

year 1865.

1,812,303 10 8

1,812,303 10 8

per Cent.

The average Rate of Interest on the Investments of the Life Assurance Fand of the Company was, on 25th March, 1866, £4 9s. 6d.; 1867, £4 10s. 7d.; 1868, £4 11s. 2d. ; 1869, £4 12s.; and 1870, £4 12s. 3d.

At the respective dates mentioned portions of the Fund were not invested ; and if it be intended by the legislature that the rate is to be calculated on the Gross Fund, including Agents' Balances, Outstanding Premiums and Interest, and Interest accrued, &c., then the Interest on the Inrested portion of the Fund was equal to the following rates on such Gross Fund, namely:-1866, £4 6s. 10d.; 1867, £4 58. 5d.; 1868, £4 28.; 1869, 44 6s, 5d.; and 1870, £4 78. öd.

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On Extra Premium." By JAMES R. MACPADYEN, of the Legal

and General Life Assurance Society, Fellow of the Faculty of Actuaries (Scotland).

[Read before the Institute, 25 March 1872.] A PAPER on Extra Premium may treat of the subject in one or other of two ways. It may be, on the one hand, an attempt to deduce tables of mortality for the various classes of underaverage risks occurring in a Life Office's transactions. Or, on the other hand, this part of the subject may be ignored and the paper be simply an enquiry into the effects resulting from the various methods employed in charging extra-an analysis of the actuarial points raised in consequence of a Society admitting not merely minimum risk lives, but others not eligible on similar terms. In short, the subject of Extra Premium may be considered either in respect to the amount to be imposed or in regard to the manner and effect of its imposition. The first question must be determined mainly by actual mortality observations—the second does not require in the same degree for its examination the light of experi

It is this latter question which I have taken up in the following paper; but as both branches of the subject are intimately connected, it may be well to make a few general remarks on the former also.





Inasmuch as the circumstances on which the rates of extra premium depend are infinitely varied, we can never have tables so complete as to be able to relegate every case that may arise in practice to a set of rates founded on observations of life in exactly similar circumstances. We can thus only administer a sort of rough justice--carry out the cardinal principle of life assur

that the many must bear the burden of the one, and place every life that is insurable at all under such premium scale as shall most nearly represent its condition.

In fact, the same thing is done in cases taken as minimum risks, and is the very essence of life insurance. The whole question then is: Where shall the line be drawn? How many tables of mortality shall be used ? Were the selection of the assured against the Office done away with, one might suffice; and according to the force of this selection, will be the number of varying premium rates required. Anything tending to diminish this force will tend to simplify the classification of assured lives. As an illustration of this, let me take the following:

It is at present the practice in many Offices to allow their policyholders free residence in any part of the world, provided the privilege be not exercised during a probationary period; this stipulation, together with a sufficiently broad foundation of general mortality, being considered adequate to check the force of selection against the Society. Similarly, any method acting as governor or regulator to the flow of assurances to an Office will obviate the necessity of a great variety of life premiums whether for disease or for climate, and will thus narrow the field of requisite enquiry as to the rates to be charged in under-average cases. With these remarks I shall leave the first branch of my subject and devote the remainder of this paper to the second form in which the question presents itself, namely, the actuarial results of the various methods now in use for charging extra premium.

Circumstances having an unfavourable tendency on the duration of human life may be divided into two great classes—the one containing those cases in which the age of the person affected has a more important bearing on the prejudicial circumstances, and the other, those cases in which less weight is attached to the time of life. I use the words "more ” and “less” advisedly, because whatever be the nature of the circumstance the age must have some weight; but there are obviously certain kinds of risks in which this is more important than in others. Following the usual mode of classification in the former section we see cases of consumptive taint or tendency, rheumatic fever, and almost all the diseases, and also what may be called the “special circumstance' extras ; in the latter, climate and occupation risks and a few maladies, such as hernia and gout, find a place. Let us now proceed to examine the first of these two sections.

The ordinary manner in which extra is charged in such cases is as follows:

The Medical Examiner, of the Office finding the proposer ineligible at the minimum rate, though not altogether uninsurable, recommends the acceptance of the risk with a certain number of years addition to the age; that, in fact, if the person be x years old, he be taken at the premium rate for, say age (x+t). Let us assume that the life is accepted on these terms, and that Px+t correctly represents the premium required to provide the benefit granted. Does it follow that on future occasions of dealing with the policy, the Office should treat it as if it were an ordinary assurance of entry age (ac+t)? I believe that this is the method generally adopted. Let us now consider how far it is justifiable.

If we have any number of quantities a, b, c, d, e, &c., such that a:b::c:d and e:6::f:d and g:b::h:d, then we know that a+e+g:6::c+f+h:d; but the converse proposition is not true —that if a +e+g:6::c+f+h:d it necessarily follows that


g:b::h:d --the reason evidently being that we can completely vary the values of a, e, and g, and also c, f, and h, without affecting their sum, so that like ratios between the individual quantities do not necessarily follow, though they may possibly do so.

To apply this to my subject. A person aged x opening a policy of assurance by annual premiums has from some cause to pay extra to such an extent that he is charged the premium which would be the normal rate at age (x+t). It does not follow that because his premium is the same as that of age (x + t) his chance of death in any individual year will be the same as that of a person of real age (x+t). Let dashes over the following symbols denote that mortality in accordance with extra is meant. Then

1 (disregarding loading) since Pxte=Px, we have

- d= 1


1 -d, or 1+0

itd., and therefore, az+e=d's; and similarly Ax+1=A'x.

1 + att

1 + Ag+t

Now, by hypothesis,
(!sto-bx+e+1)0+(!x++1 – Ix++2)v2 +

(1=-1&+1)0+(l’z+1-l’x+2)x2 +

l':(1+0's) (1x++-lx++1)0, (1x++1lx++2)v? i.e.,


Iste(1 + axte) 1x+r(1 +23+r)
(l's-lx+10 + (lx+1—1=+2)v2


l' (1 + Qxte) But as shown above, though the sum of the numerators of each side of this equation is to its common denominator as the sum of the numerators of the other side is to its denominator, it does not follow that each of the terms in one of these nuinerators is to its denominator as the corresponding term of the other numerator is to its denominator; and thus (lx+7--1x++1)

is not necessarily equal to lx+r(1+03+0)

(lx+o+1 –lx++2)v? to (l'x+1-(x+2) v2

Iz+(1+ ax+t) l' (1+03+)
Divide these ratios of possible inequality by It'orto

' I t'aste'

22 &c., respectively, and we have let-lx+e+

not necessarily equal to 's-1'3+1 le++) –lx+1+2 to lz+1–l'a+2

nor &c.; latt and generally, it does not follow that l'xta-l',


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So that if any life be charged an extra such that the premium paid is equivalent to that paid by a life t years older not subject to such extra, it does not follow that the chances of death in individual

after entry are the same in each case.

av + evé + gu3
Similarly, though

cv+fv? + hv3

it does not b

d a+e+g follow that


since, so far as we know, the b

d individual parts into which the numerators containing the powers of v may be split are quite arbitrary; and therefore, although the premiums to ages x and (2+t) are the same, the expectations of life at entry as well as afterwards in each case may be different.

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