to use these functions for the calculation of benefits, because for all ordinary tables the annuity values themselves are available. In computing tables of the annuity values, however, it is usual and convenient to form commutation columns in the first instance, and thereafter derive the final functions. Sometimes, also, when tables of temporary annuities are not available, the commutation columns afford an easy means of obtaining the required values.

QUESTIONS ON CHAPTER VII.

(1) By what two processes may we find the average present value of benefits depending on the duration of a human life? Show that they are in reality the same.

(2) Define:

(1) a pure endowment.

(2) a life annuity.

(3) an intercepted annuity.

Give the formula for each in terms of the numbers living and the rate of interest.

(3) Prove that ax = vpx(1+ax+1) and give a verbal explanation of this formula.

(4) How would you construct a table of life annuities, given the probability of living a year at each age?

(5) What are commutation columns"? State in terms of the D and N columns the values of ax, n\ax, n\max, and a(m)

(6) Find the approximate value of a(m) both by general reasoning and by mathematical process.

(7) Show how to find the value of a life annuity on a select life of age x; and state in terms of commutation symbols the value of n\ma[x].

(8) What single premium (American 31⁄2%) could be accepted in lieu of a semi-annual premium of $30 during life, at age 45? What net annual premium payable for 10 years only would be equivalent? (Ans. 1— $475.10; 2-$58.15).

(9) Given a column of the numbers living at each age and the values of v”, how would you construct by a continuous process the Dx column? (10) Give a formula for the net premium for a 20-Year Endowment Assurance at age x, assuming that no interest can be earned.

CHAPTER VIII.

NET PREMIUMS.

The Net Single Premium is the mathematical equivalent of the benefit according to the mortality table and rate of interest employed in the calculation. It is somewhat of the nature of the manufacturing cost of goods, and must be distinguished from the selling price or premium actually charged, which will be hereafter treated. In the preceding chapter annuities were discussed to an extent sufficient to explain the nature of the payments to be made for a life policy. We shall now proceed to obtain expressions for the value of the benefit or single premium, and thereafter find the net annual premium-which is the equivalent annuity. In all these calculations it is usual to assume that premiums are paid at the beginning of each year and that claims are paid at the end of the year of death. Yet, as a matter of fact, claims are usually paid as soon after the date of death as the necessary papers can be completed (see Chap. XII.). The assumption lends facility to the calculations, while the mortality table itself conforms to the same general principle.

Single The benefit under a Whole Life Policy will be Premium. best understood by considering that each one of the lx persons forming the group at age x effects a policy for I on his life. If this be done the number of deaths in the first year would be dx, and the value at the beginning of the year, of the claims by death due at the end would be vdx; the deaths in the second year would be dx+1, and the value of the claims at the beginning of the first year would be v2 dx+1; similarly for subsequent years. The total value of all the death claims would therefore be

vdx + v2 dx+1 + v3 dx+2 + &c.

If this present value be distributed amongst the lx persons we shall have as the share of each, the value of the benefit of the assurance of I at death. In technical language, we thus find the net single premium for a Whole Life assurance of 1, which is represented by Ax, A being the initial letter of the word "Assurance":

This is the value of the benefit to the life assured, represented by the payment of I at his death, and it therefore represents the smallest single premium which can be accepted by a company as the consideration for its undertaking such an obligation. In making this statement it is understood that a true rate of interest and a true mortality table will be used, both representing the actual experience to be realized by the company. The difference which these factors cause will be clear from the fact that 1000 A35 by the Northampton Table with 3% interest is 506.67 whereas the same value by the American Experience Table with 41% interest is only 293.35.

The above formula may be transformed in an instructive manner as follows:

vdx + v2 dx+1 + v3 dx+2 +

lx

v (lx−lx+1) + v2 (lx+1−lx+2) + v3 (lx+2−lx+3)+..

Assurance This shows that the value of the sum assured Relationship payable at death may be conveniently expressed in terms of an annuity payable during life, and the rate of interest. This formula again may be

to

Annuity.

mathematically transformed so as to produce other expressions giving the assurance value in terms of the annuity:—

This last formula is one of great importance, and may be explained verbally so as to assist the memory. The sum assured under the policy is I, and if it were payable at once its value would of course be I; but, as it is not payable until the end of the year of death, it follows that interest in advance for each year (d each year) must be deducted during the lifetime of (x). This interest is of the nature of an annuity-due, and its capitalized value is represented by the expression d (1 + ax), which, deducted from the full sum of I gives the present value of the payment due at the end of the year of death.

Annual Policies of life assurance are very seldom effected Premiums. by single premium: the more common form of policy is that by continued annual payments during life. In whatever form, however, the payments are received by the assurance company, the net equivalent value must be the same. If an annual premium be due at the beginning of each year, it is, as formerly explained, an annuity-due payable by the policy-holder to the company in consideration of the obligation the company incurs. If such net annual premium be represented by Px, the value of the total premiums throughout life would be Px (1 + ax), and this value must be the equivalent of the obligation incurred, namely Ax. We therefore have the formula:—

Px (1+ax) = Ax

whence the value of P, may be derived:—

(78)

Px

=

Ax

1 + ax

. (79)

The general principle above indicated, that the value of the benefit must equal the value of the payment, is an important one, and is the fundamental principle in old-line life assurance. Forgetfulness of this has been the cause of much disappointment in assessment and fraternal assurance.

It has been shown that the single premium may be expressed in terms of the annuity value and the rate of interest. By transformation the annuity value may be expressed in terms of the single premium and the rate of interest. For example, making use of formula (76) above

The annual premium P, may also be expressed in terms of annuity values or of single premiums, because we have

Dividing numerator and denominator of the right side by (I +i), we place this in a much neater form:—

Px

=

d Ax 1-A,

(83)