pating forms? What difference would you make in the case of nonparticipating policies, and why? (6) What special features are now considered important in tracing the mortality of assured lives, which were not dealt with in forming the Actuaries' or Combined Experience Table? (7) Find the annual premium for a twenty-payment Life policy, age 35, on a 3% basis, from the following data: Find qx for ages 35 to 38 inclusive, having given at 3% interest, a35 = 18.613 Ans. 935 = .00860 Ans. 937.00910 .00935 (8) Under-average lives have been assured with extra premiums payable during the continuation of the contract. Under what conditions, if at all, can the extra premium be remitted? (9) The extra hazard on account of occupation might be divided into two classes; namely, health hazard and accident hazard. Name a few occupations in each of these classes and state a method of dealing with them exclusive of placing them in a separate dividend class. (10) Mention any statistics with which you are acquainted which would be useful in rating sub-standard risks. (11) Explain what is meant by mean reserves. What assumption is made when they are used in making a valuation at the end of the year? (12) Show how the reserve on any policy at the end of n years can be expressed in terms of the net premium and the commutation symbols for age at entry and age attained. (13) A twenty-year Endowment has been in force five years and it is desired to change it to an Ordinary Life policy. What allowance should be made for the higher premium paid in the past? (14) If a policy-holder should desire to change his policy now on the twentyyear Endowment form to one on the twenty-payment Life form for the same amount and date of issue, a medical examination would be required. Why is this necessary? How would you adjust the premiums for such change? (15) A non-participating twenty-year Endowment policy was issued five years ago at a premium of $43.75 per thousand. Application is made to change it to a fifteen-payment Life fifteen-year dividend policy of the same date with a premium of $44.21. The reserve of the twenty-year Endowment is $181.72 and the reserve on the fifteen-payment Life is $154.03. Would you agree to the change and if so upon what terms? (16) Give an outline of what you consider an equitable method of distribution of dividends, and state why any arbitrary rule determined by the charter or otherwise becomes inequitable. (17) Investigate a convenient formula for ascertaining approximately the true rate of interest yielded by debentures terminable at the end of n years issued at a premium and redeemable at par. (18) Give a detailed account of the method you would adopt to calculate from the accounts of an assurance company, for use in comparison with others: (a) Expense ratios. (b) Rates of interest. State clearly why you consider your method the most equitable. (19) Deduce a formula in commutation symbols for the annual premium for an Endowment assurance for twenty years, furnishing $500 at death if in the first year, $525 at death in the second, $550 in the third, and so forth until the policy matures for $1000 if the assured survive the twenty years. (20) Under the Wisconsin law the present value of the premiums stipulated for a policy must not exceed the figures obtained by using a definite formula. Under the French law the premiums must not be less than certain results by formula. State your views of the principles involved in the two statutes above mentioned. (21) If 10% of renewal premiums be deducted from the total expenses and the balance of expense be tabulated as a percentage of new premiums of various companies, would the resulting figures give a fair comparison of the cost of new business? Give reasons. (22) The expenses of a life assurance company are to be analyzed, and the appropriate amounts charged against: (a) New Premiums; (b) Renewal Premiums; and (c) Interest and other earnings. Give the items of expense to be charged against each of these sources of income, with an outline of a process to determine the relative economy of operation. APPENDIX. PRINCIPLES OF NOTATION. The notation employed in actuarial science follows fixed rules, and if these are understood the expressions can be readily interpreted. So far as possible the initial letters of words have been employed as the general basis, such as i for interest; l for living; d for dying; p for probability of living; A for assurance; and a for annuity. In some cases it is not possible to use initial letters, and others have to be arbitrarily chosen, such as q, which represents the probability of dying, being the next letter to p, the probability of living. Further explanations are given in the text as occasion arises. These important central letters are modified by smaller letters or symbols to the left or right, and slightly below the basic letter; frequently also by letters in the upper right or left corners, and by marks placed above. The smaller letters, or subscripts, indicate the age of the life or lives under consideration, and may limit the time during which the contract would remain in force. The last letters of the alphabet, v, w, x, y, and z, are usually employed to indicate the ages of the lives under consideration. When one of these letters is placed in small brackets it relates to a person of that age; thus (x) means 'a person aged x," (y) "a person aged y," etc. lx 1x+n Px is the number living at age x. is the number living at age x + n. is the probability of living one year at age x. Px+n is the probability of living one year at age x + n. is the probability that both (x) and (y) will live one year. Note. In the above probabilities I is understood without being expressed as a subscript to the left of the basic letter. It is the time over which the probability is computed. The number of years over which a probability or benefit would extend is generally shown by the use of a letter, such as n, m, or r, either before or after the basic letter, as follows: is the probability that a life aged x will live n years. represent an annuity on (x) to run for n years, provided he live so long. is the value of the sum assured, payable either at the death of (x) or on the expiry of n years. It therefore indicates the present value of an endowment assurance. is an annuity running for n years certain, irrespective of any life. When the last survivor of two or more lives is n question, a horizontal bar is placed above the letters representing their ages, in the manner following: npxy is the probability that the survivor of (x) and (y) will live for n years. a xyz is an annuity payable during the joint lives of (x), (y), and (2), and continuing until the death of the last survivor. In order to indicate the survivorship of one life after another, such as the value of a benefit to (x) after the death of (y), an upright bar is used, separating the two lives as follows: aylx Payx is the present value of an annuity payable during the continued lifetime of (x) after the death of (y). is the annual premium for the same benefit. To indicate the order of death, a small number is used either below or above the letter representing the age. If the number is given above the letter, it also shows that the benefit is payable on the fulfilling of this condition. is the single premium for an assurance of I payable on the death of (x) if he die first, i.e. :—in the life time of (y). gives the value of an assurance payable on the death of (x) if he die third, provided that (z) shall have died first and (y) second. To indicate a deferred term, an upright bar is used to the left of the central letter. n❘ax is an annuity deferred n years, and payable during the subsequent lifetime of (x). The same function could be expressed under the rule given in a preced- . ing paragraph, thus:— To indicate the subdivision of a year, a small letter in brackets is generally used at the upper right corner. a(m) x P(m) x is an annuity to (x) payable m times a year. is the annual premium, payable m times a year, for an assurance on (x). To indicate a continuous or momentary benefit, a small bar is generally placed above the basic letter. Ax āx is the value of an Assurance payable at the moment of death of (x). is an annuity payable at infinitesimal intervals. To indicate that a benefit is complete, a small circle is placed above the basic letter. åx represents an annuity payable once each year, but with a proportionate payment from the last periodic payment to the date of death, thereby making it complete. The notation here outlined is capable of almost indefinite extension to meet varying circumstances; and, by the use of symbols, ideas can be expressed more accurately as well as much more briefly. They have a mathematical significance and rigid meaning which cannot be misunderstood, whereas the slightest turn of a phrase sometimes gives a different shade of meaning to a sentence expressed in words. It will be observed that one rule occasionally overlaps another, so that the same ideas can be expressed in two ways; but this is no disadvantage when the system is understood. |