LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Class 
 
GRADUATED EXERCISES 
 
 AND 
 
 EXAMPLES 
 
 FOR THE USE OF STUDENTS 
 
 OF THE 
 
 INSTITUTE OF ACTUARIES' TEXT-BOOK. 
 
 PART I, INTEREST (including Annuities-Certain), 
 PART II LIFE CONTINGENCIES (including Life Annuities and Assurances), 
 
 WITH SOLUTIONS. 
 
 BY 
 
 THOMAS G. ACKLAND, 
 
 FELLOW OF THE INSTITUTE OF ACTUARIES. FELLOW OF THE ROYAL STATISTICAL SOCIETY. 
 
 ACTUARY AND MANAGER OF THE GRESHAM LIFE ASSURANCE SOCIETY. 
 
 AND 
 
 GEORGE F. HARDY, 
 
 FELLOW OF THE INSTITUTE OF ACTUARIES. FELLOW OF THE ROYAL STATISTICAL SOCIETY. 
 
 ASSISTANT ACTUARY OF THE GENERAL REVERSIONARY AND INVESTMENT COMPANY. 
 
 <^7 
 
 OF THE 
 
 UNfVERSIT 
 
 LONDON : 
 
 CHARLES AND EDWIN LAYTON, 
 56, FARRINGDON STREET, E.G. 
 
 1889. 
 
INTRODUCTION. 
 
 THE following EXEECISES AND EXAMPLES are intended to supply 
 a systematic course of work for the use of Actuarial Students, 
 in practical illustration and application of the principles and 
 formulas laid down and developed in the INSTITUTE OP ACTUARIES' 
 TEXT-BOOK, PARTS I. and II. 
 
 The Examples upon Interest and Annuities- Certain have been 
 carefully selected to supplement those printed at the end of the 
 INSTITUTE OP ACTUARIES' TEXT-BOOK, PART I., and in further 
 illustration of the text and demonstrations of that work. 
 
 The Examples in Life Contingencies were originally intended 
 to be appended to the INSTITUTE OP ACTUARIES' TEXT-BOOK, 
 PART II., but their insertion was ultimately found to be 
 impracticable, owing to the already considerable bulk of that 
 treatise. It was also felt that the usefulness of such Examples 
 would bo greatly increased if they were accompanied by short 
 Solutions and references to the Text-Book. 
 
 The present volume has accordingly been undertaken, and is 
 now issued upon the sole responsibility of the joint Authors, 
 who have selected the Exercises and prepared the Solutions 
 here given, in the light of their practical experience during the 
 
 171603 
 
IV INTRODUCTION. 
 
 past six sessions (1883-4 to 1888-9 inclusive) as successive 
 Lecturers to Students of the Institute of Actuaries upon the 
 subjects of the Part II. Examination. 
 
 The Exercises and Examples have been selected from all 
 available sources in Actuarial literature, and largely from the 
 Examination Papers of past years. In a few cases, where a 
 practical illustration of some important theorem or demonstra- 
 tion treated in the Text-Book was not readily available, special 
 Examples have been prepared to meet the case. 
 
 In so considerable a body of work it is hardly probable 
 that errors will have been entirely eliminated. The greatest care 
 has, however, been taken to secure accuracy ; and the Compilers 
 desire to express their special indebtedness to Mr. WILLIAM 
 SMITH ANDERSON, A. I. A., and Mr. HARRY BEARMAN, A.I.A., 
 both of the GRESHAM LIFJE ASSURANCE SOCIETY, for valued 
 assistance in the careful examination of the Exercises and 
 Solutions, and in the revision of proof-sheets. 
 
 T. G. A. 
 G. F. H. 
 
 September, 1889. 
 
NOTES FOR THE STUDENT. 
 
 THE EXERCISES AND EXAMPLES are arranged in two main divisions 
 Interest and Life Contingencies corresponding respectively 
 with Parts I. and II. of the INSTITUTE OF ACTUARIES' TEXT-BOOK. 
 The chapters into which the Exercises are divided correspond 
 throughout with those similarly numbered in the Text-Book; 
 and the Exercises contained in each chapter are arranged, as 
 far as practicable, in the order in which the subjects are treated 
 and developed in the corresponding chapter in the Text-Book. 
 At the foot of each page of the Exercises and Examples is 
 appended a reference to the pages, later on in the present 
 volume, upon which the Solutions of such Exercises will be 
 found. 
 
 The references, in the Solutions, to figures in square 
 brackets thus, [16] indicate the numbered paragraphs in 
 the Text-Book in which the subject dealt with in the particular 
 Example is treated; and the paragraphs so referred to will be 
 found (unless otherwise stated) in the chapter of the Text-Book 
 then under discussion. Any additional hints by way of solution, 
 or references to other authorities, are appended for the guidance 
 of the Student, and the further elucidation of the points raised 
 in the several Exercises and Examples. 
 
 References to the Journal of the Institute of Actuaries are 
 indicated by the letters J.I. A. 
 
TABLE OF CONTENTS. 
 
 PART I. INTEREST (INCLUDING ANNUITIES-CERTAIN). 
 
 EXAMPLES. SOLUTIONS. 
 
 CHAPTER. Page. Page. 
 
 I. INTEREST, AMOUNTS, PRESENT VALUES, AND DISCOUNT 1 39 
 
 II. ANNUITIES-CERTAIN : 
 
 (i) Interest convertible yearly, and annuity pay- 
 able yearly 2 41 
 
 (ii) Annuity payable and interest convertible at 
 the same period, the period being less than 
 a year 3 43 
 
 (iii) Annuity payable and interest convertible at 
 
 periods of different duration ... ... 4 44 
 
 III. VARYING ANNUITIES 5 45 
 
 IV. DETERMINATION or THE RATE or INTEREST: 
 
 (i) On the determination of the rate of interest 
 where the amount of capital repaid is the 
 same as that advanced ... ... ... 6 48 
 
 (ii) On the determination of the actual rate of 
 interest paid by a borrower, where the 
 amount of capital repaid is different from 
 the amount advanced ... ... ... 6 48 
 
 VII. INTEREST TABLES 7 51 
 
 MISCELLANEOUS EXAMPLES 8 52 
 
 PART II. LIFE CONTINGENCIES (INCLUDING LIFE ANNUITIES 
 AND ASSURANCES). 
 
 I. THE MORTALITY TABLE 9 56 
 
 II. PROBABILITIES OF LIFE 10 57 
 
 HI, EXPECTATIONS OF LIFE 12 61 
 
Vlll TABLE OF CONTENTS. 
 
 EXAMPLES. SOLUTIONS. 
 
 CHAPTER. Page. Page. 
 
 IV. PROBABILITIES or SUBVIVOESHIP 13 64 
 
 V. STATISTICAL APPLICATIONS OF THE MOBTALITT 
 
 TABLE ... 14 66 
 
 VI. FORMULAS OP DE MOIVRE, GOMPEBTZ, & MAKEHAM, 
 
 FOB THE LAW OF MOBTALITT ... ... ... 15 67 
 
 VII. ANNUITIES AND ASSUBANCES ... 15 69 
 
 VIII. CONVEBSION TABLES FOB SINGLE AND ANNUAL 
 
 ASSUBANCE PBEMIUMS 18 73 
 
 IX. ANNUITIES AND PBEMIUMS PAYABLE FEACTIONALLY 
 
 THBOTTGHOUT THE YEAB 18 74 
 
 X. ASSURANCES PAYABLE AT ANY OTHEB MOMENT THAN 
 
 THE END OF THE YEAR OF DEATH 19 76 
 
 XL COMPLETE ANNUITIES 20 77 
 
 XII. JOINT-LIFE ANNUITIES 20 78 
 
 XIII. CONTINGENT, OB SURVIVORSHIP, ASSURANCES ... 21 79 
 
 XIV. REVERSIONARY ANNUITIES 22 82 
 
 XV. COMPOUND SURVIVORSHIP ANNUITIES AND ASSUR- 
 ANCES 24 85 
 
 XVI. COMMUTATION COLUMNS, AND THEIR APPLICATION 
 TO VARYING BENEFITS, AND TO RETURNS OF 
 
 PBEMIUM 24 85 
 
 XVII. SUCCESSIVE LIVES 26 89 
 
 XVIII. POLICY- VALUES 27 89 
 
 XIX. LIFE INTEEESTS AND REVEBSIONS 29 94 
 
 XX. SICKNESS BENEFITS 30 98 
 
 XXL CONSTRUCTION OF TABLES 31 99 
 
 XXII. FORMULAS OF FINITE DIFFERENCES 32 102 
 
 XXIII. INTERPOLATION 34 107 
 
 XXIV. SUMMATION 36 115 
 
Chap, I.] EXERCISES AND EXAMPLES. 
 
 GRADUATED EXEECISES AND EXAMPLES 
 PART I. INTEREST (INCLUDING ANNUITIES-CERTAIN). 
 
 CHAPTER I. INTEREST, AMOUNTS, PRESENT VALUES, AND 
 DISCOUNT. 
 
 (1). Give the two formulas which express interest for a fractional 
 period, and state what assumptions are made in each case, and the 
 advantages and disadvantages of each formula. 
 
 (2). Write down the formula for the amount of interest accrued 
 
 on a sum at the end of the th of a year, at the effective rate of i 
 
 m 
 
 per annum, and prove that such accrued interest is less than the simple 
 interest for the same period at the rate i. How would you account 
 for this ? 
 
 (3). State clearly the meaning of the term "force of interest", 
 and obtain a formula exhibiting its relation to the effective rate of 
 interest. 
 
 (4). Show that the instantaneous rate equivalent to a yearly rate 
 of * is approximately equal to the arithmetic mean between the theoretical 
 and commercial discount on 1 for a year, the rate of interest being i. 
 
 (5). Write down the present value of a sum due 3, 6, and 9 | 
 months hence. State your reasons for the formulas adopted. 
 
 (6). Given log 10 =2 '3026, show that 1 will amount to 10, at 
 1 per-cent, in 23 If years. 
 
 B 
 
 Solutions, pp. 39, 40. J 
 
PAET I. INTEREST. [Chaps. I, II. 
 
 (7). Write down expressions for the present value and amount of 
 1 at the end of n years, interest being reckoned at the nominal rate of i, 
 convertible m times a year. What do these expressions respectively 
 become when m=cc ? 
 
 (8). Given tables of the functions (1/0125)* and (T0125)-" for 
 integral values of n t how would you employ them to ascertain (a) the 
 amount of 100 in 10 years, interest being reckoned at the rate of 5 
 per-cent per annum, payable quarterly ; (/?) the present value of 100 
 due 15 years hence, interest being reckoned at the rate of 2-g- per-cent 
 per annum, payable half-yearly ? 
 
 (9). State symbolically the difference between discount at simple 
 interest, discount at compound interest, and commercial discount. 
 
 1 fj.i& 
 
 (10). Show that 8= - ~ 
 
 v x dx 
 
 (11). A gives B a bill for a due at the end of m years, in 
 discharge of a bill for b due at the end of n years. For what sum 
 should B give A a bill due at the end of p years to balance the account ? 
 
 CHAPTER II. ANNUITIES-CERTAIN. 
 t 
 
 (i) Interest convertible yearly, and annuity payable yearly. 
 
 (12). Find the relation between the discount of any sum payable 
 n years hence, and the present value of an annuity-certain for n years 
 of the same amount. 
 
 (13). An annuity-certain of 729 a year is granted for 25 years, 
 the rate of interest at 5 per-cent. Calculate the value of such annuity, 
 given, log 2 = -30103; Iog7='84510; Iog3 = '47712; log 2952=3-47012; 
 log 2953 =3-47026. 
 
 (14). What sum would have to be deducted from the first 
 payment of the annuity in the preceding question, if the first payment 
 is made three months hence, the second in fifteen months, &c.? 
 
 (15). Show how the several payments of an annuity-certain may be 
 divided into principal and interest, and demonstrate that the amount 
 remaining outstanding at the end of any year is equal to the present 
 value of an annuity-certain for the remaining term. 
 
 (16). Show that the amount of purchase-money of an annuity- 
 certain for n years which is unpaid at the end of m years, is equal to 
 
 [Solutions, 
 
Chap. II.] EXERCISES AND EXAMPLES. 3 
 
 the difference between the accumulated amount of the purchase-money 
 for m years, and the accumulated amount of the annuity in the same 
 time. 
 
 (17). An insurance company lends 50,000, repayable by an 
 annuity-certain for 25 years. How much capital is unpaid at the 
 end of 20 years, reckoning interest at 5 per-cent ? 
 
 (18). In the above case construct a table showing how much of 
 each annual instalment of annuity consists of interest, and how much 
 of repayment of capital. 
 
 (19). Show how to find the amount of each payment of an annuity- 
 certain for n years, which is purchased for a sum of V, the purchaser 
 wishing to invest his capital at a rate of interest *', and to replace that 
 capital at a rate of interest i. 
 
 (20). Show clearly into what component parts an annuity-certain 
 may be divided, and give the means of determining how much of each 
 payment goes to repay the capital originally invested, in the case where 
 the rate of interest at which that part of each payment which goes to 
 repay capital is invested, is different from the rate of interest realized on 
 the original investment. 
 
 (21). Find the present value of a perpetuity of 1, and hence deduce 
 the present value of an annuity-certain for n years. 
 
 (22). A lease of the annual value of 1 is granted for m years. 
 After it has been n years in force, the lessee requires to extend the 
 remaining term to m years. How much ought he to pay ? 
 
 (23) . Given the present value, at rate of interest , of an annuity 
 of 1 deferred d years, investigate a formula to find n, the number of 
 years it has to run. 
 
 (24) . Four persons, A, B, C, and D, contribute equal sums towards 
 the purchase of a perpetuity. Find the number of years that A, B, and 
 C may successively enjoy it, D having the absolute reversion, so that all 
 four may benefit equally. 
 
 (ii) Annuity payable and interest convertible at the same period, the 
 period being less than a year. 
 
 (25) . Find the present value and amount of an annuity-certain for 
 n years, at a nominal rate of interest of i, convertible in times a 
 year. 
 
 (26) . From the above, deduce the present values and amounts of 
 
 B 2 
 
 pp. 40-43.] 
 
4 PART I. INTEREST. [Chap. II. 
 
 * 1 
 
 continuous annuities-certain. Show how a table of the values of - - , 
 
 x 
 
 x being the argument, enables us to calculate such values. 
 
 (27). If two annuities-certain of 1 per annum for n years are 
 purchased, in one of which the annuity is payable and interest 
 convertible yearly, while in the other the instalment is payable and 
 interest convertible m times a year, the effective rate of interest being 
 the same in the two cases, what is the relation between the amount of 
 capital repaid in the till year under the two annuities respectively ? 
 
 (28) . Give a formula expressing the value of an annuity-certain of 
 1 for n years, payable m times a year, in terms of the value of an 
 annuity-certain of 1 for n years payable yearly, the nominal and the 
 effective rates of interest. What does the formula become when m=<x> ? 
 
 (29). If x be the nominal rate of interest, the value of a perpetuity 
 
 is equal to -, however often interest is convertible. Under what 
 
 CO 
 
 condition is this true, and how would you explain it ? 
 
 (30). Find the value of an annuity to run for five years, interest 
 and instalment payable half-yearly, with interest at the nominal rate of 
 
 5 per-cent, and show the amount of capital redeemed in each half-yearly 
 payment. 
 
 (31). The guardians of a poor-law union are desirous of borrowing 
 5,000, the loan to be discharged and the interest paid by an equal 
 half-yearly charge upon the rates, extending over 30 years. Assuming 
 5 per-cent interest, find the amount of the half-yearly charge, and draw 
 up a schedule showing what portions of each of the first four payments 
 are applicable for the payment of interest and the discharge of the capital 
 account. 
 
 (iii) Annuity payable and interest convertible at periods of 
 different duration. 
 
 (32). Write down the formulas for the present value and amount 
 of an annuity-certain for n years, instalment payable k, and interest m 
 times a year, in terms of the effective rate of interest i, and also of the 
 nominal rate of interest x. 
 
 (33) . Given the formula for the present value of an annuity-certain 
 for n years, instalment payable k, and interest m times a year, show 
 how the expression will be modified in the several cases where m=l, 
 
 [Solutions, 
 
Chaps. II, III.] EXE11CISES AND EXAMPLES. 5 
 
 & = 1, wi= = l, w=co, 7=co, w = & = co, M = l, & = oo, =1, 
 
 W = GO . 
 
 (34). Under what differing conditions are the following formulas, 
 (a), OS), for the present value of a continuous annuity-certain for n 
 years, correctly stated ? What does the formula (y) represent ? 
 1 f-5n 1 f-in 1 f -5n 
 
 w-nr-' cfl'-f-, w-^- 
 
 (35) A loan of X is to be discharged by an annuity (made up of 
 
 X 
 
 principal and interest) of , payable at the end of each year, the 
 
 interest thereon being at i per unit per annum, convertible half-yearly. 
 When will the debt be extinguished ? 
 
 CHAPTEE III. VARYING ANNUITIES. 
 
 (v \ r 
 J , stopping at the term involving v n . 
 
 (37) . Find the present value of an annuity-certain, the first payment 
 being 1, the second 1'2, the third 1'4, and so on, increasing '2 each year; 
 the last payment being 10. 
 
 (38) . Find the value of an annuity-certain for 21 years, the first 
 payment of which is 1, and the after-payments of which increase by 
 aVfch each. 
 
 (39) . Find the value of an annuity-certain for n years, the payments 
 of which are I 3 , 2 3 , 3 3 , &c. 
 
 (40). State a general formula for the value of an annuity-certain 
 for n years, whose successive payments are u t , u 2 , u 3 . . . u n ; and explain 
 how the values of annuities of the figurate numbers can be employed in 
 determining such values. 
 
 (41) . Deduce a formula for the present value of a perpetuity whose 
 successive payments are the figurate numbers of the rth order. 
 
 (42). Find, by the method indicated by Mr. Peter Gray (J.I.A., 
 xiv, 91), the value of an annuity to run for 40 years, the successive 
 payments of which are 1, 3, 5, 13, 33, . . ., interest being reckoned at 
 5 per-cent. 
 
 (43). Find the value, at 5 per-cent, of an annuity for 40 years, 
 whose several payments are 55, 126, 259, 484, 837, . . , 
 
 pp. 43-48.] 
 
6 PART I. INTEREST. [Chaps. Ill, IV. 
 
 (44). Find the value of an annuity for 15 years, whose successive 
 payments are 201, 303, 443, 630, 874, . . . 
 
 CHAPTER IV. 
 
 (i) On the determination of the rate of interest where the amount of 
 capital repaid is the same as that advanced. 
 
 (45) . An annuity-certain for 100 years is worth 19 years' purchase ; 
 find the rate of interest by formulas (A), (B), (C), (D). 
 
 (46). An annuity-certain for 25 years is worth 17 years' purchase ; 
 find the rate of interest by formulas (Dj) and (D 2 ). 
 
 (47). Show how to approximate to the rate of interest in an 
 annuity-certain in cases where the term is long, and hence deduce the 
 rate of interest when #941=: 27. 
 
 48. Giye at least three formulas for approximating to the rate of 
 interest in an annuity-certain, and state which you would select as likely 
 to give the most satisfactory result without excessive labour. 
 
 (ii) On the determination of the actual rate of interest paid by a 
 borrower, where the amount of capital repaid is different from 
 the amount advanced. 
 
 (49). Deduce Makeham's formula for the value, at rate of interest 
 i', of a loan bearing interest at rate *, and explain it verbally. What 
 are the advantages of the use of this formula ? 
 
 (50). A loan of s is repayable at par in n years, and bears 
 interest meanwhile at the rate i. Deduce a formula for the amount that 
 could be given by a purchaser in order that interest at the rate of j may 
 be realized upon the amount invested. 
 
 (51). Obtain a formula for the present value of the capital redeemed 
 in the successive payments of an annuity-certain of 1 for n years. 
 
 (52). A loan is repaid by a sinking fund of p per-cent on the sum 
 borrowed : find how long it will take to repay the loan, i and i' being 
 the rates of interest which the borrower pays, and at which the sinking 
 fund is invested, respectively. 
 
 (53), A debenture, redeemable in n years, and bearing interest at 
 a fixed rate i, is purchased at a premium of p per-cent. Show how to 
 find approximately the rate of interest l\ realized by the purchaser. 
 
 [Solutions, 
 
Chaps. IV, VII. ] EXERCISES AND EXAMPLES. 7 
 
 (54). If a bond of 1, repayable at par in n years, and bearing 
 interest at the rate i, be purchased at the price of l+p, so that the 
 purchaser may realize interest at the rate i' upon his investment, 
 with the return of the capital invested at the end of the term, show that 
 the excess interest received annually ['(! H-J*)*' 1 ] must be invested at 
 the rate *' to amount to p at the end of the n years. If this excess 
 interest can only be invested at the rate j, how will the value of p, the 
 premium to be paid for the bond, be affected ? 
 
 (55). A bond for 1,000, bearing interest at 3 per-cent for 20 
 years, is to be sold. What can a purchaser give to realize 5 per-cent 
 from his investment (a) supposing the bond to be repayable at par in 
 20 years, (ft) supposing the bond to be repayable in 20 annual instal- 
 ments. Given #20; at 5 per-cent=12'4622. 
 
 (56) . The tenant in possession of an estate has to pay off a charge 
 on the estate in the next n years, in the following way, viz. : M being 
 
 the amount of the charge, he has to pay back each year , and interest 
 
 at the rate of i on the amount unpaid at the beginning of the year. 
 Find the sum for which his payments might be commuted, taking 
 interest at the rate^'. 
 
 (57). A person spends in the first year m times the interest on his 
 property ; in the second year, 2m times ; in the third year, 3m times, 
 and so on ; and at the end of 2p years has nothing left. Show that in 
 the ^th year he spends as much as he had left at the end of that year. 
 
 (58). Find the rate of interest in a loan issued at 76 per-cent, with 
 interest at 6 per-cent upon the nominal amount of the loan, and repay- 
 ment of the nominal capital by an accumulative sinking fund of 1 
 per-cent per annum. 
 
 CHAPTER VII. INTEREST TABLES. 
 
 (59). Explain the methods of calculating and verifying tables of 
 present values of sums and annuities-certain. 
 
 (60). Calculate, at 5 per-cent, the present value of 1 due in any 
 number of years from 1 to 20, and also the present values of annuities- 
 certain for any number of years from 1 to 20, verifying the calculation 
 in each case. 
 
 (61). Show how tables where the argument is logo? and the tabular 
 
 pp. 48-51.] 
 
8 PABT I. INTEREST. [Chap. VII, 
 
 results log (lo?) and log(l + #) can be made available for the calcula- 
 tion of the amounts and present values of annuities-certain. 
 
 (62) . Calculate by the tables referred to in the previous example 
 the present values and amounts of annuities -certain for any number of 
 years from 1 to 10, taking 5 as the rate of interest. 
 
 (63) . State and prove the rules for finding from the tabular values 
 and amounts of annuities-certain, the values and amounts of the same 
 when payable in advance. 
 
 (64). How would you proceed to verify the columns (1 + *')** v n } 
 Sn ) and dn\ , in a printed table of such values ? 
 
 MISCELLANEOUS EXAMPLES. 
 
 (65). A 100 share, bearing dividends at 5 per-cent per annum in 
 June and December, with an annual bonus of 3 in December, is bought 
 for 130, just after the payment of the December dividend and bonus. 
 What is the effective rate of interest made upon the investment? 
 
 (66). In the previous question, what would be the equivalent price 
 just previous to the June dividend ? 
 
 (67). An assurance fund at the commencement of a year amounts 
 to 1,000,000; the income from interest is 45,000, from premiums 
 and other sources 200,000, the outgo 170,000. What is the rate of 
 interest earned by the fund 
 
 (a) assuming the fund to increase at a uniform rate through- 
 out the 3 r ear ; 
 
 ((3) assuming the income (excluding interest) , and the outgo, 
 to be evenly distributed through the year ? 
 
 (68). If 100 amounts to 106-1678 in a year, at a nominal rate of 
 interest of 6 per-cent: required to find how often interest must be 
 convertible. 
 
 (69). If 2 per-cent Consols are bought for 94, what are the 
 nominal, effective, and instantaneous rates of interest ? 
 
 (70). Show that the present value of an annuity of 1 for n years, 
 at simple interest at rate i is approximately equal to 
 
 N-{ ; 
 
 [Solutions, 
 
Chap. IH EXERCISES AND EXAMPLES. 
 
 GRADUATED EXERCISES AND EXAMPLES. 
 PART II. LIFE CONTINGENCIES (INCLUDING LIFE ANNUITIES AND 
 
 ASSUEANCES). 
 
 CHAPTER I. THE MOETALITT TABLE. 
 
 (1). Explain what is meant by a Table of Mortality, and describe 
 its usual and convenient form. 
 
 (2). If the experience of a given mortality table indicates that, 
 out of 2,000 persons alive at age 30, 29 die before attaining age 31, is 
 it theoretically correct to say that the probability of a person aged 30 
 
 29 
 
 dying before age 31 = ^-^ ? 
 
 (3). Having given a complete table of p X t accurately representing 
 the probabilities of life at all ages, show how, from the deaths taking 
 place in one year, to calculate approximately the total number living in 
 a stationary population, where there is no disturbance from immigration 
 or emigration. 
 
 (4). If m x = TTrTTN , show that 
 
 (5). (a) Explain the method of forming a table of mortality from 
 the death registers of a place, correcting for the 
 increase of population. 
 
 (/?) If such correction is disregarded, what would be the 
 effect upon the resulting Mortality Table ? 
 
 pp. 51-56.1 
 
10 PART II. LIFE CONTINGENCIES. [Chap. II. 
 
 CHAPTER II. PROBABILITIES OF LIFE. 
 
 (6). If n p x denote the probability that a person aged x will live 
 n years, and p x + r the probability that a person aged (# + r) will live 
 one year, prove that 
 
 (7). Show that l x +n=npx(d x +d x+l + .... +d M _ l ). 
 
 (8). Supposing a given number of marriages contracted between 
 males aged 30 and females aged 25, find the proportion per-cent of the 
 original number who will survive as married couples, widowers, or 
 widows, at the end of 10 years, assuming the probability of dying within 
 
 265 237 
 
 10 years at the age of 30 to be , and at the age of 25 to be 
 
 (9). Find the following probabilities, namely, that of two lives 
 
 (#) and (y) 
 
 (a) both will not survive n years ; 
 
 (/:?) either or both will survive n years. 
 
 (10). Determine (a) the probability that two persons now aged x 
 and y will both die in the nth year from the present time ; (/3) the 
 probability that one only of them will die in that year. 
 
 (11). Find the value of ' B g^p and n -i<fa. 
 
 (12). Show that the probability that (#) will survive (n) years, and 
 that (y) will survive (nY) years (*pxXn-iPv) may be expressed in 
 
 either of the forms n-\px+\-.y xpx or ' g ' y ~ 1 
 
 Pv-i 
 (13). Find the probability that one at least of three lives (or), (y) 
 
 (2;), will fail between the nth and (n + m)th year. 
 
 (14). Set down the probabilities of the various contingencies as to 
 death and survival which may happen to three lives during a year 
 (disregarding order of survivorship) ; and prove the truth of your 
 answer. 
 
 (15). Obtain the probability that out of three lives (#), (y), (z), 
 
 (a) one at least will fail in the nth year ; 
 
 (/?) not more than two will fail in the nth year ; 
 
 (y) the three lives will fail in the following order : one 
 
 before the nth year, one in the nth year, one after 
 
 the nth year. 
 
 [Solutions, 
 
Chap. II.] EXE11CISES AND EXAMPLES. 11 
 
 (16). Find the probability that at least two lives out of four lives 
 will survive a year. Give a general expression for the probability that 
 at least r lives out of m lives will survive n years. 
 
 (17). If the chance of any one of n persons dying in the course of 
 a year be represented by p, prove that the chance of exactly r of them 
 
 \n 
 dying in the course of a year will be . - >p r (I p) n ~ r . 
 
 \r \n r 
 
 (18). On the supposition in the previous question, prove that the 
 most probable number of deaths in the year is the greatest integer 
 contained in (ii + V)p. 
 
 (19). What is the probability that, out of seven individuals of a 
 given age, four at least will die in a given time ? 
 
 (20). A number n of persons, all of the same age, are each insured 
 for the same sum, 1 ; the probability of any one of them dying in a 
 year being <?, show how to find the probability of the insurance company 
 sustaining a given loss in the course of the year, and prove that the sum 
 of the products of each possible loss into its probability is nq. 
 
 (21). Two offices have each 1,000,000 assured: Office (a) by 
 100 policies of 10,000 each ; Office ((3) by 1,000 policies of 1,000 
 each. Assuming all the ages equal, and the rate of mortality to be 
 2 per-cent per annum, give an expression for the probability in each case 
 that the claims in one year will amount to 30,000 at least. 
 
 (22). If p be the probability that a person aged x will live one 
 year, find expressions for the following probabilities : That out of 1,000 
 persons aged x 
 
 (a) exactly 20 will die in a year ; 
 
 (/?) not more than 20 will die in a year ; 
 
 (y) 20 designated individuals, and no more, will die in a 
 
 year; 
 (8) 20 designated individuals, at least, will die in a year. 
 
 (23). Six persons, A, B, C, D, E, F, are of the same age, the rate 
 of mortality at that age being 1 per-cent. What is the probability (to 
 five decimal places) 
 
 (a) that they will die in an assigned order ? 
 
 (/5) that A will die first, and F last, the remaining order 
 
 not being fixed ? 
 (y) that A will die in the first year, and be the first to die ? 
 
 pp. 57-5& 
 
12 PAET II. LIFE CONTINGENCIES. [Chaps. II, III. 
 
 (24). Define "rate of mortality" (^), "force of mortality" (^), 
 "central death-rate" (wij.), and give expressions showing approximately 
 the relation between these three functions. 
 
 (25) . Assuming the force of mortality at age x to be approximately 
 
 equal to , show that when the decrements of the mortality 
 
 Alx 
 
 table are increasing, p x <q x , and when decreasing, q x <^x- 
 
 (26). Show that f i a .= - i |- l x . 
 l x dx 
 
 (27). Prove that the two expressions for the force of mortality, 
 
 7- -r-'lx and log 6 l x , are identical. 
 
 l x dx dx 
 
 (28). If the force of mortality at age (# + !)= - -, what 
 is the rate of mortality at age x in terms of it ? 
 
 (29). Show that ^=i 
 
 lx 
 
 (30). Show that /^=j 
 
 CHAPTER III. EXPECTATIONS OF LIFE. 
 
 (31). Define (a) average duration of life, (/3) probable lifetime, 
 (y) mean age at death ; and give the value of each in symbols. 
 
 (32). Given the following mortality table, deduce the average 
 duration of life at each age, the average age at death, and the "probable 
 lifetime " : 
 
 Age. 
 
 No. Living. 
 
 Age. 
 
 No. Living. 
 
 80 
 
 116 
 
 90 
 
 21 
 
 81 
 
 97 
 
 91 
 
 15 
 
 82 
 
 78 
 
 92 
 
 12 
 
 83 
 
 64 
 
 93 
 
 9 
 
 84 
 
 56 
 
 94 
 
 7 
 
 85 
 
 51 
 
 95 
 
 5 
 
 86 
 
 39 
 
 96 
 
 3 
 
 87 
 
 33 
 
 97 
 
 1 
 
 88 
 
 29 
 
 9fe 
 
 
 
 89 
 
 24 
 
 
 
 (33). The rate of mortality at each age being given, show how to 
 construct therefrom a table of the mean duration or expectation of life. 
 
 (34). Given a table of mortality, show how to find (a) at what age 
 it is most probable a person of a given age will die ; (/3) how many 
 years he has an even chance of living. 
 
 [Solutions, 
 
Chaps, in, IV.] EXERCISES AND EXAMPLES. 13 
 
 (35). Show, upon De Moivre's hypothesis, (a) that the average 
 duration of life at any age equals the probable after-lifetime ; (/?) that 
 at any age the probability of dying in each of the subsequent years is 
 the same; (y) that the force of mortality is equal to the rate of mortality 
 at all ages. 
 
 (36). Prove that 
 
 (37). Find the curtate expectation of a life (#) after n years, and 
 
 during the following m years ( n \m^x) What correction would you apply 
 to obtain the expression for the complete expectation during the same 
 term? 
 
 (38). Deduce p x in terms of e x . Prove that if c x -i=e x =e x+l1 
 then p x _ l =. p x ; but that if p x -i=p x =p x +i, then e x -\ will not be equal 
 
 1 
 
 to e x unless e x +% = - 
 
 (39) . Show that the average duration of life 
 
 (40). Find an expression for the average age at death of l x 
 persons. 
 
 (41). Explain what is meant by the curtate expectation of two 
 joint lives (x) and (y). Prove that it is equal to 
 
 CHAPTER IV. PROBABILITIES or SURVIVORSHIP. 
 
 (42). Find the probability that (#) will die in the nih year, (y) 
 being alive at the moment of (#)'s death. 
 
 (43). Show that the probability of (#) dying before (y) in the nih 
 year, when no assumption is made as to the distribution of deaths 
 throughout the year, may be expressed in the form 
 
 dx+nlly+n-l . dy+ 
 
 l2l x ly 
 
 (44) . Find the probability that (#) will die before (y) . 
 
 pp. 59-64.] 
 
14 PART II. LIFE CONTINGENCIES. [Chaps. IV, V. 
 
 (45). Assuming that the deaths in each year are uniformly dis- 
 tributed, and supposing that two persons O) and (y) both die in the 
 same year, prove that the chance of (x) dying before (y) is accurately i. 
 
 (46). Show that the probability of a life (x) dying before a life (y) 
 may be expressed by the formula 
 
 (47). Find the probability of a person aged x dying before one 
 aged y, or within t years after. 
 
 (48). Find expressions for the following probabilities : 
 
 (a) That (y) will be alive t years after the death of (x) . 
 () That (y) will be alive at the end of the tih year 
 succeeding the year of the death of (#) . 
 
 (49). Find the value of <, <, and QJ, 3 . 
 
 CHAPTER V. STATISTICAL APPLICATIONS OP THE MORTALITY TABLE. 
 
 (50). Given the values of l x and e x at every age, show how to 
 construct the columns L^, N'a?, T#, and Y^, and explain generally the 
 uses to which such columns may be applied. 
 
 (51). A community proposes to establish pensions for such of their 
 number as attain a certain age, to be provided by equal annual subscrip- 
 tions payable up to that age. How would you proceed to find the 
 amount of such subscriptions ? 
 
 (52) . A body of men, kept of uniform strength by annual entrants 
 engaged at age 20, and superannuated at age 60, are grouped in three 
 classes, the members in the classes being as three in the junior, two in 
 the intermediate, and one in the senior. Assuming promotion to go by 
 seniority, how can a mortality table be used for finding the average age 
 of promotion from one class to another? What causes may be in 
 operation to vitiate the results ? 
 
 (53) . Show that in a stationary population the average present age 
 at any time is equal to the average expectation of life at age and 
 upwards. 
 
 (54) . In a stationary community there are 209 deaths annually to 
 each 10,000 inhabitants. What is the average age at death ? 
 
 [Solutions, 
 
UNIVERSITY 5 
 j^x 
 
 Chaps, VI, VII.J EXERCISES AND EXAMPLES. 15 
 
 CHAPTER VI. FORMULAS OF DE MOTVHE, GOMPERTZ, AND 
 MAKEIIAM, FOR THE LAW OF MORTALITY. 
 
 (55). Explain any advantages which would accrue from being able 
 to express the number living at any age, out of a given number born, as 
 a function of the age, and give examples of any formulas that have been 
 suggested for this purpose. S 
 
 (56). if a mortality table were such that the numbers living formed 
 a series in geometrical progression, what would be the nature of the 
 tables giving the chance of dying in a year, and the expectation at 
 any age ? 
 
 (57). Upon what hypothesis as to the law of mortality is 
 Gompertz's formula based? Prove that if this hypothesis is assumed, 
 the formula of Gompertz for the number living at a given age x will be 
 obtained. 
 
 (58). Upon what modification of Gompertz's hypothesis as to the 
 law of mortality is Makeham's formula based ? Show that by adopting 
 this modified hypothesis, the formula of Makeham for the value of l x 
 will be obtained. 
 
 (59). Given l x =ks x (g) c * , find the values of log^ and of p x . ^ 
 
 (60). Given a table of the values of log^ at all ages, state 
 generally how you would proceed to deduce values for the constants 
 k, 5, g, and c, involved in Makeham's formula, l x -=Jcs x {g) cX . 
 
 CHAPTER VII. ANNUITIES AND ASSURANCES. 
 
 (61). Find by the Life Table given in the "Text-Book", taking 
 interest at 3 per-cent, the present value of a sum to be received by a 
 person aged 10 provided, he be living at age 21. 
 
 (62). Find the value of an annuity on a single life, and show how 
 the annuity at any age can be expressed in terms of that at the next 
 higher age. 
 
 (63). Prove that the value of 1, to be received immediately upon 
 the decease of a person of a given age, is greater than that of 1 
 receivable at the expiration of a number of years certain equal to the 
 complete expectation of life at that age. 
 
 pp. 65-69, J 
 
16 PART II. LIFE CONTINGENCIES. [Chap. VII, 
 
 (64). Assuming that the present value of a life annuity has been 
 demonstrated, show clearly and logically what is the present value of 1, 
 payable at the end of the year of death, finding equations connecting a x 
 and A x in terms of v, d, and a m respectively. 
 
 (65). Given the formula \=ia x +(~L + i)K x , find an analogous 
 expression involving the values of temporary benefits, and thence deduce 
 the value of a term assurance. 
 
 (66). Give a verbal interpretation of the formulas 
 
 (a) \ n A x = 
 
 ((3) A X n\ = v dln-tdx. 
 
 (67). Prove that the single premium for an assurance payable on a 
 life now aged cc years attaining the age of (x + f) or dying previously 
 
 may be expressed by the formula . , and show that this is equal 
 to M *- 
 
 (68). Describe the construction of the D and N columns for joint 
 lives proposed by Professor de Morgan, and applied by Dr. Parr to the 
 English Life Table No. 3, and point out in what respects it differs from 
 that previously used. 
 
 (69). Give formulas for the folio wing, using commutation symbols : 
 
 (a) Annual premium payable during m years only for an endow- 
 ment assurance on (#) payable at (x+m + ri) or previous 
 death. 
 
 (ft) Single premium for a temporary assurance, payable in the 
 event of (x) dying within n years. 
 
 (y) Annual premium for a whole-life assurance on (#), deferred 
 n years, the premium payable (i) for n years, (ii) for 
 the whole of life. 
 
 (70) . If there be n lives all of the same age x, find the value of an 
 annuity on the last survivor. 
 
 (71). Find, in terms of joint-life annuities, an expression for the 
 value of an annuity payable only while exactly r lives out of m lives 
 survive. 
 
 (72). Obtain the value of an annuity, payable so long as two at 
 least out of three lives (#), (y), (2) are alive. Obtain this value also 
 by general reasoning. 
 
 [Solutions, 
 
Chap, VII,] EXERCISES AND EXAMPLES. 17 
 
 (73). Find a formula to express the amount to which a life-annuity- 
 due will, on the average, accumulate at the end of the year of death, 
 supposing each payment to be invested and accumulated at compound 
 
 interest. Explain the difference between this value and . 
 
 M-x 
 
 (74). Find the value of an annuity on (#) during the lifetime of 
 (y), and for t years after the death of (y), if (#) live so long. 
 
 (75). Given a table of annual premiums, show how to construct a 
 corresponding table of mortality. 
 
 (76). Obtain the values of a x and Aa, upon De Moivre's hypothesis 
 as to the rate of mortality. 
 
 (77). Give an algebraical proof of the formula 
 
 xyz a abcxyz 
 
 and show by argument that the result is correct. 
 
 (78). Give formulas for the value of an endowment, payable 
 
 (a) If (x) survive n years. 
 
 (/?) If both (x) and (y) survive n years. 
 
 (y) If either (#) or (y) survive n years. 
 
 (8) If (x) survive, and (y) die within n years. 
 
 (79) . Investigate formulas for annuities payable until the death of 
 
 the last survivor of (x) and (y), the payments being reduced to of 
 
 n 
 
 their former amount (a) at first death, (/?) in the event only of (x) 
 dying before (y). 
 
 (80). Prove that A^.= k xxx 3 (A^-AJ . Does the annual pre- 
 mium follow the same rule ? Prove your answer. 
 
 (81). Find the value of a temporary annuity payable for m years 
 if (x) survive, or for n years if (y) survive, (x) having died within the 
 n years. Assume n<m. 
 
 (82). The yearly rents of an estate belong in equal shares to four 
 sisters, being equally divided at the end of each year amongst all of the 
 sisters who survive that year, the last survivor taking the whole for the 
 rest of her life. Find in terms of the single and joint-life annuities an 
 expression for the present value of the interest of any one of them. 
 Write down the general formula for the same thing, when the number 
 of persons is n. 
 
 c 
 
 pp. 69-73,] 
 
18 PAET II. LIFE CONTINGENCIES. [Chaps. VIII, IX. 
 
 CHAPTER VIII. CONVERSION TABLES FOR SINGLE AND ANNUAL 
 ASSURANCE PREMIUMS. 
 
 (83). If P represents the present value of 1 to be received at the 
 end of the year in which a certain status may fail, and Q the present 
 value of 1 per annum to be received during its continuance, prove that 
 
 P= _i9 . Hence show that 
 
 (84). Give a general formula for the class of benefits the single 
 and annual premiums for which can be derived from Conversion Tables, 
 and explain it verbally. 
 
 (85). Show how Conversion Tables can be employed in finding the 
 single and annual premiums for an endowment assurance. 
 
 (86). Show how to construct a table from which may be found the 
 annual premium corresponding to any given single premium on a whole 
 life assurance. 
 
 (87). Given the values of the single and annual premiums for the 
 insurance of 1 on a life (#), find the rate of interest. 
 
 (88). By the H M Table, the value of an annuity on a life aged 30 
 is 17 "1309, and the single premium for an assurance on the life is 
 302658. Find, without referring to any book, the rate of interest at 
 which these values are calculated. 
 
 (89). The mortality by a given table is such that the values of 
 annuities at all ages, at the rate of interest i per-cent, are identical with 
 those of another table at the rate of i' per-cent. Will the corresponding 
 tables of annual premiums also coincide? Give a reason for your 
 answer. 
 
 CHAPTER IX. ANNUITIES AND PREMIUMS PAYABLE FRACTIONALLY 
 
 THROUGHOUT THE YEAR. 
 
 (90). Assuming a uniform distribution of deaths throughout the 
 year, prove the formula for the value of an annuity payable half-yearly 
 
 ' 
 
 [Solutions, 
 
Chaps, IX, X.] EXERCISES AND EXAMPLES. 19 
 
 and show that the common approximation (O? + T) agrees with the above 
 to two decimal places, but not in the third place. 
 
 (01). The common rule for finding the value of an annuity payable 
 half-yearly or quarterly is to add i or to the value when payable 
 yearly. What is the amount of the error ? 
 
 (02). Show that the value of a life annuity payable m times a year, 
 
 the first payment being made after the interval , is approximately 
 
 equal to (a x + \) whatever value is assigned to m. 
 
 (03). Given the formula for converting an annuity payable yearly 
 into one payable in times a year, investigate a rule for finding a premium 
 payable m times a year from the annual premium. 
 
 (04) . If P be the annual premium for the insurance of 1 on a life 
 
 (#), show that the half-yearly premium is nearly equal to - - 
 
 1 
 
 (05). State a general formula for the value of an annuity on a 
 single life or combination of lives, payable m times a-year, and apply it 
 to determine the value of a continuous reversionary annuity to (#) 
 after (y). 
 
 (06). Find the annual premium for a half-yearly annuity on (#) 
 deferred n years, the first payment of the annuity being made six months 
 after the payment of the last premium. 
 
 (07). Assuming the usual correction for the value of an annuity 
 payable quarterly, prove that the value of a temporary annuity on a 
 
 vi. / N i Ntf Na?+n i~D x+n 3 
 
 life (#) payable quarterly for n years is - - + - . 
 
 DX 8 
 
 CHAPTER X. ASSURANCES PAYABLE AT ANT OTHER MOMENT THAN 
 
 THE END OF THE YEAB OF DEATH. 
 
 (08). Find the single premium for an assurance on a life (#) 
 payable at the instant of death. 
 
 (00). What will the formula A x = * become when the, annuity 
 
 is payable (1) half-yearly, (2) m times a year, (3) continuously? 
 
 (100). Prove that, upon the assumption of a uniform distribution of 
 deaths throughout the year, the value of an insurance payable at the 
 
 c 2 
 
 pp. 73-77.] 
 
20 PART II. LIFE CONTINGENCIES. [Chaps. X-XIL 
 
 ^A 
 instant of death is equal to - - - . Expand this expression in a 
 
 series as far as terms containing i 3 . 
 
 (101). Find the value of A,^. 
 
 (102). Obtain an expression for the value of the force of mortality 
 in terms of the expectations of life. 
 
 CHAPTER XI. COMPLETE ANNUITIES. 
 
 (103). Find an expression for the value of a continuous annuity on 
 a life aged #, and thence deduce a formula for the value of e x . 
 
 (104). What is the nature of the error involved in the approximate 
 formula for the value of a complete annuity payable yearly 
 
 Deduce a more correct expression, on the assumption of a uniform 
 distribution of deaths, and thence obtain an approximate value for the 
 amount of the error involved in the above formula. 
 
 (105). What addition should be made to the value of a curtate 
 annuity payable yearly (a x ) to obtain the value of a complete annuity 
 payable half-yearly (d ( J) ? 
 
 (106). Given the formula for a complete annuity payable m times 
 a year 
 
 obtain an expression for the value of (^d^,), a complete annuity deferred 
 t years, and payable quarterly during the joint existence of (or), (y), 
 and (z). 
 
 (107). Show that approximately 
 
 = 
 
 CHAPTER XII. JOINT-LIFE ANNUITIES. 
 
 (108) . State some of the methods which have been suggested for the 
 approximate calculation of the value of an annuity on three joint lives. 
 Which would you select as likely to give the most satisfactory result 
 without excessive labour ? 
 
 [Solutions, 
 
Chaps. XII, XIII.) EXERCISES AND EXAMPLES. 21 
 
 (109). Show that if a yz =:a w , the value of an annuity on the last 
 survivor of three lives, (a?), (y), (2), is approximately equal to 
 
 ^ + 5 
 
 (110). Show that, if Gompertz's formula l x =k(g) cX be assumed, the 
 probability of two joint lives (#) and (y) surviving w years is identical 
 with the probability of a single life (to) surviving the same period, 
 where c w =c x + cv. 
 
 (111). Show that, if tp =s t (ff) cX ( ct - l \ where s, g, and c are constants 
 independent of x and t, the probabilities of life of n joint lives 
 
 x, y, z, , are identical, for all values of t, with the probabilities of 
 
 life of n joint lives of a uniform age w, where nc w =c x -{-cy -\-c 2 -}- 
 
 (112). Prove that, upon Makeham's hypothesis as to the law of 
 mortality l x =ks x (g) cX , the value of an annuity on m joint lives #, y, 
 
 2, , calculated at the rate of interest i t is equal to the value of an 
 
 annuity on m joint lives of a uniform age to, calculated at the rate of 
 
 . , , ., , ., 1 + i - 
 interest I. where ^ = 1. 
 gi i 
 
 (113). How may the relation shown in the previous example be 
 practically applied in the approximate calculation of the value of an 
 annuity on three lives at a rate of interest of 3 per-cent, by means of 
 tables of the values of annuities on two joint lives, calculated at 3, 3^, 
 and 4 per-cent ? 
 
 CHAPTER XIII, CONTINGENT, OR SURVIVORSHIP, ASSURANCES. 
 (114). Prove the formula for a contingent assurance 
 
 What supposition is made, in obtaining this formula, as to the deaths 
 which take place in the course of any year ? 
 
 (115). Find the value of an assurance on the life of (x) provided he 
 die after (y). For which of the two lives would the usual medical 
 examination be required ? 
 
 (116). Find the single premium for an assurance for n years of 1 
 on the failure of a life aged #, provided that another aged y survive him. 
 
 (117). Write down, in commutation symbols, the single premium 
 for a survivorship assurance, using Professor de Morgan's method of 
 
 pp. 77-79.] 
 
22 PART II. LIFE CONTINGENCIES. [Chaps, XIII, XIV. 
 
 constructing the joint commutation columns, and also the ordinary 
 method previously in use. 
 
 (118). Having given complete D and N columns for single and joint 
 lives, and also a table of M^ for x<y\ find |^A^, l^J, \ t A? xa and |^A^. 
 
 (119). Find the single and annual premiums for an assurance payable 
 in the event of (x) dying in the lifetime of (y), or within t years from 
 the death of (y). 
 
 (120). How would you approximate to the annual premium for an 
 assurance payable on the death of a person aged 48, in the event of his 
 dying before the survivor of two persons aged respectively 75 and 70, or 
 within one year after the death of such survivor ? 
 
 (121). Find a formula for the annual premium to assure a payment 
 on (#) attaining the age of (x + n) or at previous death, provided (y) be 
 living in either case ; and show how it can be adapted for the use of the 
 values tabulated in the Institute Life Tables. 
 
 (122). Show how to find the single premium for an assurance payable 
 on the death of B aged y, provided he die after A aged #, and before C 
 aged z. If the premium is to be paid annually until the risk determines, 
 by what annuity would you divide the single premium ? 
 
 (123). Determine A 1 ^, the present value of 1 payable upon (x) 
 failing either the first or last of three lives (#), (y), and (z). 
 
 (124). Show that A^= p x a xy ~ , and hence deduce the approxi- 
 mate expression for the value of the annual premium corresponding to 
 
 , i i . IT, d'X 1 t/ ~"~~ ^x "HI*'?/ A^u? 
 
 this single premium AJ.^, namely, p x + % - . 
 
 CHAPTER XIV. REVERSIONARY ANNUITIES. 
 
 /1or , dxNy + d x 
 
 (125). Show that a y a xy = - 
 
 l x \j y 
 
 (126). Find the value of an annuity for such portion of a term of t 
 years certain from the present time as will remain after the death of a 
 person aged #, the first payment to be made at the end of the year of 
 his death. 
 
 (127). Grive an expression for the annual premium for a contingent 
 annuity to commence at the death of A, aged #, and to continue as 
 long as either B or C, aged respectively y and z, is living. 
 
 [Solutions, 
 
Chap. XIV.] EXEBCISES AND EXAMPLES. 23 
 
 (128). An annuity on the longest of three lives, A, B, C, aged 
 respectively #, y, and z, is to be enjoyed by A during his life : after 
 his decease it is to be divided equally between B and C during their 
 joint lives, and the survivor of them is to have the whole. Find an 
 expression for the value of B's interest. 
 
 (129). A sum S is applied in the purchase of an annuity ori three 
 lives aged (#), (y), (z), the annual payment being A while all three 
 are alive, A while two are alive, and f A while one is alive. Find A. 
 
 (130). Required the single and annual premiums for an annuity 
 payable to the last survivor of (#) and (y), to commence at first death 
 if within n years ; or at the end of n years, if both be then living. 
 
 (131). Find single and annual premiums for an annuity of 1 during 
 the term which may remain between the period of y's death and that of 
 x completing the age (x + n) . 
 
 (132). Find the value of an annuity on (#), to commence at the 
 death of (y) and to continue payable till the end of t years from now, 
 and as much longer as (x) may live. 
 
 (133). Find the value of an annuity on a life aged y, the first 
 payment to be made at the end of the year of death of a person aged X, 
 but the annuity to continue for t years certain whether (y) survive 
 or not. 
 
 (134). Find in a convenient form for computation the single and 
 annual premiums for an annuity to commence on the death of (y) and 
 continue payable during the remainder of the life of (#), but to be 
 payable only if (y) dies within t years. 
 
 (135). There are two expressions for the value of a reversionary 
 annuity to (#) after (y), namely: 
 
 and S^- 
 
 Give an algebraical demonstration that these are identical. 
 
 (136). How would you proceed to obtain the value of a reversionary 
 annuity to () after () in the case where the two lives are now resident 
 in India, but (a) has the intention of living in England after the death 
 of (b) ? 
 
 (137). (a) Provethat(^-^):(A,/-A-)::(l + 0:^ (ft) Explain 
 clearly in what respects the assumptions involved in the ordinary formula 
 for a reversionary annuity to (#) after the death of (y), namely, 
 (a x a xy ) y do not agree with the conditions of practice, and give 
 particulars of the various formulas devised to remedy this. 
 
 pp. 79-84.] 
 
24 PART II. LIFE CONTINGENCIES. [Chaps. XIV XVI. 
 
 (138). The formula (a x a xy iA x J) has been given for the value 
 of a half-yearly reversionary annuity payable during the life of (#) after 
 the death of (y). On what assumption is this solution approximately 
 correct ? 
 
 (139). Deduce a formula for the value of a complete annuity to (#), 
 payable m times a year, to be entered upon at the moment of the death 
 
 of (y), =t. 
 
 CHAPTER XV. COMPOUND SURVIVORSHIP ANNUITIES AND 
 ASSURANCES. 
 
 (140). Find the present value of an annuity on a life (#) after the 
 failure of the joint existence of (y) and (z) provided that event take 
 place by the death of (z). 
 
 (141). Show that, upon Grompertz's hypothesis as to the law of 
 mortality, the relation l*$i=C)j*|f?y is accurately true. 
 
 (142). Find the value of a contingent life interest for the lives of (#) 
 and (y) and the survivor, after the death of (z), subject to the condition 
 that (x) shall survive (z) . 
 
 (143). Investigate a formula for the single premium for an assurance 
 payable on the death of (#) if he die third of the three lives, (#), (y), 
 and (z), (z) having died first. (A^ 3 ). 
 
 CHAPTER XVI. COMMUTATION COLUMNS, AND THEIR APPLICATION 
 TO VARYING BENEFITS, AND TO RETURNS OF PREMIUM. 
 
 (144). Explain a commutation table, and the relations of the 
 columns to each other. Why is it called a commutation table? In 
 what way does the N of the English Life Table differ from Davies's N ? 
 
 (145). Given only the D and N columns, show how to deduce all the 
 rest from them. 
 
 (146) . The value of a deferred annuity is -~ . Explain fully the 
 
 D x 
 
 ,, . Na._* 
 
 meaning 01 the expression . 
 
 U# 
 
 [Solutions, 
 
Chap. XVI.] EXERCISES AND EXAMPLES. 25 
 
 (147). Having given a x , and the value of the annuity on (#) com- 
 mencing at k payable at the end of the first year, and increasing Ji per 
 annum ; find the value of the corresponding assurance commencing at k 
 if death occur in the first year, and increasing Ti per annum. 
 
 (148). Find the present value of an annuity on (#) payable half- 
 yearly, commencing at 1, the payments to be doubled every 10 years. 
 
 (149) . Investigate a formula for the annual premium for an assurance 
 on the life of (#) which is to be 1, 2, 3, ... n pounds, according as the 
 death of (#) shall take place in the 1, 2, 3, ... nth year, and to remain 
 constant at the last-named amount during the remainder of life, (a) for 
 a premium payable n times if (x) live so long, (ft) for a premium payable 
 throughout life. 
 
 (150). State a formula for the net annual premium for an assurance 
 on a life (#) commencing with 1 and decreasing '05 every year until its 
 extinction at the end of 20 years ; the premium being also reduced 
 by gVkh f i^ s original amount each year. 
 
 (151). Explain the method of calculating a table of ascending 
 premiums, giving the proper formulas. What precaution must always 
 be observed in practice with regard to the magnitude of these premiums, 
 and how would you fix the premium to be charged during the first term 
 of years ? 
 
 (152). Deduce a general formula (in terms of the commutation 
 columns) for the value of an annuity on a life (#), whose successive 
 payments are u , u\, u%, &c 
 
 (153). Find the annual premium for a deferred annuity on (#) to 
 commence n years hence, it being stipulated that if (x) die before the 
 annuity commences, the net premiums paid are to be returned with 
 compound interest. 
 
 (154). Show in the previous question that if the rate of interest at 
 which the premiums are calculated is the same as that at which they are 
 
 accumulated, say i, then the net annual premium = ta *+ n ^ 
 
 (l + i) n I 
 
 (155). Prove that the single premium for an assurance of 1 on the 
 life of #, with the return of the premium along with the sum assured, is 
 equal to the annual premium to secure a perpetuity of 1 per annum, 
 of which the first payment is made at the end of the year in which # 
 shall die. 
 
 (156). Find the net annual premium for an assurance of 1 payable at 
 the end of (m + n) years if (#) be then living, or at his death if that 
 
 pp. 84-88.] 
 
26 PART II. LIFE CONTINGENCIES. [Chaps. XVI, XVII. 
 
 happen after m years but before (m + n) years: it being stipulated that 
 if (#) die within m years all the office premiums are to be returned 
 except the first. 
 
 (157). Find the net annual premium for an assurance on (x) of 1, 
 with return of all the office premiums paid, the premiums being payable 
 for t years only. 
 
 (158) . Find the annual premium payable for n years for an assurance 
 on the life of (#), with return of the excess of the limited premium over 
 the ordinary whole of life rate (pure premiums only) in the event of 
 death occurring in the first n years. 
 
 (159). A survivorship annuity of 100 payable half-yearly till death 
 on (26) after (32) is to be paid for by an annual premium, returnable 
 if (26) die before (32). Required the annual premium which shall give 
 to the grantor a profit of 15 per-cent, the rates of mortality and interest 
 being assumed as Carlisle 4 per-cent. 
 
 (160). Find the single premium for an annuity to (#) after (y), with 
 the condition that the premium be returned if (# ) die before (y) . 
 
 CHAPTEE XVII. SUCCESSIVE LIVES. 
 
 (161) . Show that the value of 1 to be received on the failure of the 
 successive lives x and y is A a . x A. y . When is the second life supposed 
 to be nominated ? 
 
 (162). An estate is held for a single life, renewable for single lives 
 successively, by payment of a fine of 1 at the end of the year in which 
 the life in nomination fails : assuming that the present life is now aged x, 
 and that the succeeding lives are all nominated at the age y, find the 
 present value of all the fines to perpetuity. 
 
 (163). A copyhold estate is held on three lives aged severally 
 61, 50, and 45, each renewable at the end of the year in which it drops 
 by a life of 7 years of age on payment of a fine of 5. Give an 
 expression for the present value of all the fines for ever. 
 
 (164). A perpetual annuity is to be enjoyed, first by a person aged x 
 for his life, afterwards by a successor to be appointed at his death, and 
 when the second life fails, by a third to be then appointed, and so on. 
 Determine the present value of the annuity for the first n successive 
 lives, the ages being all different. 
 
 [Solutions, 
 
Chaps. XVII, XVIII.] EXERCISES AND EXAMPLES. 27 
 
 (165). Investigate an expression for the value of the wth presentation 
 
 to a living. 
 
 (166). Find the value of the second, and every third succeeding 
 presentation to a living, assuming all the nominees to be of the same 
 age on presentation. 
 
 CHAPTER XVIII. POLICY- VALUES. - 
 
 (167.) Show that the values of policies which have been n years in 
 force are equal to the accumulated premiums less the accumulated 
 claims. 
 
 (168) . If l x persons each buy an endowment at an annual premium, 
 show that the total amount which will be paid to the survivors is made 
 up of an accumulation of (a) the premiums paid by the survivors ; 
 ((3) the premiums paid by those that die. 
 
 (169). Find the value of a policy in terms of (a) annual premiums 
 and rate of interest ; ((3) annuity and annual premiums ; (y) single 
 premiums at entry and at valuation. 
 
 (170). Explain under what conditions n Vx<n-iVx+i and give an 
 example from some known table of mortality where the anomaly occurs. 
 
 (171) . Give a verbal interpretation of the expression 
 
 State what this becomes when the premiums are payable for a limited 
 number of years. 
 
 (172) . If P^ denote the annual premium for a whole-life insurance, 
 P x ^\ the annual premium for a term insurance for n years, and P^] the 
 annual premium for an endowment payable at the end of n years, prove 
 that the value of a policy effected at the age x which has been n years in 
 
 P# Pznl 
 
 force may be expressed by the formula n ~V x = - . 
 
 (173). If a whole-life policy be valued by the "hypothetical" or 
 "re-assurance" method, show under what circumstances the reserve will 
 be (a) greater than, ((3) less than, (y) equal to the policy- value by the 
 ordinary net-premium method. 
 
 (174). Given two mortality tables, how would you proceed to 
 ascertain within what limits as to entry-age and duration the policy- 
 
 f UN.VERSITY) 
 
28 PAKT II. LIFE CONTINGENCIES. [Chap. XVIII. 
 
 values brought out by the one table are greater or less than those 
 brought out by the other table ? 
 
 (175). Prove that, if the annuity- values by two several tables of 
 mortality are in the relation a' aJ =(l + K)a a ;, where K is a constant, then 
 the policy-values by the two tables will be equal for all ages and 
 durations. What will be the relation between the probabilities of living 
 a year at any age x (p x ) by the two tables ? 
 
 (176). In two mortality tables, (A) and (B), the probability of 
 living a year by the latter table is throughout less in a constant ratio 
 than by the former. What relation will exist between the policy-values 
 by the two tables ? 
 
 (177). If the rate of mortality in one table be throughout greater 
 than that in another table, should you expect the values of policies 
 obtained in the ordinary way from the first table to be greater or less 
 than those obtained from the second ? State your reasons. 
 
 (178). Given a table of policy- values (^V^) for all integral values of 
 x and ^, how would you proceed to ascertain the value of a whole-life 
 
 policy, effected by yearly premiums at age #, after a duration of In H 1 
 years? On-lV,). 
 
 (179). What modification should be introduced in the usual formula 
 2Aa; +w 2P#(i + 0;c+tt) for valuing a group of policies effected n years 
 ago with yearly premiums at age #, in the case where, from an unequal 
 distribution of premium income, it is ascertained that the yearly 
 payments next falling due will, on the average, be payable 7i months 
 hence ? 
 
 (180). Show to what extent the payment of claims by an assurance 
 company immediately on proof, instead of at the end of the year of 
 death, affects (a) the liability under the sum assured, (ft) the asset in 
 respect of the annual premiums : and state what additional reserve is 
 necessary in each case. 
 
 (181). Prove that the value, after n years, of an endowment- 
 assurance policy effected on a life (#), payable at (x + n + m) or previous 
 death, may be expressed by either of the following formulas : 
 
 (182). Apply the retrospective method to the valuation of an endow- 
 ment policy effected n years ago at age #, payable at age 
 
 [Solutions, 
 
Chaps. XVIII, XIX. I EXERCISES AND EXAMPLES. 29 
 
 with the condition that the whole of the office premiums are to be 
 returned in the event of the life failing during the term of the endow- 
 ment, and prove the identity of the expression thus obtained with that 
 which would be arrived at by valuing separately the future benefit 
 assured, and the future net-premium payments. 
 
 (183). Show that the value of an endowment policy taken out at 
 age x, payable in n years, is, after it has been t years in force, 
 
 . Show that the amount of a free or "paid-up" policy which 
 may be given in exchange for a whole-life assurance effected on a life 
 
 (or), after n years' duration, is equal to (1 p-^-J, and give a verbal 
 
 interpretation of this expression. 
 
 (185). In a society consisting of n members, 1 is payable at the 
 death of each member by the survivors. Find an approximate expression 
 for the present value of the future payments. 
 
 (186). How would you proceed to value as for the 19 May 1877 
 a policy dated 19 August 1848 for 100 on a life then aged 41, at a 
 premium of 2. 5s. 2d. for the first seven years, 3. 9s. Qd. for the 
 second seven years, and 4. 14s. 4d. for the remainder of life ? 
 
 (187). The values of all the policies in an office at the beginning of 
 any year and the premiums for that year, both accumulated at interest 
 for one year, are equal to the claims of the year and the values of the 
 policies at the end of the year. Under what conditions is this tru ; 
 and show what modifications would be required on the assumption that 
 the premiums are payable throughout the year ? 
 
 CHAPTER XIX. LIFE INTERESTS AND REVERSIONS. 
 
 . Investigate a formula for the market value of a life interest 
 in a term annuity. 
 
 (189). Give the formula for the value of a life interest of s per 
 annum, and state for what amount the life policy shall be effected. 
 What is the value of such policy to the purchaser of the life interest, 
 after the expiration of t years ? 
 
 pp. 91-96,] 
 
30 PAET II. LIFE CONTINGENCIES. [Chaps, XIX, XX, 
 
 (190). If a' x represent the value of a life-interest secured by an 
 assurance effected at a given annual premium, show that at the expiration 
 of n years the value of the policy (i.e., the value of the sum assured minus 
 
 the value of the premiums payable) is S( 1 -- - f+M w here S denotes 
 
 V 1 ~t~ a x J 
 
 the amount of the assurance; and further, that this expression resolves 
 itself into (a' x a x +^)- 
 
 (191). A common formula for the value of a reversionary life- 
 interest is 
 
 If 1 is advanced in consideration of a reversionary charge so calculated, 
 set out the theoretical sum assured, the theoretical cost of the annuity 
 purchased, and the value of the charge at death of (y) coupled with the 
 policy. 
 
 (192). Find a formula for the value of a reversionary life-interest so 
 as to return one rate of interest while the life-interest is in reversion, 
 and another rate when it is in possession. 
 
 (193). A, aged x, is entitled to a reversionary life interest contingent 
 on his surviving (y) and (2), and neither leaving issue. Give the formula 
 for valuing A's interest, allowing for a single whole- world premium of m 
 per-cent, and a single issue premium of n per-cent. 
 
 (194). A, aged #, is entitled to a sum S provided he survive B, aged y. 
 Find the amount which a purchaser could give for the reversion. 
 
 CHAPTEE XX. SICKNESS BENEFITS. 
 
 (195). In a given community the rate of sickness (average number of 
 weeks' sickness per annum) at each age, z x , =A-f-B<2 f a ;, where ^=the 
 rate of mortality at age x. Express the value of the sickness benefit 
 ceasing at age 70. 
 
 (196). If, in two sickness tables, (A) and (B), the rates of sickness 
 (2j.) are throughout identical, while the probabilities of- life at each age 
 (p'x) in table (B) are less than those in table (A) in the ratio 
 <p' x = (L K)P X , what relation will hold between the values of the sickness 
 benefits under the two tables ? 
 
 (197). -If the law of sickness be such that at any age two are 
 
 [Solutions, 
 
Chaps. XX, XXI,] EXERCISES AND EXAMPLES. 31 
 
 constantly sick for one that dies, show that the single premium for a 
 sick allowance of 10s. per week at age #, to cease at age 65, is 
 
 (198). Given a table showing at each age the number living (fc) 
 and the average number of weeks' sickness (2^), how would you proceed 
 to deduce the present value of a sick allowance of 1 per week (a) payable 
 throughout life, (/?) payable up to age 70, (y) to commence at age 60 
 and continue throughout the remainder of life ? 
 
 (199) How would the formula for the present value, at age #, of a 
 sick allowance throughout life, be modified, in the case where it is 
 assumed that a member will be permanently on the sick list after 
 age 70 ? 
 
 (200). A friendly society grants the following benefits to its 
 members : (a) a sum of 10 at death ; (/3) a weekly allowance of 1 
 during the first six months' sickness, 10s. during the second six months' 
 sickness, and 5s. during subsequent sickness up to age 70 ; (y) a 
 permanent allowance of the minimum sickness rate from age 70 until 
 death. State a formula for the weekly contribution necessary to provide 
 the above benefits at age #, and for the reserves which should be in hand 
 after n years, assuming that 30 per-cent of the weekly contribution is 
 absorbed in expenses. 
 
 CHAPTER XXI. CONSTRUCTION OF TABLES. 
 (201). Give a practical method of forming a table of the values of 
 
 (202). Give a method of calculating the D and N columns by a 
 continued process. 
 
 (203). What, in your opinion, would be the best way, as regards 
 speed and accuracy, to construct a table of the values of annuities, having 
 given the usual l x column of a life table ? If you are acquainted with 
 more than one method of constructing the annuity table, what do you 
 consider to be the special advantages of each ? 
 
 (204). Give De Moivre's method of calculating the values of 
 annuities, and show the applicability of Gauss's logarithmic tables. 
 
 pp, 96-99.] 
 
32 PART II. LIFE CONTINGENCIES. [Chaps. XXI, XXII. 
 
 (205). Explain the formula 
 
 and apply it to the construction of a reversion to 1 upon the decease of 
 a single life. 
 
 (206). What conditions must the benefits fulfil to which the 
 demonstration of the above formula applies, and in regard to which it 
 subsists ? 
 
 (207). Griven a complete table of values of deferred annuities (%]##) 
 and of temporary annuities (\ n a x ) , for all values of x and n, how would 
 you proceed to verify the results ? 
 
 (208). Describe how to construct a table of policy-values of whole- 
 term assurances. 
 
 (209). Prove the formula n V x =l (7r x -\-d) (l + a x+n ), and point out 
 its advantages in calculating a table of policy-values. 
 
 (210). Show how to construct by a continued method tables of 
 policy-values of endowment assurances. What precautions would you 
 take to check the results ? 
 
 (211). Show how by a continued process to construct commutation 
 columns M* y for survivorship assurances. 
 
 (212). Deduce a formula for the continued construction of a table of 
 single premiums for survivorship assurances on (#) against (y) . 
 
 (213). Describe at length a method of calculating a complete table 
 of last-survivor annuities. 
 
 (214). Explain how you would construct tables of the values of 
 d x and Aa;. 
 
 (215). Show how to construct a table of premiums for the assurance 
 of 1 for the term of one year by a continued process. 
 
 CHAPTEE XXII. FORMULAS or FINITE DIFFERENCES. 
 
 (216). If u x be any function of x of n dimensions, prove that 
 is constant, and hence show how to form a table of cubes of 
 the natural numbers expeditiously. 
 (217). Prove that & n x n =\n. 
 
 (218). Find A 71 ^, when x is variable, the increment of x being unity. 
 (219). By means of the formula 
 
 [Solutions, 
 
Chap. XXII.J EXERCISES AND EXAMPLES. 33 
 
 or otherwise, form a table of the differences of the powers of nothing, 
 as far as A~0~. 
 
 (220). Investigate the expressions 
 
 (a) for u x +n in terms of Azr. r , A 2 z^, A 3 ^, &c. 
 
 (/?) for A*%. r in terms of u x and its successive values. 
 
 (221). Show that 
 
 Hence show that, second differences being constant, 
 
 (222). If A#=, and &u x =u x +aUa;i prove that 
 
 n n(n a) 
 
 v } 
 
 (223). Obtain a formula for expressing u x+m in terms of u x and the 
 successive finite differences bu x , &?u x , &c. If Wi=4, / 2 =30, 3 =120, 
 ^4=340, ? 5 =780, and A 4 ^ is constant, find an algebraical expression 
 for u x . 
 
 (224). Express u^+i in terms of u x and its successive differences. 
 
 (225). (a) Given (n + 1) consecutive equidistant values, U Q , uj^ 
 u^i, . . . Unht of Ux-> which is a rational integral algebraic 
 function of x of degree n, find the general expression 
 for u x . 
 
 (ft) When u x is an expression of the 4th degree in #, and 
 2^0=0, Awo=l, A 2 w =14, A% =36, A% =24, find u x , 
 the increment of x corresponding to the A differences 
 being A, so that u x +h u x -=&u x . 
 
 (226). In the series # 5 , (x + li)\ (o: + 2A) 5 , (a?+3A) 5 , Or + 4A) 5 , 
 let u x represent the first term, and k be written A#, and show that 
 (x+n) 5 , where n is a multiple of A, is equal to 
 
 n &u x n(n h) b?u x n(nh)(n2h) & 3 u x 
 Ux + 
 
 n(nh) (n- 
 
 ~ 
 
 |6 
 
 pp. 93-106. j 
 
34 PAET II. LIFE CONTINGENCIES. [Chaps, XXII, XXIII. 
 
 If Aa? be made infinitely small, the value of n remaining unaltered, 
 what does this expression become ? 
 
 (227). Express the second difference of the product of two functions 
 in terms of the separate functions and their respective differences, that 
 is, show that 
 
 A 2 (? Wr ) =u x &v x + 2Aw a .(At> a . + A 2 y. r ) + A 2 w. r (v x + 2 v x + A 2 r. r ) , 
 
 and by means of the result, find A 2 (#log#). 
 
 (228). If u x be a function of x of the form ^. r =6 h T + & 2 # 2 + &c., ad 
 inf., show that it can also be expressed in the form 
 
 
 
 (three orders of differences will suffice) . 
 (229). Show that 
 
 and hence determine a series of such a nature that the terms after the 
 first shall be respectively double the first terms of the successive orders 
 of differences (w 2 =2A% 1 , $ 3 =2A 2 zf 1 , and so on). 
 
 CHAPTER XXIII. INTERPOLATION. 
 
 (230). Investigate an expression for & n u x in terms of u x and its 
 successive values. Using the formula thus found, if in the series 1, 6, 
 21, 56, K, 252, 462, &c., the sixth differences vanish, find K, and the sum 
 of the series to 10 terms. 
 
 (231). Given UQ, 2^, u 2j u 3 , u, and w 5 , and assuming fifth differences 
 to be constant, show that 
 
 where a=u -\-u 5 , ~b=. 
 
 (232). Given wo= 100,000, "4=98,391, t* 5 =98,011, 6=97,615, 
 
 construct the series from w 10 to ^15, assuming that third differences are 
 constant. 
 
 (233). If w =100,000, z* 7 =97,624, w 8 =97,245, and w 9 =96,779, find 
 u\ to UQ inclusive by means of constant third differences. 
 
 [Solutions, 
 
Chap, XXIII.] EXERCISES AND EXAMPLES. 35 
 
 (234). The H M premium at age 40 is at 3 per-cent= -025891 
 
 3i , 
 
 , =-024654 
 
 1 
 
 , =-023517 
 
 4 
 
 , =-022470 
 
 5 
 
 , =-021509 
 
 6 
 
 = 019811 
 
 Interpolate the corresponding at 5^ per-cent (a) using two 
 of these values; (ft) using four; and (y) using six. 
 
 (235). Having given log50=T698970 
 log 52=1716003 
 log 54= 1-732394 
 log 55=1-740363 
 
 find, as accurately as possible from the above data, the value of log 53. 
 
 (236). Find w 12 and also u 2 , when ^ 5 =55, w (J =126, w 7 =259, t/ 8 =484, 
 w g =837, and A 4 is constant. 
 
 (237) .Given log 235 = 2-3710679 
 
 log 236=2-3729120 
 
 log 237=2-3747483 
 
 log 238=2-3765770, 
 
 find log 23563. 
 
 (238). Find the Northampton 3 per-cent annuity for age 30, from 
 the following table : 
 
 A _. Northampton 
 
 3 per-cent Annuity 
 
 21 18-4708 
 
 25 17-8144 
 
 29 17-1070 
 
 33 16-3432 
 
 37 15-5154 
 
 (239). (a) Given every nfh term of a series of values, i.e., u x , u x + n , 
 UX+ZH, &c., show at length how the intermediate terms u x+ i, u x +z, &c., 
 may be obtained by interpolation. 
 
 (ft) Given that in the series u x , 
 
 ^=9936675-4 
 
 S!=+ 12767-62 
 S 2 = 3013-725 
 8 3 =+ 422-8247 
 8 4 = - 34-72847 
 S 5 = + 1-254221 
 
 pp. 105-113.] 
 
36 PART II. - LIFE CONTINGENCIES. [Chaps. XXIII, XXIV. 
 
 you are required to construct the series as far as the term u x +\o- What 
 assumption is necessary ? 
 
 (240) . If u x be a function whose differences, when the increment of 
 x is unity, are denoted by $u x , S%. r ..... , and by Aw^., A%. r ..... , 
 when the increment of x is n\ then if &u xy 8 2 ?/. r+1 ..... , are in 
 geometrical progression, with common ratio q, show that 
 
 (241). Having given the values of annuities at the following rates of 
 
 interest, namely, at 
 
 3 per-cent= 15-863 
 
 3| =14-941 
 
 4 =14-105 
 4i =13-343 
 
 5 =12-648 
 
 find the value at 4'328 per-cent. 
 
 (242). Define a "differential coefficient", and find expressions for 
 
 . du x d*u x d*u x . , ., 
 
 the values of - , -7 , -rt ..... > in terms of the successive finite 
 dx dx- dx* 
 
 differences Su x , &u x , &MX ..... 
 
 (243). Show that when fourth differences are constant 
 
 (244). Prove that 
 
 0^ + 8)= ^ - approximately. 
 
 
 CHAPTER XXIV. SUMMATION. 
 
 (245). Show that finite integration is equivalent to the summation 
 of an infinite number of infinitely small terms. 
 
 (246). Explain the meaning of, and the necessity for, the introduction 
 of a constant in the process of integration. 
 
 (247). Show that 5^ +% =(A- 1 )(l + A)^ a; , and thence deduce the 
 formula for the sum of n terms of a series, 
 
 [Solutions, 
 
Chap. XXIV.] EXERCISES AND EXAMPLES. 37 
 
 (248). If ^=000? + cits 1 + c&*+ . . ., show that 
 
 , n(n+l)(2n 
 + - -^ ~ 
 
 ~30~ 
 
 (249). Let HI, u<i, u 3t . . . denote a series of quantities, and let S^ 
 denote the sum of the first x of them (S being = 0) . Having given 
 the values of S w , S 2 , . . . S rn , show how to find the values of u\ t u^ t 
 u a , . . . U rn . 
 
 (250) . Find an expression for u x in terms of ascending powers of # , 
 where 85^ = 1,865, S 1( ^i = 5,155, 8^1=13,370, and 8^1=28,635; 
 A 3 being constant. 
 
 (251). (a) Assuming the formula for the force of mortality, 
 
 Hx= j- ~ , find l x in terms of //,#. 
 I'x doc 
 
 (/?) How can you tell, by mere inspection, that there is a 
 mistake in the following approximate formula for the 
 value of /,# ? 
 
 What is the correct expression ? 
 
 (252). State Lubbock's formula for approximate summation, and 
 explain generally the process by which it may be deduced. What are 
 the advantages and disadvantages of this formula, and the limits within 
 which it may usefully be applied, in the computation of life benefits ? 
 
 (253). Show that 2,u x may be developed in a series proceeding by 
 the differential coefficients of u x , and prove that no differential coefficient 
 of an even order will appear in the expansion. 
 
 (254) . Deduce Woolhouse's formula for approximate summation, 
 
 1 i duo , i PO 
 -w - , -=- + w^-r^ ~ &c -' 
 
 2 12 dx 720 dx 3 
 
 by the method of separation of symbols. 
 
 (255). Write down Woolhouse's formulas for approximate summation 
 (a) to deduce the value of the continuous benefit from that of the yearly 
 
 benefit ; (ft) to deduce the value of the benefit at intervals of th from 
 
 m 
 
 pp. 113-119.] 
 
38 PART II. LIFE CONTINGENCIES. [Chap. XXIV. 
 
 that of the yearly benefit ; (y) to deduce the value of the yearly benefit 
 from that of the benefit at intervals of n years. 
 
 (256). Prove the identity of Lubbock's and Woolhouse's formulas 
 for approximate evaluation of the yearly benefit in terms of the value of 
 the benefit at intervals of n years. 
 
 (257). State some of the formulas of approximate summation 
 suggested by G-. F. Hardy for the calculation of benefits. Which 
 would you select for practical purposes, in order to obtain a sufficiently 
 close approximation without excessive labour ? 
 
 (258). Apply Lubbock's formula for approximate summation to the 
 computation of the value of a life annuity payable half-yearly, a table of 
 the commutation column D x being given for all integral values of x. 
 
 (259). Apply Woolhouse's formula of approximate summation to the 
 computation of the value of a continuous annuity on x (d x ), deduced 
 from the value of a yearly annuity (a^) . 
 
 (260). Calculate by Lubbock's formula of approximate summation 
 the values of #20.30.40 an( l A 20-30 . 40 according to the Life Table printed in 
 the Text-Book. 
 
 (261). Find the value of the following benefits, by the use of one of 
 G. F. Hardy's formulas of approximate summation : 
 
 A 1 A 2 A 2 
 
 ^20:40:60? -"-20:40:605 -^-20:40: 
 
 1 
 
 _ 
 
 20:40:60) 
 
 (262) . Find the value of a reversionary life annuity to the survivor 
 of (30) and (40) after the death of (20) = 2 oio7S5 > using one of the 
 formulas of approximate summation. 
 
 (263). Calculate by Woolhouse's formula of approximate summation 
 the value of an annuity to a life aged 40, to commence on the failure of 
 a life aged 50, provided a life aged 30 be then alive (a so ^\ 40 ) . 
 
 (264). Calculate the value of d 30 .J \ 4() by a formula of approximate 
 summation. 
 
 (265) . Obtain the annual premium for an assurance payable on the 
 death of (#), provided that event happen before the failure of the 
 survivor of two lives (y), (z), or within t years after such failure. 
 
 NOTE. For further practical examples and illustrations of the subject of this 
 chapter, the student is referred to Text-Book, Chap, xii, [53]-[63]; xiii, 
 [48]-[59]; xiv, [36]-[44]; xv, [10], [11], [l7]-[46]; also to Journal of 
 the Institute of Actuaries, vol. xxiv, p. 95; xxvi, 276; xxvii, 122. 
 
 [Solutions, pp, 119, 120.] 
 
Chap. I. 39 
 
 t 
 
 SOLUTIONS. 
 PART I. INTEREST (INCLUDING ANNUITIES-CERTAIN), 
 
 CHAPTER I. 
 
 (I). Text-Book, [4] ; (J.I. A., vol. iii, pp. 335-338; iv, pp. 61, 
 72, 243,253). 
 
 (2).- [6]. 
 
 (3). [7]. The force of interest or force of discount is the 
 rate per annum at which each unit of capital is momently increasing by 
 
 the operation of interest. It is usually denoted by 8. Thus, if is a 
 
 * 
 
 small fraction of a year, a capital of 1 will become H -- at the end of 
 
 M 
 
 o / X \ 2 
 
 such interval : and, at the end of of a year, ( H ) ; and, at the 
 
 in \ in / 
 
 (8 \ m 
 H 1 , the value of which, when m is indefinitely 
 mj 
 
 increased, becomes *=! + ; where i is the effective rate of interest: 
 therefore 
 
 (4) [8]. 
 
 (5). Let x be the nominal annual rate of interest convertible 
 quarterly, and i be the corresponding effective yearly rate : then the 
 present values of a sum due three, six, and nine months hence are, 
 respectively, in terms of the nominal rate #, 
 
PART I. INTEREST. [Chap. I. 
 
 or in terms of the effective rate i, 
 
 (! + )-*, (1 + 0-*, (1 + 0-*, 
 
 the relation between the nominal and effective rates being exhibited by 
 the equation 
 
 (6). (1-01)*=10 
 
 _ 2-3026 _ 2-3026 
 
 " X " -^ =281f 
 
 (*)- [9], [11]. 
 
 ( 
 
 (8).-[12]. (a) l+ 10> =(1-0125)* 
 
 ( i5x2) 
 
 =(l-0126)- 
 
 (9).- [15]. 
 (10). Theory of Finance, Chap, i, (23). 
 
 (11)- Value of bill given by A to B =v m a. 
 Value of bill given by B to A =v n b. 
 Let x be the amount of the bill due p years hence, then 
 
 a v 
 
 whence x= - =v m ~P.a v n ~P b 
 
 VP 
 
 An approximate value of x may be obtained as follows : 
 
 (1 + ) - J -f (1 + i) ~vx= (1 + -, 
 or approximately, 
 
 whence or= - approximately 
 
Chap. II,] SOLUTIONS. 41 
 
 ClIAPTEE II (i). 
 
 (12). Discount (D)= s(l-v n ) 
 
 Annuity (a^]) = . - . 
 
 whence 
 
 cin\ : D = l : i. 
 (13,.- 
 
 = 10,274. 9s. Sd. 
 
 (14). At the epoch of ^he first payment, three months hence, the 
 value of the annuity as modified would be 
 
 while the value of the original annuity at the same date would be 
 
 The difference of these two values (to be deducted from the first 
 payment) would be 
 
 or 1 + - 1 + J 25X729=387. 12s. 7d 
 
 (16). Accumulated capital = 
 
 l)~w> i)fi 
 
 osi (! + )=- - 
 Accumulated payments = 
 
 (1 + Q W 1 _ v~ m -l 
 i i 
 
 1 v n- 
 
 Difference = -- : 
 
42 PART I. INTEREST. [Chap. II. 
 
 which has been proved in Example (15) to be the amount unpaid at the 
 end of m years. 
 
 (17). 50,000(l-P^s 2 - 0) )=50,000-^ =15,359. 7s. 5d. 
 
 #251 
 
 (18).- [24]-[26]. (J.I.A., vol. xi, p. 172.) 
 
 (19). [27]. For each unit invested the annual payment is 
 
 = P^I+t'= -!.+'= + (*'-*) 
 
 SH\ n| 
 
 The annual payment in respect of a capital of V is thus 
 
 where P^j , 8n\ , and a^\ are calculated at the rate i. 
 
 The second formula enables us to ascertain the value of the annual 
 payment by means of an ordinary table of " the annuity which 1 will 
 purchase" at the rate i. Theory of Finance, Chap, ii, (41), (42). 
 
 (20). - [27]. 
 
 (21). [28]. 
 
 a-^\=a a ,n\a< a = - v n ' - : 
 i i i 
 
 \ v n 
 
 X O O \ ... 
 
 (23).- dkra=:**-_f_ =V 
 
 i 
 
 it 
 
 then : =1 v 7 
 
 Ay 
 
 (24). Let x, y, z be the number of years during which A, B, and C 
 may respectively enjoy the annuity : then 
 
Chap. II.] SOLUTIONS. 43 
 
 CHAPTER II (ii). 
 
 (25) .-[33]. 
 
 (26). [36]. (J.I.A., xv, 437.) 
 
 (27). [34]. Let x be the nominal and * the effective rate of 
 interest : then the capital repaid in the tih year is, in the case of the 
 annuity with yearly payments, =v n ~ t+1 , and in the case of the annuity 
 
 with mihly payments, =v n ~ t+l x - . 
 
 (28). [35]. a^=-a^, d^=\a^. 
 
 X O 
 
 (29). [38]. The required condition is, that the payments of 
 the perpetuity are made as often as interest is convertible. See also 
 Chap, v, pp. 119, 120. 
 
 (30) . [42] . (Numerical illustration of [34] .) 
 
 1 (1'025)~ 10 
 Value of Annuity = - ^Tr* 
 
 Amount redeemed in the (p + l)th instalment = ^t> 5 -f, where =(1-025)~ 2 ; 
 or =%V I -P, where ?=(1'025)- 1 . 
 
 5 000 
 (31). Equal half-yearly charge = , where 5o| is computed at 
 
 60| 
 
 2 per-cent. 
 
 Sinking fund =5,OOOPeol (computed at 2^ per-cent). 
 Amount repaid in (j9 + l)th instalment =5,OOOP6o|(l + ' 
 
PART I. INTEREST. 
 
 [Chap. II. 
 
 CHAPTER II (iii). 
 
 (82). - [41]. 
 
 (33). Table showing the formulas for the present value of an 
 annuity-certain for n years, instalment payable /</', and interest convertible 
 m times a year. 
 
 (a) In terms of the nominal rate of interest x (or 8, when in or 
 *=). 
 
 (/3) In terms of the corresponding effective rate of interest i. 
 
 ^=00 
 
 w \ 
 
 w> i 
 
 to 
 
 ( 
 
 (a) 
 
 JK 
 
 (Here = z) 
 
 Ci+Oi-i 
 
 (18) 
 
 (/8) 
 
 In all cases, 
 
 _ discount on 1 for n years, 
 
 n ' number of instalments of annuity per annum x interest on 1 for each instalment period 
 
 Theory of Finance, Chap, ii, (20). 
 
Chaps. II, III.] SOLUTIONS. 45 
 
 (34). (a) Here 8 = the nominal rate of interest convertible 
 
 momently. 
 (/3) Here i = the nominal rate of interest convertible 
 
 momently. 
 
 (y) Here the annuity is payable yearly, and interest con- 
 vertible momently, and 8 = the nominal rate of 
 interest ; while i = the effective rate of interest. 
 
 2J-1 
 
 Let ( 1+ I) = ( 1 + i/ ) : then ' dividin g bT x . 
 
 =1, 
 
 whence n= 
 
 CHAPTER III. 
 
 [In the following Solutions, the symbol t^\r\ is used to represent the 
 rath term of the rth order of figurate numbers, and the symbol a^\^\ is 
 used to represent the present value of an annuity for n years, whose 
 successive payments are the terms of the rth order of figurate numbers] . 
 
 (36).- f^ = - --_. [45], p. 68. 
 
 The value of this when #<r=0, and giving to x the values r, r 
 r + 2, &c., the successive payments of the annuity become 
 
 1=1 
 
 r.r-I 2 
 
 = r 
 
 t=i 
 
 r + l.r.r 1 3 r + l.r 
 
 L 2 
 
 &c. &c. 
 
PAET I. INTEREST. [Chap. III. 
 
 |2 \n-r 
 
 the expansion stopping at v n . 
 
 (37). (a) From first principles 
 
 =am + -2 |1=*1> + ^d-" 44 ) + . + ^(1-^-1 
 L * i i 
 
 = 46j + 2 - : -- . 
 
 (b) By Makeham's formula (J.I.A., xiv, 189), 
 
 46|2l =46) + '2 ^ 
 
 (c) By Gray's formula (J.I.A., xiv, 91, 182, 397), 
 1 '2 /lG-2 -2 
 
 (88).- 
 
 (39). 
 
 (40).-[43]-[45]. 
 
 (41). ^= -. ^eory of Finance, Chap, iii, (20)-(22). 
 
Chap. III.] . SOLUTIONS. 47 
 
 (42). To employ Gray's formula, the values of w 41 , A 4 i, A 2 w 41 ..... 
 must first be ascertained. 
 
 By the following method, the general law of the series and its 
 differences for any value of in can readily be ascertained. 
 
 We have 
 
 |3 
 
 
 . . . 
 
 e 
 
 . . . 
 
 Inserting the values of u { , Aw b A^ b A%! . . . and reducing, we have 
 
 Ww=w 3_6w 2 + 13ra-7. 
 (This formula exhibits the general law of the series) 
 
 Or for w= 
 
 A% 41 =6. 
 
 Now, applying Gray's formula, we have, for the value of the variable 
 annuity, 
 
 126 /59361 4682 240 
 
 = (when ^=-05) 117066. 
 
 (43). _ w = 55 126 259 484 837 ... 
 A = . . 71 133 225 353 .... 
 A 2 = ... 62 92 128 ..... 
 A 3 = .... 30 36 ...... 
 
 A 4 = . 6 
 
48 PART I. INTEREST. [Chaps. Ill, IV, 
 
 Value 
 
 = 55ff-4o| + 7140i 21 + 62040J 1 + 30#4Q| 5| + 6040, 5] 
 
 = (at 5 per-cent) 1,521,443. 
 (44) .2010151 + 102i5l 2| + 38ai5 j s| + 90IF > *! + i5 1 5] 
 
 CHAPTER IV (i). 
 
 (45). (A) -052310; (B) '052310; (C) -052310; (D) -052310. 
 (46). (DO -032167; (D a ) "032167. 
 
 1 v n 
 (47) . 0^1 = : . The nearer n approaches to oo , the more nearly 
 
 does dn\ approximate to 0^=-, .-. when n is large 0,7;=- approxi- 
 
 2- Z 
 
 mately, and i=. approximately. Inserting this value in the formula 
 
 approximately, 
 
 0^= we have 
 
 and i= 
 
 (Text-Book, Example (5), p. 171.) 
 For the case 094] =27, we have 
 
 ' = ~ 7 l + 7 = '' 3582 a PP roximatelv - 
 (48).- [54]-[63]. 
 
 CHAPTER IV (ii). 
 
 (49). [66], [67]. (J.I.A., xviii, 132.) 
 (50). Here the formula 
 
 -CO ([67]) 
 
 becomes v' n + (1 - ?/ w ) , where t?' w = (1 +j) ~ n , 
 
 J 
 
 1 (l+j)~ n 
 or l-0'^O' *), where 0^=- ^-r - 
 
Chap, IV. J SOLUTIONS. 49 
 
 (51). - [69]. 
 
 (62) f 
 
 100 
 
 whence n= ** - , which is independent of the rate i. 
 
 (Text-Book, Example (28).) 
 
 (53). By Makeham's formula, we have 
 
 rhence 
 
 Insert a value of v' n near to the true rate, and then deduce i' by 
 successive approximations. 
 Or, as follows: 
 
 but a'i' v' n = I 
 
 and 
 
 =:^ 
 
 100' 
 
 from which i' may be obtained by successive approximations. 
 (54). (Text-Book, Example (6), p. 134). 
 Here we have 
 
 Let 
 then 
 
50 PART I. INTEREST. [Chap. IV. 
 
 (55). (a) l,OOo{t>'+(l-V) ^JH = l,000{l-a'2o;(-05--03)} 
 = 750. 15. Id. 
 
 = 849. 4s. lie?. 
 (56). By Makeham's formula 
 
 A=C'+(C-C')4,, 
 
 we have 
 
 where '! is computed at the rate/ 
 
 The problem may also be solved as follows 
 The successive payments are 
 
 M .,/ 1\ M 
 
 -+,M( )= - 
 
 = . 
 
 n / n 
 
 = 
 
 The present value of these payments at the r&tej is 
 
 
 
 = Mr 
 
 L 
 
 as before. (This example illustrates the advantages of Makeham's 
 formula in dealing with similar problems.) 
 
Chaps. IV, VII.] SOr.UTTOXS. .") ] 
 
 (57). Lot P be the amount of the property at the commencement, 
 and P!, P 2 , P 3 , ..... the amounts at the end of the 1st, 2nd, 3rd, ..... 
 years. Then 
 
 P, = 
 
 Therefore 1 + (1 2pm) i = ; 
 
 whence 1 -f / = 2pm? . 
 
 Amount spent in^tli year 
 
 =pmiP p _ l =~-~P p _ l . 
 
 Amount left at end of j9th year 
 
 (58).- 76= (6 + 1) 
 
 whence 10*85714= a^ at rate x. 
 
 Also 
 
 whence n=33 approximately. 
 
 Then we have 10-85714=^ at rate x ; 
 
 whence, by formula (C), [57], #= -086265 
 
 CHAPTER VII. 
 
 (59), (60). [84], [85]. Tables and Formula, (Gray), Chap, ii, 
 (47)-(55), (68)-(76). 
 
 (61), (62). Theory of Finance, Chap, v, (28)-(30). Tables and 
 Formula, Chap, ii, (63)-(65), (74), (75). 
 
 E 2 
 
52 PART I. INTEREST. [Chap. VII. 
 
 (63). Let &n\ be the present value of an annuity-certain for n years 
 payable in advance, then 
 
 Similarly, let s^| be the amount of an annuity-certain for n years payable 
 in advance, then 
 
 Swi=*7+I| 1- 
 We have also 
 
 Chap, ii, [29]. 
 (64). Theory of Finance, Chap, v, (19), (23), (26). 
 
 (!+>!-* 
 
 MISCELLANEOUS EXAMPLES. 
 
 (65). Let a? be the effective rate of interest, and let it be assumed 
 that the half-yearly dividends can be immediately re-invested at the 
 rate cc : then the accumulated payments in respect of any year are 
 approximately equal to 
 
 2-5 + 2-5 l+ + 3 = (8 + 1-25*), 
 
 which, by the terms of the question, = 130# , 
 whence #='062136 nearly. 
 
 If the dividends can be re-invested at a rate of interest of i per 
 annum, the formula becomes 
 
 8+l'25*=180#, 
 
 8 + 1-25* 
 whence -^j , 
 
 in which, by inserting the value of *', that of sc can be obtained. 
 
Miscellaneous.] SOLUTIONS. 53 
 
 (66) . Here the accumulated payments in respect of any year, at the 
 effective rate x (the re-investments of dividend being made at the same 
 rate), become approximately equal to 
 
 But, in the solution to Problem (65), it was shown that 
 
 -25^=130^ 
 
 and the price to be given is therefore 134. 
 
 If the re-investments of dividend are made at the rate /, we have, 
 for the accumulated payments of any year, 
 
 -25*'. 
 But, in the solution to Problem (65), it was shown that, in this case, 
 
 5-25^=130^ + 4^, 
 
 in which, by inserting values for x and i, the required price may be 
 obtained. 
 
 It is also evident that, where re-investments of dividend are made 
 at the rate #, the value of the share just before payment of the June 
 dividend, is equal to the value just after payment of the December 
 dividend and bonus, plus six months' interest thereon, 
 
 = 130 (!+!) = (130 x 1-031068) = 134 approximately. 
 
 (67). (a) Dividing the income from interest by the " mean fund" 
 in the middle of the year, we have 
 
 = -043373 =4. 6s. 9d. per-cent. 
 
 1,037,500 
 
 This result represents the "force of interest" (8), and 
 the yearly rate realized may be deduced by the usual 
 relation, 
 
 8=log(l + =log, (l+ *) x 2-302585, 
 
54 PAHT i. 1KTEBEST. [Miscellaneous. 
 
 043373 
 whence 2.302585 =' 018837 = lo ^io( 1 + J 
 
 and ^='04433 = 4. tots. 8J. per-eent. 
 
 (J3) If i be the effective rate of interest, the fund of 1 ,000,000 
 becomes, at the end of the year, =1,000,000(1 + 0; 
 and the income (excluding interest), less the outgo, 
 forms a continuous annuity for one year of 30,000, 
 the amount of which, at the end of the year, 
 
 Hence, the interest earned in the year 
 
 = 1,000,000* + 30,000^-^ 4- . - .)=45,000; 
 \2 1J / 
 
 =4. 8s. Sd. per-cent approximately. 
 (68). Let n= the number of times interest is convertible 
 
 then (l + '^Y = 1-061678 
 
 n J 
 
 06V* /-06 -0036 -000216 
 
 whence lo g a'061678= 06- - - + 
 
 0018 -000072 
 
 or, -0598505 = -06 -- - + - -- ... 
 
 n n 2 
 
 aud 
 
 n n 
 
 from which it is obvious that n=l2 nearly; and substituting 12n for 
 n 2 , we have 
 
 .0001495=^- 
 
 whe " ce 
 
 = (-0018- -000006) 
 
 ^l/ 
 
 001794 
 
Miscellaneous.] SOLUTIONS. 55 
 
 '025 
 (69). Nominal rate = = '02(560 = 2. 13*. 2d. per-cent. 
 
 Effective rate (with quarterly dividends) 
 
 = (1 '00(5(55 ) 4 l = '026865 =2. 13s. 9c?. per-cent. 
 Instantaneous rate = 8=log c (l + t), where i = above effective rate 
 =log* r02G865=log 10 r026865 x 2-302585 
 
 = 0265 13 =2. 13s. Od. per-cent. 
 
 (70). We have (Alyelra Exponential and Logarithmic Series) 
 HI (mn l/mn\ 3 . 
 
 therefore 
 
 , V_i_V , I 
 
 3V2 + 2W 
 
 If i is a small fraction this becomes approximately, 
 
 1 1 + 0* + *)*'= *' 
 '!+( ^) I + ni' 
 
 Giving to n successively the values 1, 2, 3, ... n, we have therefore, 
 
 -i- ^~ + -^ + +- 1 
 
 l + t l + 2t l + 3 1-fi 
 
 0*+*)* 
 
 -JM 
 
 , 
 
 = - loge r^ p-r- approximately. 
 
56 PART ii. LIFE CONTINGENCIES. [Chap. I, 
 
 SOLUTIONS. 
 
 PART II. LIFE CONTINGENCIES (INCLUDING LIFE ANNUITIES 
 AND ASSURANCES). 
 
 CHAPTER I. 
 
 (1). Text-Book [Y\-[4>-]. 
 
 (2). If out of (m + n) trials, the result A lias happened m times, 
 and the result B n times, then the probability that the next trial 
 
 11 j-i, u- A 4. ^1 m 
 
 will produce the result A is strictly - = - , or, in 
 
 J ' 
 
 SO 
 the present case, - . (De Morgan on Probabilities, Chap, iii, 
 
 p. 65.) 
 
 This result is, however, based upon the assumption that all values 
 of the required probability are, a priori, equally likely, which cannot be 
 said to be true with regard to the probabilities of death. 
 
 (3). From the conditions laid down, the total deaths in any year 
 will be equal to the number of annual births. If this =1 we can obtain 
 by successive multiplication by p , p lt p% ..... , the values of Zi, Z 2 , 
 Z 3 ..... the survivors at the several ages. If we assume the numbers 
 
 living between ages and 1, 1 and 2, &c., to be - , - - 2 , &c., the 
 
 2 - 
 
 sum of these numbers will represent the total population. 
 (4).- [18]. 
 (6).-(a) [15]. 
 
 (/?) The effect would be to exaggerate the mortality, especially 
 at the younger ages. (J.LA., xviii, 107.) 
 
Chap. II. SOLUTIONS. 57 
 
 CHAPTEE II. 
 
 (6) . Expressing the given formula in terms of the number living 
 at each age, we have 
 
 (7). We have 
 
 (8). Proportion per-cent of married couples = 100 X - - X - 
 
 2501 2oll 
 
 = 81-29. 
 2236 237 
 
 = 8-12. 
 
 265 2374 
 = 10 X 2501 X 26IT 
 = 9-63. 
 
 (The remaining '96 per-cent represent the deceased couples.) 
 
 9).-() [15]. \- npxy . 
 
 (/J) [12]. l-(l-,,^)(l- ft ). 
 
 .-(a) [18]. n _ lto x ._,| ft = 
 08) [21]. w -H^(l- W - 
 
 .- [11]. M -= | ^x|,^= 
 
 [20]. n-ife=l?5-l-iffw- 
 
58 PAIIT II. - LIFE CONTINGENCIES. [Chap. II. 
 
 (12).- 
 
 _ PX+tt 'y+111 _ "x+n vy+n\ x+l _ 
 
 npx X n-\Py ~j J 7 -- J - ~j n-lpx+\ :y Xpx, 
 I'x ly ".v+l 'y ' 
 
 also 
 
 _lx+n ly + n-i _ lx+n 'i/+nl *y\ _ npx:y-\ 
 npxKn-ipy] -, ~ ' - - 
 
 '// 
 
 (13). 1 {1 (^. B M+W 
 
 (14). If the ages of the three lives are severally a?, y, 2, the 
 following are the various contingencies : 
 
 Die. 
 
 Survive. 
 
 Probability. 
 
 None 
 
 x.y.z 
 
 Px Xp v *pz 
 
 X 
 
 y.z 
 
 (l-p x )p y Xp 2 
 
 z 
 
 **9 
 
 (1~P*)P**P9 
 
 xyz None (1 p x } (1 p y ) (1 p z ) 
 
 It will be found on expanding these several probabilities that the total is 
 unity, which shows that all possible cases have been included. 
 (15). (a) l-(l- w _ 1 i^,)(l-_i|^)(l-_ 1 |^). 
 
 (ft) I(n-i\Sx X n-ilS V X n-l\tjz) . 
 
 (7) (l n-lpx) \_(n-lpy np y ')npz+ (n-\p z npz)npy\ 
 
 \px npx)npz+ (n-\p 2 np z )upx~\ 
 
 (16).-[27], [28]. 
 
 (17).- [34]. 
 
 (18). [34]. If , however, (n + l.)p is an integer, say K, the rth 
 and (r + l)th terms are equal, and the occurrence of either (/!) or r 
 deaths is equally probable, and more probable than that of any other 
 number. 
 
 (19) . Let p be the probability of any one of the lives surviving the 
 given period : then the required expression is the sum of the last four 
 terms of the expansion of {p-\-(\ p)} 7 , that is, 
 
 (20). [35]. 
 
Chap. II. J SOLUTIONS. 59 
 
 (21). (a) 1-(- 
 08) 1-(- 
 
 (22). Let =(1^), represent the probability of dying during 
 the year. 
 
 1000 
 
 1000 11000 
 
 (0 iT+V*V*i+ 
 
 (y) 2 20 ? 980 . 
 
 = snce 
 
 (23).-(a) =-00139. 
 
 08) = = -03333. 
 b 
 
 (24). If a number of persons are exposed to risk of death at the 
 same age #, 
 
 The "rate of mortality" (q x ) is the ratio of those dying 
 
 within a year to the number living at age x. 
 The "force of mortality" (/u. a .) is the annual rate at which 
 the lives are dying at age x. It may also be denned as 
 the limit of the expression 
 
 when M is indefinitely increased. 
 
 The " central death-rate " (m x ) is the ratio of the numbers 
 dying within a year to the average number living during 
 the year. 
 
 (Iv 20 a . 
 
 in x = ^ = /**+* approximately = -*- 
 '.c+i ^ <lx 
 
 x . 
 
 </.=- = - approximately. 
 
60 PART II. LIFE CONTINGENCIES. [Chap. II. 
 
 (25). 
 
 d x -\ + d x d x 
 
 according as -- > = < 
 
 
 
 55 ,5 
 
 (26). [37] (7:Z-4.,xvi,450.) It has been shown (Ex. 24) that ^ 
 
 l x l x +L i 
 
 is equal to the limit of the expression - when vanishes. By the 
 
 1 m 
 
 m* 
 
 definition of l x , it is the limit of - - . Hence ,<.= -r- l x . 
 dx 1 l x dx 
 
 m 
 
 (27). This arises from the general principle of the Differential 
 Calculus, that, where <f> x is any function of x 
 
 Id d . 
 
 (28).- ^=-. (Chap, i, [17].) 
 
 jJ*+t 
 (29) .-We have 
 
 d 1 
 
 di 
 
 and 
 
 whence - C l x+t 
 
 - C 
 
 ^c/ l x 
 
 (30). [44]. 
 
Chap. III.] 
 
 SOLUTIONS. 
 
 61 
 
 CHAPTER III. 
 
 (31). (a) The "average duration of life" (e x ) is the number of 
 years which persons of a specified age, taken one with 
 another, survive, according to the given table of 
 mortality. 
 
 ,=*+ 
 
 (/?) The "probable lifetime" (vie probable) is the number 
 of years which a person of given age has an even 
 chance of surviving according to the experience of a 
 particular mortality table. 
 
 = deduced from the equation -= =|-. 
 
 l>x 
 
 (y) The " mean age at death " is the average age to which 
 persons of a given age, taken one with another, will 
 survive, according to the experience of a particular 
 mortality table. 
 
 [8]. 
 
 Age 
 
 80 
 81 
 82 
 83 
 84 
 85 
 86 
 87 
 88 
 89 
 90 
 91 
 92 
 93 
 94 
 95 
 96 
 97 
 
 Average 
 
 Duration 
 
 of Life. 
 
 5-19 
 5-11 
 5-23 
 5-27 
 4-95 
 4-38 
 4-58 
 4-32 
 3-84 
 3-54 
 2-98 
 2-97 
 2-58 
 2-28 
 1-79 
 1-30 
 83" 
 50 
 
 Average 
 Age at 
 Death. 
 
 85-19 
 86-11 
 87-23 
 88-27 
 88-95 
 89-38 
 90-58 
 91-32 
 91-84 
 92-54 
 92-98 
 93-97 
 94-58 
 95-28 
 95-79 
 96-30 
 96-83 
 97-50 
 
 "Probable 
 Lifetime " 
 
 ft 
 
 3-75 
 4-21 
 4-00 
 4-25 
 4-20 
 3-70 
 4-25 
 3-75 
 3-17 
 3-00 
 2-50 
 2-75 
 2-50 
 2-25 
 1-75 
 1-25 
 ' '75 
 50 
 
62 PART IT. LIFE CONTINGENCIES. [Chap. III. 
 
 (33).- ^=(1-2,0(1 + ^+0 or '*=+ (1 -2*) (! + ^+0- 
 
 The values for q_ x being given for all ages, and the expectation at the 
 limiting age being zero, that for all younger ages could be found by 
 successively applying one or other of the above formulas. 
 
 (34). (a) At age (# + r), where d x+r is the greatest value of d 
 from age x to the end of life. 
 
 08) /years, where ^ = -L 
 '. 
 
 (35). -(a) [9]. 
 
 (/?) The probability of a life aged x dying in the wth year 
 =-i!2* = -y-^' But, by De Moivre's hypothesis, 
 
 'x 
 
 ' o ! 2?=i ! 2^= .... =n-\\%x=n .'2r=&c.= '=- for all values of n. 
 
 <x 
 
 1 ^ 
 .'. ^a.= - = = j' a .. 
 
 ^.r '.t- 
 
 This equality also follows from the reasoning in Solution (25), bearing 
 in mind that on De Moivre's hypothesis d, v -i = d x for all values of x. 
 (36). 2^+^(1 + ^+0 +2^(1 + ^+2)+ ..... 
 
 _ _1_ >r+1 4. ^' T +l ^+1 _, 'a^+2 | %+2 J?+J , 
 
 I J~ ~7 " 7 ~7 ' T~ ' 7 T ..... 
 M ^.r *aN"l 'J? '.c '.r+2 
 
 /o?7\ 
 (37) . - 
 
 '.r 
 mpx+nC.v+ n+m) 
 
 1 l>x+n 
 
 (38) .-[18]. 
 
 We have e. r =|+p. T (| + c. r+1 ). [Example (33).] 
 
Chap. III. I SOLUTIONS. 63 
 
 Whence p x = 
 
 I f <'. r - ! = <. r = <'.r + 1 = <, the above becomes 
 
 Also 6>. r _! = i +^. r _i (| + t' 
 
 If ^. r _ 1 =^ >r =^ a , +1 =^, these become 
 
 Then <af-i = t'a? n ly if ^=t'.r+ 1=^+2) and we have 
 
 ^+2=i+^(i + ^+2), whence e x+2 = - - - . 
 
 Z 1 p 
 
 (39). [1]. Assuming that the deaths are equally distributed 
 through the year, 
 
 mq x persons originally aged x who die in the first year live ^mq x years 
 mi\q x second ^m\\C[x 
 
 271-1 I 
 
 m n -\\<lx ,, nth -jr**-i\<l* 
 
 Thus the persons altogether live 
 
 and the average duration will be 
 
64 PART IT. - LIFE CONTINGENCIES. [Chaps. Ill, IV. 
 
 (40). Assuming the lives to die upon the average in the middle of 
 the year the mean age at death is equal to 
 
 I* 
 
 [21]. 
 
 The curtate expectation of two joint lives (#) and (y) is the 
 integral number of years which lives at those ages will, upon the average, 
 jointly survive according to the given table of mortality, and is obtained 
 by the formula 
 
 which is obviously equal to the expression given in the question. 
 
 CHAPTER IV. 
 
 (42).- [1], [6]. 
 
 (43). [6]. (J.I. A., xxvii, 158.) 
 
 ().- [8], [8]. 
 
 The following is an additional formula involving the ordinary 
 approximation for the value of //,#, but without making any assumption 
 as to the distribution of deaths. 
 
 /lx+t ly+t 
 T - { -^x+t' 
 f'x 
 
 but px+t= - 177 l nearly 
 
 /*ly+t'lx+t-l ly+t-lx+t+l j, 
 
 .-. the above = / ^-^ - at 
 
 */ Alsxy 
 
 _ 1 f*fl<y+t-lx+t-l ^ lx-\ _ ly+t-lx+t+l ^ ^+l \ 
 2*x \ lx-\ly lx Ix+l-ly '.v J 
 
Chap. IV. | SOLUTIONS. 65 
 
 (45). Let the year be divided into m parts. Then the probability 
 of (#) dying in the first of such parts and of (#) dying afterwards, is 
 
 -I m ~ l 
 m m 
 
 The probability of (x) dying in the second interval and (y) 
 in a later interval, is 
 
 - I 7 ^T_? 
 in t?i 
 
 The probability of (#) dying in the nih interval and ( t y) 
 in a later interval, is 
 
 1 mn 
 m m 
 
 The probability of (#) dying in the (m l)th interval and 
 (y) in the wth, is 
 
 1. !_ 
 
 Vfl til 
 
 Summing these expressions, we obtain 
 
 \fni- 1 m 2 1\ m 1 
 
 / _ I I I j 
 
 ~ ' ' * ' ' / ~ct -- 
 
 m\ in m mj 2m 
 
 If m be infinitely large this becomes in the limit =. 
 
 Or, if t represents the fraction of the year elapsed from its com- 
 mencement to the death of (y), then, since (#) is equally likely to die in 
 any portion of the year, the probability that he will die in the interval t 
 (.<?., before y) == ; since t may have any value from to 1. the total 
 
 /* J 1 
 
 probability that (x) dies before (#)= / t.dt=-. 
 
 *J o " 
 
 (46).- [8]. Qi,= \(\ - e ^= + * 
 
 2 v Py-\ PX 
 
 Since the expectation of life is equal to the annuity when the 
 
 N' 
 
 rate of interest =0, e x y= -- and the above becomes 
 
 _i/ N^. y -! N 
 
 2V /y 
 
66 PAET IT. - LIFE CONTINGENCIES. [Chaps. IV, V. 
 
 (47). _ [13] . if, i n the expression 
 
 i #(i Qb!i.f)i 
 
 the approximation given in the solution of Example (44) be adopted, we 
 have, as an approximate value, 
 
 The value of this probability may also be approximately 
 found by substituting another life z, whose expectation of life exceeds 
 that of y by t years ; then 
 
 Qi. 2^0= Qi* approximately. 
 
 (48) .-[14]. 
 
 (49).- [18], [21], [22]. 
 
 CHA.PTEE V. 
 (50). [7]- 
 
 (51). Assuming that the age-distribution of the community, that 
 is, the ratio of the number living at any age x to the total population, 
 
 = -^ , does not vary at different periods for any given value of #, let 
 lo 
 
 L , L!, . . . represent the number of persons living between the ages 
 and 1, 1 and 2, . . .; then if x be the age at which the pension is to com- 
 mence, the number of persons above that age would be L^ + L x+1 + &c. 
 = T X , and the number of persons below age x will be T T^: hence 
 (upon the supposition that money does not yield interest) the contribu- 
 tions required from each of the latter to provide a pension of 1 per 
 
 T 
 
 annum for each of the former = x . 
 
 lo la; 
 
 (52). Let x be the age at which promotions are made from the 
 junior to the intermediate class, and y that of promotion to the senior 
 class ; then 
 
Chaps. V, VI.] SOLUTIONS. 67 
 
 The first equation determines the value of #, and either of the others the 
 value of y. 
 
 The above solution will only hold good provided the mortality 
 experienced agrees with that of the table used, and that there are no 
 withdrawals otherwise than by death. 
 
 (53) . The average age of the population at any time will be 
 
 xH) + (L 2 x2j)+ ..... 
 
 Lo+In+Ls+ ..... 
 and the total number of years to be lived by 
 
 the L persons will be=|L + L]-hL 2 -f 
 theLi = i 
 
 the L 2 = 
 
 for the whole population 
 
 and the average expectation 
 
 (54) . Since the number of annual births must equal the number of 
 deaths, 70=209, and 
 
 rn -i f\ (\(\(\ 
 
 The average age at death = = = 47 '85. 
 
 /ft ZOi) 
 
 CHAPTER VI. 
 
 (55). Provided the formula employed to represent the number 
 living is such that its sum for successive values of x can readily be 
 represented by an algebraical expression, the values of the various 
 annuity and assurance benefits can be similarly expressed, thus obviating 
 the use of numerical tables. Such an expression is that given by 
 
 De Moivre, 
 
 ^=86-tf. 
 
 Messrs. G-ompertz and Makeham have also given formulas 
 
 k 
 
 and l x 
 
 F 2 
 
68 PART II. LIFE CONTINGENCIES. [Chap. VI. 
 
 The sum of these expressions for successive values of x can only be 
 expressed in the form, of series. 
 
 For advantages in the calculation of joint-life annuities, see Chap, vi, 
 [69], and Chap, xii, [20], [26]. 
 
 (56). Let 
 
 Ixt lx+\i Ix+z, ..... he represented by r r , tfr r+1 , <7r r+2 , ...... 
 
 d x 
 
 then q x = - = (l r) 
 
 l x ar 
 
 d x +n 
 
 "4" 4+8"T ..... 
 
 ~T~ 
 
 i> x 
 
 + ad inf. 
 
 ~ 2 a 
 
 = - + r+r^ + r*+ ..... ad inf. 
 
 I r_ (1 + r) 2-g 
 
 * 
 
 2 1-r 2(l-r) 
 (57).- m, [8], [10]. 
 
 (59).- 
 
 =logs+ (c x + l 
 =logs + c 
 To find the value of p x , we have 
 
 It will be seen that this expression is of the form 
 
 where A, B, and c are constants ( [14], [15]). 
 (60).- [19], [20]. 
 
Chap. VII. J SOLUTIONS. 69 
 
 CHAPTER VII. 
 
 '10 
 
 (82).- [5], [6]. 
 
 (63). In [25] it has been proved that 
 
 But A.=l 
 
 while i,<*+i)= 1 - d(l + |) 
 
 .'. A x >^+ 1 ^ 
 
 multiplying both sides by (1 + *')*, 
 (1+)*A> (*+ 
 
 Or Aa- >?<*) 
 
 that is, the value o an assurance payable at the instant of the death o 
 (#) is greater than that of a sum certain payable at the expiration of 
 the complete expectation. 
 
 (64).- [30], [39]-[44]. 
 
 (65) . From the investment of a unit in connection with a life (#) 
 we can obtain an annuity of i per annum for (n l) years, and an 
 assurance of (! + &') payable upon the attainment of age (x+ri) or at 
 previous death : that is, 
 
 (| A* + *) 
 
 = i\n-\<*x+ (1 +*') (|nAo; + lu, 
 
 = (1 + i) (\ n k x -f \ n a x ) \ n -\a x 
 whence | W A^= t; (1 + 1^-!^) - \ n a x . 
 
 (66). (a) v(\ + \ n -ia x ) represents the value of an annuity of 1 for 
 n years payable at the end of each year, provided 
 the life (#) be in existence at the beginning of the 
 year. 
 
 \ n a x represents the value of a similar annuity, provided 
 the life (#) be in existence at the end of the year. 
 
 The difference between the value of these two annuities 
 represents the present value of 1 payable at the end 
 of the year in which the life (or) fails, provided that 
 event happen within the n years. 
 
70 PART II. - LIFE CONTINGENCIES. [Chap. VII. 
 
 (/?) If an assurance on (#) were payable at the end of the 
 first year, its value would he v ; hut, under the 
 conditions of the given formula, the sum is payable 
 after n years or at the end of the year of previous 
 death : we must therefore deduct from v the value 
 of the interest thereon for the term of (n 1) years 
 during the life of (#) that is, iv\ n -\a xt or d\ n ^a x 
 
 (67) . A capital of 1 will provide a payment of i at the end of each 
 of the ( 1) years while the life (#) is living, with a return of the 
 capital and one year's interest at the end of the period or at the end of 
 the year in which the life fails : thus, 
 
 whence 
 
 (68).-[53], [54]. 
 
 It should be noted that in Dr. Farr's tables 
 
 (69).- (.) *=* 
 
 08) c ~ 
 
 Ux 
 
 (r) (0 JCT 
 (ii) W- 
 
Chap. VII.] SOLUTIONS. 71 
 
 (70). -[82]. 
 
 The annuity is equal to 
 
 
 I. [85]. 
 
 = Sfl* \jPx u 4- #? j-s 4- fe typxyz] 
 
 If an annuity of 1 be purchased on the joint existence of each possible 
 pair of lives, we shall evidently have a total annuity payment of 3 so 
 long as all the lives are in existence, and as each of the lives is included 
 in two pairs, the failure of any life will reduce the annuity to 1, and the 
 failure of a second life will extinguish it. It is evident, therefore, that the 
 required case will be exactly met by the purchase of an annuity of 1 on 
 each possible pair of lives, less an annuity of 2 during the joint existence 
 of all three lives that is, 
 
 (a xy + a xz + a yz 2a xyz ) . 
 
 >. [93]-[97]. (J.I.A., xxiii, 244.) 
 
 - [99]. The value of this annuity is approximately equal to 
 a xz where the value of z is deduced from the formula e e =e y + t. 
 (75). [103]. 
 (76).-[105]-[107]. 
 (77).- [113]. 
 
 (78) ._ (o) ^j 
 
 08) -- 
 
 Uxy 
 
 T> x+n ^y+n 
 (7) -t P ~i 
 
72 PART II. LIFE CONTINGENCIES. [Chap. VII. 
 
 (79). (a) a xy + - (a x -a xy ) + - (a v - ajev ) 
 
 m 
 ax v ~^ ^ \ a u~ a xy) + ( a x a xu) 
 
 IV 
 
 = a x +-(a y -a xy ). 
 
 (80) . A^= 1 - 
 
 The annual premiums would not follow the same rule, for 
 
 -f 
 
 - wlm w^a; T |w^^ 
 
 (82). Let the several ages be w, x, y, z, then the interest of (w) 
 is equal to 
 
 which may be symbolically expressed ( [85]) 
 
 ;py + W2 + a lvxz ) 
 For two lives (w), (#), we have 
 
 %a wx + ( w - fl wa? ) =a w ia,a.= w (Z i-Z 1 ) . 
 For three lives (?), (^7), (y), 
 
Chaps. VII, VIII.J SOLUTIONS. 73 
 
 For four lives (as above) , 
 
 The expression for n lives is thus evidently 
 
 which may be symbolically expressed as 
 
 , rio ge (i+Z)-i 
 
 " w L~~z J 
 
 and is equal to 
 
 ..... ) + ..... =fc - (ciwxyz ..... (n)) 
 
 If all the lives are of the same age (# ) , the expression becomes 
 n-l Q-l)Q-2) ( M 
 
 * i^ ^au' rr 
 
 ~ Uxxxxx ..... (n) 
 
 CHAPTER VIII. 
 
 (83) . A capital of 1 will produce an income of I per annum during 
 the continuance of any status, the value of which is by hypothesis iQ; 
 and, at the end of the year in which the status fails, a payment of 
 > * ne present value of which is P(l-M), hence 
 
 
 If the status be a single life (#), we have 
 ! 
 
 whence (multiplying by D 
 (84).- [33]. 
 
74 PAST II. LIFE CONTINGENCIES. [Chaps. VIII, IX. 
 
 (85).- A^ = l 
 
 Hence, applying the principle set out in [33], the tables must be 
 entered with the value of a temporary annuity on the life (x) for (iil) 
 years. 
 
 (86).- [32]. P,= -^L = ^L . 
 
 hence, by calculating the value of this expression for successive values of 
 Aa., we have the table required. 
 
 cl=P 
 
 d 
 and 
 
 (88).- 
 
 697342 
 
 = '04. 
 
 K 17-433558 
 (89). [24]. (Chap, xviii, [52].) 
 
 CHAPTER IX. 
 
 (90). [6]-[9]. 
 
 (91). It is shown in [21], [22], that the value of an annuity 
 upon a life (a?), payable m times a year, 
 
 and (4) = o? + f 
 
IE 
 
 NIVERSITY 
 
 / 
 
 'SOLUTIONS. 75 
 
 Chap. IX.] 
 
 the amount of the error is therefore +yV(/ jt # + ^) ^ u the case ^ a half- 
 yearly annuity, and + -/*-(/*.& + 8) in the case of a quarterly annuity. 
 
 (92)._ [38]. i>L JH) = approximately J(4 m) + a * l) ) 
 I/ m 1 . w + l\ 
 
 = (^a? + i) approximately. 
 (93). [39]-[41]. 
 (94). From the result in [41], we have 
 
 TVO\ *-X 
 
 and the half-yearly premium 
 
 JX2) 
 
 = -f == 1 _i/p g . ^ a Pproximately. 
 
 -i 2 ' 
 
 If m=cc , this becomes 
 
 (96).. =+ n imatel 
 
 ... 
 
 This expression may also be written 
 
76 PART II. LIFE CONTINGENCIES. [Chaps. IX, X. 
 
 (97). i4 4) =4 4) -^j 
 
 , 
 
 8 
 
 , 
 8 
 
 This expression may also be written 
 
 CHAPTER X. 
 
 (98).- [12], [16], [17]. 
 
 (99) . Let i be the effective rate of interest, and let 
 
 be the nominal rates when interest is payable half-yearly, or m times a 
 year respectively. 
 Then we have 
 
 whence (1) Aj?ss 
 
 (2) A? l) = 
 
 -i /2] 
 
 If m=oo , 7' (m )=8, and ^ } =0, and we have 
 M 
 
 (3) A=l-8*. 
 
 (100). Chap, ix, [13] ; Chap, x, [7]. 
 
 Let the year be divided into m equal intervals ; then, as it is 
 equally likely that a given life (x) will fail during any of these intervals, 
 
Chaps. X, XL] SOLUTIONS. 77 
 
 the present value, at the commencement of the year of death, of the sum 
 payable at the instant of death will be, upon the average, 
 
 m-1 m 
 
 Let m=cc , then this expression becomes = ^ (v). 
 
 o b 
 
 If the sum be payable at the end of the year of death, the 
 present value at the commencement of that year will be uniformly v. 
 Thus 
 
 A X :A X = g (v):v ; 
 
 ^ f i i 2 i 3 \ 
 
 or Aa.= g ( A. r ) = (I + - - + ^J A. r approximately. 
 
 (101). l=z| w _ 1 
 
 whence A.xn = 1 oi^,^ . 
 
 (102) .-[20]. 
 
 CHAPTER XI. 
 
 (103). Chap, ix, [22]. 
 
 m 1 m 2 1 
 
 If w=oo, 
 
 whence, eliminating the element of interest, 
 
 (104).- [3], [5], [7]. 
 
 (105).- 4 2) =4 2) + ^ - ^ , formula (11), 
 
 f T~4i 
 
 4 12 
 
78 PART II. LIFE CONTINGENCIES. [Chaps, XI, XII. 
 
 (107).- [11]. 
 
 CHAPTER XII. 
 
 (109) .-[12]. 
 
 but npw-S^^l 
 
 therefore if w is deduced from the equation 
 
 or from the equivalent relation 
 
 the probabilities for the single life (w) will be identical for all values of 
 n, with those for the joint lives (#), (y). 
 
 (111). Chap, vi, [28] ; Chap, xii, [26], [27]. 
 
 (112). [43]-[45]. (J.I.A., xv, 401.) 
 
Chap. XIII, ] SOLUTIONS. 79 
 
 CHAPTER XIII. 
 
 (114). [3] -[5]. The single premium may also be approximately 
 obtained by the following process, in which no assumption is made as to 
 the distribution of deaths. [See also Solution of Ex. 44, p. 64.] 
 
 V X ly 
 
 f f ly+t "x+tl ''X+t+l 7 , j i 
 
 = /# r . dt, approximately 
 J ly M x 
 
 = f ay:x - 1 p x dy.. x+l ) approximately. (J.I.A., xv, 123.) 
 J \ p x -\ / 
 
 (115). A^=A x -Ai y 
 
 The life y must be medically examined, but if y is much older than x, 
 both lives should be examined. 
 
 (116).- 
 
 . \n^x-\:y _\nfh 
 
 I. [IB]- 
 (118). ] t A^ l> = -^ 
 
 D X y 
 
 V(l^xl:yl ^x+tl:y+tl) (-N#: y' 
 
 then 
 
80 PART II. LIFE CONTINGENCIES. [Chap, XIII, 
 
 An approximate value for the single premium can also be 
 found as follows [see Solution to Ex. (114)] : 
 
 - T L /<t+n 7 lx+t+n-l lx+t+n+l ^ 
 
 \tA-x +J v n YJ- l>y+n*- ^ ~ ' " n approximately 
 
 v t 
 
 If t=\ this becomes 
 
 The value of this assurance may also be found approximately by 
 substituting another life 2, whose expectation of life exceeds that of y 
 by t years, then 
 
 Ai : ^jj)=Ai approximately. 
 
 (120). We have 
 
 If ^75: 7o=#W} this becomes approximately 
 
 Then, if ^=^ 75 + l, y=e 70 rl-l, e s =e w -\-l the expression becomes 
 approximately 
 
 Or we may proceed as in the previous example (119), where it is 
 shown that 
 
 T"i i -D#+l fO"y-x \ 
 
 A. Sf (jO= t ^+ -^- \ : -- px+}-d y:a; +2j approximately. 
 
 In the present case y=75.70 and ^=48; the above expression there- 
 fore becomes 
 
Chap. XIII.] SOLUTIONS. 81 
 
 which becomes, if #70:75=^05 
 
 4 g r;o: 48+^75: 48 W:48 /- 
 
 r- -- _>.19(70: 50+^75: 50 w:5o) 
 
 48 L ^48 J 
 
 D 4 
 
 A more accurate solution may be obtained by the use of one of the 
 formulas of approximate summation given in Chap. xxiv. [See 
 Example (265)]. 
 
 The divisor for the annual premium will be either 
 
 (1 ' ^48: 70: 75 / ^*"T ^48:70 ^48 : 75 ^48:70:75* 
 
 in the case where it is payable until the failure of the joint existence of 
 (48) and the survivor of (70) and (75) ; or 
 
 ^48:7^501)) =(l + ^4s) FT" 0*49 ^49:75 ^49:70+ ^49:70:75) > 
 
 in the case where it is payable until the failure of the joint existence of 
 (48) and the survivor of (70) and (75) and for one year longer if (48) 
 live so long. 
 
 (121). The single premium for the benefit required is equal to 
 
 D 
 
 x+n-. 
 
 , 
 n ' 
 
 , Pa?+n!y+7tF 1 If 
 
 -^- I 1 - 2! 
 
 , 
 
 
 Mr+w %+wf" 1 lf n 7/T . \ . a?+-l:y+n ^+n : y+w-i)~| 
 
 ^fji T~ 1 o) 1 " a ( L + ^+n:y+n)-\ -- 
 -Ua: *y L ^ < ^a?+i-l Py+n-i I.J 
 
 all of which values will be found in the Institute Tables. The annual 
 premium would be obtained by dividing the above expression by 
 
82 PART IT. LIFE CONTINGENCIES. [Chaps. XIII, XIV. 
 
 (122). [25]. The divisor for the annual premium would be 
 (123). A 1 i; f ,= A x -AJ : ^ 
 
 A A 1 A 
 ** -"-xy " 
 
 (124). [32], [33]. (J.I.A., xv, 119.) 
 
 CHAPTER XIV. 
 (125). [17]. -***= 
 
 . . _ - 
 
 (126). [11]. f--|^. 
 
 (127). a x \ 7z =a--a x .- 
 
 = ay + GZ &yz axy a xg + a xyz . 
 The divisor for the annual premium is =(l + ff a: -) 
 
 (128). B's interest is equal to 
 
 ^ x \y z -\-a~\y 
 
 (129). 
 
 _ A 
 
 5S 
 whence A= 
 
 2(40 + y +v) *' 
 
Chap. XIV.] SOLUTIONS. 83 
 
 (130). The value for the first n years is equal to 
 
 and after n years is equal to 
 The total value is thus 
 
 ty a xjl \ n a xy = a-j \ n a xy . 
 The annual premium would be obtained by dividing the above expression 
 
 (131) . Single Premium = \ n a x \ n a yy . 
 
 Divisor for Annual Premium=(l + f M _ 1 ^7,, v/ ). 
 (132). <i y \ri=axi\ a\:y 
 
 = a-x + a t\ \ t a x a xy \ t a y + \ t a xy 
 
 (133) . For the first t years, reckoning from the end of the year of 
 death of (#), the value of the annuity is evidently equal to 
 
 After t years the value is equal to 
 rn w 
 
 a y+ tax:y+t)=-^ 
 The total value is thus equal to 
 
 Another solution may be obtained as follows : The annuity for the 
 first t years is payable if (#) be dead, irrespective of the life (y), and its 
 value is equal to 
 
 Z 7 
 
 2>{v n = Ctt\ \tffx 
 
 l x 
 
 G 2 
 
84 PART II. - LIFE CONTINGENCIES. [Chap. XIV. 
 
 After t years, the annuity is payable (i) in the event of (x) having died 
 not more than t years previously, irrespective of the life (y) ; or (ii) in 
 the event of (#) having died more than t years previously, provided (y) 
 be living : its value is therefore equal to 
 
 -S! 00 nf x+n ~ t '*-H _ * 'x+n-t l.t/+n\ 
 
 *t+l*> I- 7 j -j ) 
 
 \ I'x * * ' 
 
 Adding to this the value for the first t years, we have 
 
 This is equal to the value previously deduced, for 
 
 /I N l i)* 
 \i J d 
 
 (134) . The required single premium is equal to 
 
 ^X+t:y+t f N 
 
 "VW f) \ a y + t\x+t) 
 
 _, x )x+t-.y+t , 
 
 ^>xy 
 
 The divisor for the annual premium is equal to 
 (135) .-[18]. 
 (136).- [19], [37]. 
 (137).- 
 
 . Ay A 1 dsiyl + da,- d(a~ a y ) 
 (a) _ a - = 
 
 _ d(a x a xy ) _^__ i 
 a x a xy 1 + i 
 
 08) [21]-[85]. 
 
 (138). [24]. It is assumed that the first payment of the annuity 
 will be made at the end of the year of death of (y), and that there will 
 be no proportionate payment at the death of (a?) . 
 
 (139).- [28]-[88]. 
 
Chaps. XV, XVI.J SOLUTIONS. 85 
 
 CHAP TEE XV. 
 
 (140).- [6]-[ll], [18], [19]. 
 
 (141). - [8]. 
 
 (142). The value of the annuity is evidently equal to 
 
 1 /* 
 
 j-j- I V 
 
 lijl z \s 1 
 
 This expression may be summed by one of the formulas of approximate 
 summation in Chap. xxiv. 
 
 The annuity is also equal to that of (i) a reversionary annuity to (#) 
 after (z) ; (ii) a reversionary annuity to (y) after the last survivor of 
 (#) and (2) provided (2) fail first, that is 
 
 The value of a\ u may be obtained by reference to [6]-[ll], 
 [18], [19]. 
 
 (143). [12], [13]. 
 
 CHAPTER XVI. 
 
 (144).- [3]-[5], Chap, vii, [14]. 
 (145).- [6]. 
 (146).-[9], [10]. 
 
 = 
 
 ~ 
 
 From this expression it will be seen that -^^ represents the accumulated 
 
 -Da? 
 
 amount, after t years, of the value of an annuity on a life aged (x ), 
 allowing for mortality and interest. If deduction be made of the 
 
PART II. LIFE CONTINGENCIES. [Chap. XVI. 
 
 accumulated amount of the payments made under the annuity (also 
 allowing for mortality and interest) 
 
 the difference is equal to 
 
 (147). Chap, vii, [48]. We have 
 
 P, 
 
 P; 
 
 But M^N^-N*; E,+ 
 
 .-. (vA)o,= 
 
 (148).- 
 
 1>! 2P. r+ao /N a . +ao 1 
 
 
 *' 
 
 If' 5 
 
 and S 
 
 the above becomes 
 
 (149). The single premium is equal to 
 
Chap. XVI.] SOLUTIONS. 87 
 
 the annual premium (a) payable for n years 
 
 (/?) payable throughout life 
 
 n r in - (Ma? Man-sap) in 
 
 This may be also expressed as 
 
 _ Na,- T y(S*+i--S*+) P^-Affl^-N^+M) , 
 
 ' 
 
 (151).- [27]-[37]. 
 (152).- [39]-[43]. 
 (153). Equating the benefit and payment side, we have 
 
 = 7r(N x _i Nx+n-i) [where C and N are calculated at the rate 
 
 of interest /], 
 which becomes 
 
 [ri4-f^* n 
 where 6^, = ^- ^4-- J, 
 
 hence 
 
 (154). If ./=*", the denominator in the above expression becomes 
 _0 - { (1 + (^N,-_^N,) +' 
 
88 PART II. LIFE CONTINGENCIES. [Chap. XVI. 
 
 which becomes, after simplification and reduction, 
 
 =!> 
 
 This result may also be expressed in the form - , which is obviously 
 
 *5) 
 
 equal to the annual premium for a deferred annuity when the element of 
 mortality is entirely eliminated during the term of n years. 
 
 (155).- [53], [54]. 
 
 (156). If TT be the net annual premium, and 7r(l + &)+ the 
 corresponding office premium, we have 
 
 whence 
 
 (157). Making the same assumptions as in the previous solution, 
 we have 
 
 
 /iro\ 
 
 whence 
 
 (159).(7".J.-4., xxi, 67.) 
 (160).- [92], [93]. 
 
Chaps. XVII, XVIII.] SOLUTIONS. 89 
 
 CHAPTEK XVII. 
 
 (161).- [6]. 
 (162).- [7]-[ 
 
 o 
 
 (061 + 050+045) 
 
 1-A 
 
 (164).-[20]-[22J. 
 
 (165). Let x be the age of the present incumbent, s the annual 
 income, K the stipend of the curate-in-charge, /"the expenditure at entry, 
 and y the age at entry of the new incumbent : then the value of the nth 
 presentation is equal to 
 
 },-! j ( s _ K )dy f] = A x ALy n ~ 1 { (s K) (<% + i) f} approximately. 
 (166). Ax{(s- K )d y 
 
 = approximately 
 
 CHAPTER XVIII. 
 
 (167). Chap, xvi, [14], [15] ; Chap, xviii, [4], [5]. 
 
 (168). If TT XII \ be the net premium paid by the l x persons, the 
 accumulated amount paid (a) by the l x + n survivors will, at the end of 
 n years, be equal to 
 
 and the accumulated amount paid (/?) by those dying in the several years 
 will be equal to 
 
90 PAIIT II. - LIFE CONTINGENCIES. [Chap. XVIII. 
 
 The whole accumulation is thus equal to 
 
 
 PJ--+M Nj-! N-p+M-i _ 
 
 (169).- [18], [22]. 
 (170). 
 
 _ 
 
 ' 
 
 As regards adult ages, at which alone policy-values are practically in 
 question, the value of a x decreases as age increases ; but at the com- 
 mencement and end of the mortality table the value of a x+ \ is frequently 
 greater than that of a x , as in the Northampton Table from age to 7, 
 in the Carlisle Table from age to 6, and 91 to 94, and in the Life Table 
 in the Text-Book from age to 4. 
 
 (171). [17]. 
 
 MAT 
 32 -UN j; _J- % | 
 
 v 
 
 n v x 
 
Chap, XVIII. ] SOLUTIONS. 91 
 
 This relation may be proved also from the following considerations : 
 An annual payment of P x for n years may be regarded as made up of 
 (i) the annual premium to cover the risk of death during the term 
 = P :cu ; (ii) the annual premium to secure a sum equal to the value of 
 the policy at the end of the term, if the life be then in existence 
 =Va.Pa.,7j; that is, 
 
 P M 1 xn\ ~\~ n x tain.' 
 i IT- x -tjcwl 
 
 whence n^= - 
 
 p _ p 1 -. 
 
 The formula - ^~ a ^ so exhibits the value of a policy, effected 011 a 
 
 ITCH? 
 
 life (#) after n years, as the accumulation, allowing for mortality and 
 interest, of the amount paid in excess of that required each year for the 
 actual risk of death. 
 
 (173).-[38]-[44]. 
 (174).-[49]-[65]. 
 (175).- [51], [56]. 
 (176). [69]. We have 
 
 f A ^ V 
 
 /i3\ Y' xxn _ x x ...) (vcp 
 
 ' 
 
 In the second equation vc takes the place of v in the first equation 
 throughout : it is therefore evident that the policy-values by Table (B) 
 will be equal to those under Table (A) when the rates of interest adopted 
 are such that the present value of 1 due a year hence equals v in the one 
 case and vc in the other; that is, the policy- values by Table (B), 
 computed at the rate of interest i, will be equal to the policy-values by 
 
 Table (A), computed at the rate of interest (- -- 1 ). 
 
 (177). From [45]-[68], and especially from Tables F, G, and H, 
 it will be seen that no conclusion as to the relative policy-values at 
 different ages and durations can be arrived at from the fact that the rate 
 of mortality in one table is throughout greater than that in another 
 table. 
 
 (17S).-[76]-[79]. 
 
92 PART II. LIFE CONTINGENCIES. [Chap. XVIII. 
 
 (179) . [81] . The modified formula would be 
 
 -- tf *(!+'*+). 
 
 (180).- [94]-[100]. 
 (181).- 
 
 _ A x+n. m\ -t. 
 ?x+ 
 
 We have also 
 
 
 -\<*>x+n) 
 
 1 A.x-\-n.m\ 
 
 d _ A. x -}-n.m\ A-x.n+m 
 
 (182). [108]-[112]. 
 
 The value of the endowment policy after n years, by the 
 retrospective method, is equal to the accumulated net premiums, less 
 the accumulated claims in respect of premiums returnable at death, 
 that is, 
 
 Rx ~ Rx+n nM-x+n 
 
 The value by the prospective method will be 
 
 r- /-i _j_ \ . -| ^(^-a:+re ^-a:+n+m) "I" h'a 
 1 
 
 If the identity of these two expressions be assumed, and the value of TT 
 deduced therefrom, it will be found that 
 
 This being the correct expression for TT (see Chap, xvi, formula 47) the 
 identity of the above expressions is established. 
 
Chap. XVIII.] SOLUTIONS. 93 
 
 (183). 
 
 (184).- [19], [122]-[124]. 
 
 (185). [129]-[134]. 
 
 (186). It would be necessary in the first instance to ascertain the 
 net premium to be valued, and to do this some assumption must be made 
 as to the loading ; <?.y., if we assume this to consist of m per-cent upon 
 the gross premium and a constant of c per-cent, we have the following 
 equation to determine the net premium 
 
 100M 41 = 
 
 Hence, the net premium after the second seven years will be 
 /100m 
 
 V 100 c )~ say ' 
 
 and the value of the policy, the age of the life being now 69f, and the 
 next annual premium due in three months' time would be 
 
 28| V 41 Ag9f ( 4" ~f" ^69f ) -P 
 
 (187).- [14], [15]. 
 
 (1) The premiums must be net, without expenses, or the 
 
 loading exactly absorbed by expenses. 
 
 (2) The premiums must be payable at the beginning of each 
 
 year. 
 
 (3) The rate of interest assumed must be identical with that 
 
 realized. 
 
 (4) The rate of mortality assumed must be identical with 
 
 that experienced. 
 
 (5) The claims must all be payable at the end of the year. 
 
 Assuming that l x policies were effected n years ago at age #, the 
 following equation exhibits the relation stated in the question : 
 
 or, dividing throughout by l x + n -\ , 
 
 -iVa.(l + *) + P. r (l + i) = 
 
 which may be written 
 
94 PART TT. LIFE CONTINGENCIES. [Chaps. XVIII, XIX. 
 
 In the case where the premiums are payable at intervals throughout 
 the year, the above expressions will be modified as follows : Making the 
 usual assumptions that a group of lives entering in the course of the nth 
 year prior to the date of valuation at office age (#) , were on the average 
 of the precise age (# -i) at entry, entered on the average (n%) years 
 ago, and completed the average age (x + ii} at the date of valuation, we 
 have (assuming yearly payments throughout) the following expressions 
 for the value of the policies a year ago and at the present time 
 respectively, 
 
 (i + a?+ w-i) 
 
 i i 
 
 *-A 
 2 ) 
 
 . 
 
 V 1+**-* 2 
 
 whence, from the relations, 
 
 and P{r _ i 
 
 we can obtain the expression 
 
 showing the desired relation. 
 
 CHAPTER XIX. 
 (188). We have 
 
 The assurance required is therefore an endowment assurance on (or), 
 payable at age (x + n + V) or at previous death. The sum assured is 
 equal to 
 
 and the annual premium to 
 
Chap. XIX.] SOLUTIONS. 95 
 
 The total outlay is made up of the first premium and the value of the 
 life interest 
 
 -*- - 
 
 The life interest provides 
 
 as the annual premium, and 
 
 d 
 
 as interest on the total outlay, the sum of these two expressions making 
 up unity, the amount of the annuity. 
 
 The sum assured provides the return of the total outlay, with interest 
 thereon for the last year. 
 
 (189). [3], [10]. The value of the life interest, the age of the 
 life tenant being (#), is equal to 
 
 The sum to be assured is equal to -- - , and the value of the policy 
 
 Xjp + a 
 
 o 
 
 after t years is equal to -- -j[l ( 
 "art" a 
 
 The value of the life interest at the end of the t years is = (s x a x+ t). 
 The value of the policy and of the life interest together is therefore 
 
 = s t - 1), a constant value irrespective of t, and equal to the 
 
 \lo?~J~ d / 
 
 value of the life interest at the commencement. 
 
 If the value of the life interest and of the policy, after t years, be 
 computed upon office rates of premium throughout, we shall have 
 
 Market value of the life interest after t years 
 
96 PART It. - LIFE CONTINGENCIES. [Chap. XIX. 
 
 Value of the policy after t years, computed by the "re-assurance" or 
 "hypothetical" method (Chap, xviii, [33]-[36]) 
 
 The sum of these two values being, as before, 
 
 (190). We have 
 
 Sum assured = - 
 
 i x~\ 
 
 Value of policy after n years 
 
 (191).- [28], 
 
 Since the value of a reversionary life interest of 1 is equal to 
 
 an advance of 1 will represent the value of a life interest of 
 
 The sum assured for a reversionary life interest of 1 is equal to 
 
 /. the sum assured for a reversionary life interest of 
 
 l- 
 
 is equal to - - 
 
 The cost of the annuity purchased during the joint lives is evidently 
 
 ' 
 
 The total outlay is made up of (1) the value given for the 
 
Chap. XIX.] SOLUTIONS. 97 
 
 reversionary life interest=l; (2) the cost of the annuity purchased 
 during the joint lives = ^ x (3) the first premium 
 
 jj t 
 for the assurance = - , the sum of these values being 
 
 l~(P+dt)(i+*i) 
 
 equal to - - ~ , the total outlay. 
 
 The annuity will provide (1) the premium for the assurance 
 p 
 
 :; /-p _i_ J\/i i \ 5 (2) the interest upon the total outlay 
 * (,-t x~ - 
 
 The sum assured will provide the return of the total outlay, with one 
 year's interest thereon, at the end of the year of the failure of the 
 life O). 
 
 Let the age of (#) at the end of the year of death of (y) be 
 ; the value of the annuity is then equal to 
 
 and the value of the policy is then equal to 
 
 the sum of these values being equal to 
 
 an expression independent of n, and which, when put in the form 
 
 ( 73 ^ 1 )( n 7^~~T^7T~ \l W} ^ ^ e seen ^ represent the value 
 \P x + d /U--(!Vh0(l+r)/ 
 
 of an immediate life interest on (^p), the amount of the life interest 
 being equal to that involved in the question. (J.I.A., xiv, 426-7.) 
 
 (192).- [26], [27]. 
 
 (193). The net value is =a x a x: -, or in market values, and allowing 
 for whole-world and issue premiums, 
 
 m+n 
 
 1- 
 
 (194).-[41]-[46]. 
 
98 PAET II. - LIFE CONTINGENCIES. [Chap. XX. 
 
 CHAPTER XX. 
 (195). The value of the sick allowance to age 70 is ( [5]) 
 
 But z x = by hypothesis A + B^.r, whence we have 
 * 
 
 +r> 69 ) 
 
 70 * 
 
 - 
 
 (196). We have, by Table (A), 
 
 s x =v i *\_ 
 and, by Table (B), 
 
 Comparing the expressions within brackets [ ], we see that v, in the 
 first formula, is throughout replaced by #(1 K) in the second formula; 
 therefore the value of 4(1 K )* by Table (B), where 4 is computed at 
 the rate *, is equal, for all values of #, to the value of s x by Table (A), 
 
 computed at the rate f - -- 1 ). 
 \1 K J 
 
 (197). If at any given age x there be d x deaths, there will by 
 hypothesis be 2d x persons constantly sick throughout the year ; that is, 
 
 there will be -- - - d x weeks' sickness ; the cost of which will be at 
 
 QOpf 1 
 
 10s. per week = d x =52'l&d xt and the value of the sick allowance 
 to age 65 will be =^52'l8d x =52'lS 
 
Chaps. XX, XXI.] SOLUTIONS. 
 
 (198). [l]-[5]. (J.I.A, xxvii, 280, 281.) 
 
 K.-K, 
 
 99 
 
 (y) 
 
 _ -tv. r jx 70 
 
 ~DT~ 
 
 Jv fin 
 
 where K;,,= 
 
 (199). [8]. The formula (for the present value of an allowance of 
 1 per week) becomes 
 
 (200). (J.I. A., xxvii, 281.) 
 
 KK* + -K 7 3 ) + 13N 70 + 10M, 
 
 . 
 
 N r -N 70 
 
 and the weekly contribution, loaded for expenses, is = ^ . The 
 
 "7 x 52 
 reserves after n years are equal to 
 
 + (K x | n -K 70 ) +i(KJ H -K 7 2 ) + i(K x j >t - 
 
 CHAPTER XXI. 
 
 (201).- [43], [44], [101]. 
 
 (202).-[49], [52]. 
 
 (203) .-[54], [57]-[60]. 
 
 (204).-[57]-[59]. 
 
 (205). Gray's TrtiZes and Formula* Chap, vi, (193), (222). 
 LEMMA. If B denote the present value of a benefit of 1 upon a given 
 life or combination of lives, and such that, in the case of a combination 
 of lives the risk is determined by the failure of any one of them ; and if 
 B! denote the present value of a similar benefit on a life or combination 
 of lives respectively one year older than those on which B depends ; if, 
 
 * This Treatise being out of print, and practically inaccessible to students, the 
 solutions to problems (205) and (206) have been extracted in full. 
 
 IT 2 
 
100 PART II. LIFE CONTHS'GEIS'CIES. [Chap. XXI. 
 
 moreover, p denote the probability of a payment of B being received in 
 the first year, and IT the probability of the single life, or of all the lives 
 on which that benefit depends, surviving a year; then will the following 
 equation always subsist : 
 
 For, if the benefit makes its payments at the end of the year in which 
 they respectively become due (as is always the case unless it be otherwise 
 expressly stated) the value in respect of the first year is obviously vp. 
 And the value in respect of the years following the first is v-n-Bi ; for B! 
 is the value, at the time it is entered upon, of the remaining portion of 
 
 the benefit, TT is the probability of its being entered upon, and vl = - - . J 
 
 is the ratio in which the value is diminished on account of the time 
 that has to elapse before it is realized. Hence, a whole being equal to 
 the sum of its parts, 
 
 or 
 
 In this expression we have, for the benefit 
 
 ir=p x , B=A^ 
 
 p=(lp x ), Bi=Aa. +1 
 
 (206). Gray's Tables and Formula* Chap, vi, (194), (197), 
 (198). The benefits to which the demonstration applies and in regard 
 to which, consequently, the equation deduced subsists, must fulfil these 
 two conditions : First, that the payment in respect of any one year in 
 which it may become due shall be always 1. The fulfilment of this 
 condition obviously precludes the application of the formula to increasing 
 or decreasing benefits. Secondly, the benefit must be such that in the 
 case of two or more lives the risk will be determined by the first failure 
 that takes place from amongst those lives. The fulfilment of this 
 condition excludes from the application of the formula benefits depending 
 
 * This Treatise being out of print, and practically inaccessible to students, the 
 solutions to problems (205) and (206) have been extracted in full. 
 
Chap. XXL] SOLUTIONS. 101 
 
 upon specified orders of survivorship amongst three or more lives. 
 Thirdly, the benefit B l5 it is stated in the Lemma, must be "similar" to 
 the benefit B. The only restriction here implied, beyond that of its 
 being of the same amount, and subject to like conditions in respect of 
 the life or lives on which it depends as Bj , is, that its duration shall 
 extend to the same period of life : in other words, that - both benefits 
 shall cease at the same age. It is no matter what that age is, whether 
 the limiting age of the table or any less tabular age ; the formula will 
 still apply, provided only the age of cessation in respect of both benefits 
 is the same. The formula consequently holds in the case of temporary 
 as well as whole-life benefits. Fourthly, the formula holds also in the 
 case of deferred benefits. To adapt it to such it is only necessary to 
 make p=0 for those years in which no payment is made. This reduces 
 the formula, in the case of deferred benefits, to B = y7rBi. 
 
 The following formulas show the application of the Lemma to the 
 case of the benefits most frequently required : 
 
 x+ i) , !;= vp x (l + |_ 
 
 n a x =.vp x X n-il*+i, 
 
 (207). -[69]. 
 (208).- [72H81]. 
 
 (209). Chap, xviii, [17]. 
 
 (210). [82]. 
 (211). [95]-[98]. 
 (212).-[99]-[101]. 
 
 (213). Institute of Actuaries' Life Tables (Introduction, pp. 
 Iviii-lxviii) . 
 
102 PAliT II. LIFE CONTINGENCIES. [Chaps. XXI, XXII. 
 
 (214). By the formula 
 
 a table of values of a x can be computed. We then have 
 
 
 from which the continuous construction of the values of a x can be 
 completed. 
 
 A table of values of A^ can be constructed by the formulas 
 
 (215). 
 
 CHAPTEE XXII. 
 (216). [10]. 
 
 (218). [22]. 
 
 (219). [11]. Sunderland's Notes on Finite Differences (Chap. 
 iii, 8). 
 
 (220). (a) [21]. * w = 
 
 e 
 
 Writing u x+n for w w , and u x for w 0) we have 
 
Chap. XXII.] SOLUTIONS. 103 
 
 n(nl 
 (P) 
 
 Writing u x for w , and u x + n for u n , we have 
 
 *(*!) 
 
 (221). [18]. If second differences are constant, A 3 ^.=0 ; but 
 
 A 3 w a .=A 2 w u . + i A 2 ^ J . 
 (222) . We have u x u x 
 
 ^a, + A (u x + A^ 
 
 = KJ. + 2 A Wa- 
 
 iind evidently u x + w*=u x + RAv, 4 
 
 3 
 
 W 
 
 Writing now n for w, and - for , we have 
 
 5(5 -i) 
 
 M \ / 
 
 n(nd) 
 
104 PART II. LIFE CONTINGENCIES. [Chap. XXII. 
 
 (223). [21]. The required formula is 
 
 (m 1) 
 
 A%, 
 
 |3 |4 
 
 Then we have, if a?=l, #&:=(# 1), 
 
 e 
 
 From the values given in the question, we have Wi=4, Aw!=26, 
 =64, A% 1 =66, A 4 ^ 1 =24 ; and inserting these values, and reducing, 
 the above formula becomes 
 
 ,= 4+ (,-1)26 + <= 
 
 |4 
 (224). Chap, xxiii, [13]. From the formula 
 
 |3 
 
 
 we have, when n = - , 
 
 _ 
 
Chap. XXII.] SOLUTIONS. 105 
 
 (225). (a) From Example (222), putting h for , we have 
 
 n n(nH) 
 
 Putting now #=0, rc=#, we have 
 or 
 
 Inserting the values given in the question, 
 
 9 ^ 
 
 (226). The common difference of x being =A, 
 
 I W A. , n ( U ~ K) 
 
 n(nh) 
 
 H 
 
 |1 A* 
 
 If Aa;, and therefore A, be infinitely small, the above expression 
 becomes 
 
 n du n 2 d^U 
 
 where -= ; = ; &c., are successive differential coefficients of u x , 
 cix d/x 
 
 i.e., of x 5 . In this case, 
 
106 PART II. LIFE CONTINGENCIES. [Chap. XXII. 
 
 Substituting these values, 
 
 u x + n # 5 + 5 n& + 10 n?x 3 + 10 ri*tf + 5 n*x + n 5 
 =(#+*). 
 
 (227). Sunderland's Notes on Finite Differences, Chap, i, (3, (2) ((3) 
 
 Repeating the operation of A on each term, 
 
 If u x =x, v x =log^ this becomes 
 
 = A 2 (a? log x) = x A 2 log # + 2 ( Alog a? + A 2 log x) . 
 (228). u x 
 
 Writing now b 2 ^ 
 
 &c. &c. &c. 
 
 X 
 
 i/UU\ WLlUi &"L\"U\ 
 
 :j P Y\ ^2 TlT ^3 
 
 (229). We have, by successive substitutions, 
 
 = U n -i 
 
 and ultimately, continuing the same process, 
 
 Let Ui=a 
 
 =2Awi by question ; 
 
Chaps. XXII, XXIII.j SOLUTIONS. 107 
 
 whence &ui = Ui=a and ^ 2 =2Aw 1 = 2. 
 
 Similarly w 3 =^ 2 + Awi + A% 1 = 2A 2 w 1 by question; 
 
 whence &?Ui = u<2+ Aw 1 =3Aw 1 = 3, 
 
 and u 3 =2A?Ui=6a. 
 
 Proceeding thus, we find ^ 4 =26#, &c. 
 
 The series and the successive orders of differences are thus as 
 follow : 
 
 u = a, 2a, 6a, 26a 
 
 A = a, 4ia, 20a 
 
 A 2 = 3, I6a 
 
 A 3 = 13 
 
 From which it will be seen that 
 
 w 4 =2A%, &c. 
 
 CHAPTEK XXIII. 
 
 (230). The required expression is (Chap, xxii, [18]. formula 3), 
 writing u x for u , and u x+n for w w , 
 
 n(n 1) 
 
 - jS - 
 
 r 
 
 whence ^u l =u 7 
 
 =462-1512 + 15^-1120 + 315-36 + 1. 
 But A 6 w 1? by hypothesis =0 ; 
 therefore 15/c=1890, and K=126. 
 
 The sum of the first ten terms of the series can be readily ascertained 
 by the formula (Chap, xxiv, [17]) 
 
108 PAIIT II. - LIFE CONTINGENCIES. [Chap. XXIII. 
 
 whence, inserting the values of Ui , bui , A 2 ^ ..... , deduced from the 
 given series, we have 
 
 This result could also be arrived at by expanding the series to ten terms 
 and summing the results : thus 
 
 X 
 
 1 
 
 2 
 3 
 
 1 
 6 
 21 
 
 An* 
 
 5 
 15 
 
 ** 
 
 10 
 20 
 
 10 
 
 5 
 
 
 A 6 w 
 
 
 
 35 
 
 
 15 
 
 
 1 
 
 
 4 
 
 56 
 
 
 35 
 
 
 6 
 
 
 
 
 
 
 70 
 
 
 21 
 
 
 1 
 
 
 
 5 
 
 126 
 
 
 56 
 
 
 7 
 
 
 
 
 
 
 126 
 
 
 28 
 
 
 1 
 
 
 6 
 
 252 
 
 
 84 
 
 
 8 
 
 
 
 
 
 
 210 
 
 
 36 
 
 
 1 
 
 
 7 
 
 462 
 
 
 120 
 
 
 9 
 
 
 
 
 
 
 330 
 
 
 45 
 
 
 1 
 
 
 8 
 
 792 
 
 
 165 
 
 
 10 
 
 
 
 
 
 495 
 
 
 55 
 
 
 
 
 9 
 
 1,287 
 
 
 220 
 
 
 
 
 
 
 
 715 
 
 
 
 
 
 
 10 
 
 2,002 
 
 
 
 
 
 
 
 (231). We have, by Maclaurin's theorem (Chap, xxii, [29]), 
 
 Similarly, 
 
 Adding these two expressions together, the terms involving differential 
 coefficients of the odd orders vanish, and we have 
 
 Similarly, 
 
 
Chap. XXIII. ] SOLUTIONS. 109 
 
 which becomes 
 
 81 
 
 _- x j-a)+5(*-*) 
 
 whence 
 
 /ly __ 
 
 \dsfJ 
 
 /A 4 3 ( 
 
 UJ ^ = ~ 
 
 48 
 
 2 
 
 and inserting these values in the above formula for c, we obtain 
 
 the expression given in the question. 
 
 (232). Sunderland's Notes on Finite Differences (Chap, ii, 7). 
 From the values given in the question, we have 
 
 A^4= o80 
 ^=98,011 
 
 Then from the formula 
 
 we have, making ir=4, n= 4, 
 
 Wo =^ 4 4Aw 4 + 10A% 4 -20A 3 w 4 ; 
 
 or inserting the value of UQ given in the question and the other values as 
 deduced above, 
 
 100,000=98,391 + l,520-160-20A 3 7/ 4 ; 
 whence A 3 w 4 = 12'45, 
 
110 PATCT TT. LIFE CONTINGENCIES. [Chap, XXIII. 
 
 Then by the formulas 
 
 u w = u 4 + 6 Aw 4 + 15 A% + 20A% 4 = 95,622 
 A MH>=A ^4+6A% 1 + 15A^ 4 =-662-75 
 A 2 w 10 =A% + 6A% 4 = 90-70 
 
 A 3 w 10 =A% 4 =- 12-45 
 
 the values of UIQ to w 15 can be readily deduced in a tabular form, as 
 follows : 
 
 A 3 A 2 A u x x 
 
 -12-45 90-70 662-75 95,622-00 10 
 
 -103-15 753-45 94,959-25 11 
 
 -115-60 - 856-60 94,205-80 12 
 
 -128-05 972-20 93,349-20 13 
 
 -1,100-25 92,377-00 14 
 
 91,276-75 15 
 
 (233). [9], [12]. Sunderland's Notes on Finite Differences 
 (Chap.ii, 7). 
 
 (234). [8]. Let the values at 3, 3, 4, 4, 5, 5|, and 6 per-cent be 
 represented by ^ , u\^ u<i, %, ^4, %, and U G . Then to find ?^ 5 , (a) two 
 values ^ and u 4 being given, we have 
 
 A^ 4 =!(^ 6 O = -000849 
 and u & = u 4 +A 4 =- 020660. 
 
 (^8) Using four values ^, u 3 , u, and w 6 , we must assume A% 2 =0 ; 
 then 
 
 whence 
 
 4 
 (y) Using six values, and assuming A 6 M =0, 
 
 whence u 5 = 
 
 - -020628, 
 
Chap. XXIII.] SOLUTIONS. Ill 
 
 (235). Let the terms given be u , u^, u 4 , u$ ; required z/ 3 . As there 
 are four given terms, we must assume A%. r =0 ; 
 
 then 
 
 To eliminate u\, multiply the last equation by 4, and add the result 
 to the first expression. Then 
 
 4w 5 15^4 + 20^310^2 + ^0=0 ; 
 
 T ,70 
 
 whence 1(3= - =1'72 
 
 20 
 
 (286). We have 
 
 u 5 =55 ; A 5 =71 ; 
 A%=62 ; A% 5 =30 ; A 4 %=6. 
 Then w 2 =^ 5 -3A^ 5 + 6A% 10A^ 5 +15A% 5 
 
 = 4 
 
 =3,114. 
 
 These values may be readily proved by extending the series from 
 to Uiz, applying the given differences, thus : 
 
 ^x 
 
 6 
 
 12 
 
 8 
 
 Aw* 
 5 
 
 u x 
 
 4 
 
 a? 
 2 
 
 6 
 
 18 
 
 20 
 
 13 
 
 9 
 
 3 
 
 6 
 
 24 
 
 38 
 
 33 
 
 22 
 
 4 
 5 
 
 6 
 
 30 
 
 62 
 
 71 
 
 55 
 
 6 
 
 36 
 
 92 
 
 133 
 
 126 
 
 6 
 
 6 
 6 
 
 42 
 
 48 
 54 
 
 128 
 
 225 
 353 
 
 259 
 
 484 
 837 
 
 7 
 8 
 9 
 10 
 
 170 
 218 
 
 272 
 
 523 
 741 
 
 1,360 
 
 
 
 
 1,013 
 
 2,101 
 
 11 
 
 
 
 
 
 3,114 
 
 12 
 
112 PART IT. LIFE CONTINGENCIES. [Chap. XXHL 
 
 As an exercise, the student may apply Lagrange's theorem, 
 , to ascertain the desired values. We have 
 
 u 5 =. 55 =5 #=2 
 
 c=7 
 
 w 8 =484 d=8 
 
 w 9 =837 e=9 
 
 Then, by formula (3), 
 
 (-4)(-5)(-6)(-7) (-3)(-5)(-6)(-7) 
 
 (Z^yr^FSK 11 *) ' l.(-l)(-2)(-8T 
 
 (_3)(-4)(-6)(-7) (_ 3 )(-4)(-5)(-7) 
 
 2 .l(_l)(-2) " ~3.2.1(-1) 
 
 (-3)(^4)(-5)(-6) 
 -4^271" 
 
 = (55 x 35) - (126 X 105) + (259 x 126) - (484 x 70) + (837 x 15) 
 =47,114-47,110=4. 
 
 In a similar manner, putting x =12, the value of u\^ may be 
 ascertained. 
 
 (237) . From the values given in the question, we have 
 
 ^235= 2-3710679 A^235= '0018441 
 
 2 W235= _ -0000078 A%235= '0000002. 
 
 .rjo _ -i \ 
 
 Then ^235 . 63= ^235 + '63 Aw^s H 
 
 , 
 
 I?. 
 08063- 
 
 
 =2-3722306, 
 
Chap. XXIIL] SOLUTIONS. 113 
 
 (238). Let 
 
 A A 2 A :} A 4 
 
 = .= 18-4708 
 
 = , = 17-8144 -- 0510 
 
 29 = Uz = 17-1070 - -0564 -0022 
 
 - -7638 - -0076 
 33 = u 3 = 16-3432 -0640 
 
 - -8278 
 37 = u 4 = 15*5154 
 
 9 95 
 
 Then 30 = u^ = u + A w fl + A2w 
 
 9.5.1 
 
 = 1 8 . 47 08- C'6564)- (-0510)- (-0054) 
 
 = 16-9216. 
 
 (239). (a) [15]-[25]. (/8) Assuming fifth differences to be 
 constant, the series of values from u x+l to U X +IQ can readily be obtained 
 by algebraic addition of the successive differences. 
 
 (240). By the terms of the question, we have 
 
 But by hypothesis $ 2 u x+ t=q t 8 2 u x , .-. we have 
 
 whence - 
 
 (241). We have 
 
 = '922 
 ^=14-941 
 
 A% 6 = -086 
 u 8 = 14-105 
 
 A% 6 =-'012 
 ^=13-343 
 
 A% 6 = -005 
 ^,0=12-648 
 
114 PART II. - LIFE CONTINGENCIES. [Chap. XXIII. 
 
 71 
 
 where u n = the annuity-value at - per-cent. Then the value at 4'328 
 per-cent =u 8 . G56) and 
 
 2-656 x 1-656 x -656 2'656 x 1-656 x -656 x (--344) 
 
 = 13-5973. 
 
 Applying the method of central differences, [26] -[32], we have 
 the following values : 
 
 A 2 
 
 -922 
 14-941 -f-086 
 
 -836 --012 
 
 ^ = _. 762 
 13-343 +-067 
 
 -695 
 12-648 
 
 and by formula (10), 
 
 H 
 
 or, inserting the values of #( = '656) and of the successive differences, 
 
 ^-656= 14-105- (-656 x -762) - <656 ^' 844 (-074) 
 
 2i 
 
 . 1-656 x -656 x -344 1-656 x -656 x -344 x 1-344 
 
 6 - C'007) + 24 
 
 whence 
 
 ^56=13-59732. 
 
 (242).-[39]-[46]. 
 (243).-[40]-[42]. 
 
Chaps. XXIII, XXIV.] SOLUTIONS. 115 
 
 (244). Since 
 
 Id Id 
 
 we have (^ + 8) = --. 
 
 But by [47], formula (25) 
 
 Wj._ ! ) approximately 
 T> a ;-i) approximately, 
 and ( ^4-3 ) = _..D,==5ziZ approximately. 
 
 CHAPTER XXIV. 
 (245). We have, by the definition of a differential coefficient, 
 
 u x = limit of z when h is infinitely small. 
 
 dx h 
 
 Similarly, 
 
 = 11 11 7 J5 > >? 
 
 *+* : h 
 
 
 The sum of the right-hand terms is clearly = - ; therefore 
 
 we have 
 
 /**** 
 
 J -^. u x'dx=u x+n 
 
 = Limit of k \ u x + u x +j t + -=- u x+ ^ t + . . . + y- WJT+W-/I r 
 v.a# a? aiP wo? ; 
 
116 PAHT II. - LIFE CONTINGENCIES. [Chap. XXIV. 
 
 that is, the integration is equivalent to the sum of an infinite number of 
 infinitely small terms. 
 
 (246).- [3], [6]. 
 
 (247) .Since u x+n = (1 + A)^, (Chap, xxii, [20]) 
 
 and 2u x+n =&- l u x+n (Chap, xxiv, [2]) 
 
 Expanding (1 + A) W , we have 
 
 >--^C 
 
 and 
 
 / -j \ 
 
 ^. ^.. O /., L V ' A ,. 
 
 (248). Macdonald's Calculus of Finite Differences (Trans. Act. 
 Soc. jEdin., 1876), page 12. 
 We have 
 
 Summing perpendicularly, we have 
 S w ^=S w (^o + S w (a? 1 )ci+S tt 
 and inserting the values of S w (a?), S w .(^ ! )j & c ...... 
 
 -~ 
 
 30 
 
 (249). [20]. Sunderland's Notes on Finite Differences, Exercise 
 (22). 
 
/ 
 
 Chap. XXIV.] SOLUTIONS. 117 
 
 (250). From the formula deduced in the solution to Ex. (248) 
 S n u x =nc + - 
 
 we have, giving to w the successive values, 5, 10, 15, and 20, 
 
 55<? 2 + 225c 3 = 1,365 
 385c 2 + 3,025c 3 = 5,155 
 = 15c + 120C?! 4- 1,240^ + 14,400c 3 = 13,370 
 ,870^+44,100^3=28,635 
 
 from which four equations the values of the four unknown quantities, 
 C Q , GI, c 2 , c 3 , can be deduced by successive subtraction: thus 
 
 55c 2 + 225c 3 = 1,365 
 330c 2 + 2,800c 3 = 3,790 
 855^+11,375^3= 8,215 
 5c + 90^ + 1,630^2 + 29,700^3= 15,265 
 
 275c 2 + 2,575^= 2,425 
 525c 2 + 8,575^= 4,425 
 775^+18,325^= 7,050 
 
 6,000c 3 = 2,000 
 250c 2 + 9,750^= 2,625 
 
 3,750^= 625 
 
 whence c 3 = 
 
 120 
 80 
 
 _ 1,075 
 30 
 
 3,420 
 
 C = 
 
 and 
 
 30 
 
 5*3 + 120# 2 + 1,075# + 3,420 
 30 
 
 Macdonald's Calculus of Finite Differences (Example xiii). 
 
118 PAKT II. - LIFE CONTINGENCIES. [Chap. XXTV. 
 
 and l x=e - 
 
 where K=e c , and is obviously =Z 0j the arbitrary value 
 called the radix of the mortality table. If, as in 
 Makeham's formula, p x is in the form A + Bc^, then 
 
 G C 10g e C lOgeC 
 
 B 
 
 which, since e c and e lo < c are constants, is in the form 
 
 See Chap, vi, [15]. 
 
 When d x is constant for all ages, fju x is known to be equal 
 
 to -r^ = g^ x , but the expression given in the question 
 * 
 
 . 
 
 would in this case become ~^ ; hence it is clear that 
 
 2l x 
 
 the coefficients, either in the numerator or denominator, 
 are in error. The correct expression is 
 
 _ 
 
 (252).- 
 
 (253). [32]. 2^ may be written symbolically as A-%# (see 
 
 The expression in brackets may obviously be expanded in a series of 
 
 terms involving the powers of the symbol , which, when combined 
 
 dx 
 
 with the symbol of quantity, u x , become the successive differential 
 coefficients of u x 
 
Chap. XXIV.] SOLUTIONS. 119 
 
 Assume that, for all values of - , 
 
 ax 
 
 then 
 
 and 
 
 Y+ () 
 
 involving only the constant B, and even powers of . 
 
 But 
 
 ledx_ 
 
 ~~~ 
 
 hence, in the series (ii), B= 1, and the coefficients of the remaining 
 terms (D, F, Ac.) all =0. 
 
 (254).- [39]. 
 
 (255). (a) Formula (24), [34]. 
 08) Formula (25), [36]. 
 (y) Formula (26), [37]. 
 
 (256) .- [40]. 
 
 (257). [42]-[55]. J.I. A., xxiv, 95. Owing to the simplicity of 
 the coefficients in formulas (33) and (36), they can be readily applied to 
 numerical calculation, and will generally be found to give accurate 
 results. 
 
 (258). Chap, ix, [19]. 
 
 (259).-Chap. ix, [21], [22], [25]. 
 
 (260). Chap, xii, [58], [61], [62]. 
 
 (261). Chap, xiii, [51]-[58]. 
 
 (262). Chap, xiv, [41], formula (27). 
 
 (263). Chap, xv, [11]. 
 
 (264). Chap, xv, [20], [25]. 
 
 (265). We have 
 
 lz+n 
 
120 PART IT. LIFE CONTINGENCIES. [Chap. XXIV. 
 
 The value of the corresponding continuous annuity is 
 
 ff *** 
 
 _ _ / j., 1 fly+n , 'z + n 
 
 ax:^m=W x +J v*+> - \^L- + -- - 
 
 The values of the continuous annuity, and of the single premium for the 
 corresponding assurance, can be obtained in one operation by the use of 
 one of the formulas of approximate summation given in Chap. xxiv. 
 The divisor for the annual premium (payable until the expiration of the 
 risk) may be deduced from the above continuous annuity by the formula 
 
 *** 
 
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