INTERPOLATIONS. LII. To find, by interpolation, the intermediate value of an annuity of £1 on the joint existence of any two lives, not contained in the Table. Find the value of an annuity of £1 during the existence of two sets of joint lives, of whichThe eldest given age, and the nearest tabular difference above the difference of the given ages deducted from the eldest given age, are the ages of the first set of joint lives; andThe eldest given age, and the youngest age of the first set of lives, increased by five, are the ages of the second set of joint lives. Divide the difference of the values of these two sets of joint lives by five; Multiply the quotient by the difference between the youngest of the joint lives, and the youngest given life; This result, subtracted from the value of the youngest set of joint lives, will be the intermediate value required. What is the value of an annuity of £1 during the joint existence of two lives, aged thirty-nine and forty-two respectively, reckoning interest at 3 per cent. per annum? The diff. between the given ages, 39 and 42, being 3, nearest tab. diff. above is 5 years; What is the value of an annuity of £1 during the joint existence of two lives, aged thirty-three and forty respectively, reckoning interest at 4 per cent. per annum? The diff. between the given ages, 33 and 40, being 7, nearest tab. diff. above is 10 years: The above examples are here subjoined in their interpolated form; the intermediate values being successively found by continually deducting the common difference between each value. LII. INTERPOLATIONS-continued. What is the value of an annuity of £1 during the joint existence of two lives, aged twenty-six and thirty-nine respectively, reckoning interest at 5 per cent. per annum? The diff. between the given ages, 26 and 39, being 13, nearest tab. diff. above is 15 years; Then 39, eldest life, -15, the diff., 24; giving 24 & 39 for ages of first set of joint lives: = And 24, the youngest age of the first set of lives, increased by 5,=29; Therefore 29, and 39 the eldest life, are the ages of the second set of joint lives. The value, by Table X., of an annuity of £1, at 5 per cent., on two joint lives, aged 24 and 39,=£9.895; and 29 and 39,=£9.703; diff. of these values=£.192; The diff. between 24, youngest of the joint lives, and 26, youngest given life, is 2; Therefore (9.895, the value of the youngest set of joint lives,)-(.192÷5) × 2= £9.8182, the intermediate value required. What is the value of an annuity of £1 during the joint existence of two lives, aged forty-nine and sixty-five respectively, reckoning interest at 3 per cent. per annum? The diff. between the given ages, 49 and 65, being 16, nearest tab. diff. above is 20 years; The value, by Table X., of an annuity of £1, at 3 per cent., on two joint lives, aged What is the value of an annuity of £1 during the joint existence of two lives, aged Fifty-three and fifty-six. Answer, £ 7.1110. Answer, £10.8046. Answer, £12.3298. Answer, £ 4.6446. at 3 cent.?. at 5 per cent.?. at 4 per cent.?. at 5 per cent.?. . Answer, £ 4.7786. Answer, £10.6768. . at 3 per cent.?. Answer, £ 2.1446. Answer, £9.9000. Twenty-eight and thirty-seven at 5 per cent.?. Calculate also each of the above examples in its interpolated form, in the manner shown in the preceding page. IMMEDIATE ANNUITIES. LIII. To find, by means of Table X., the present value of an annuity, to continue during the joint continuance of any two lives, at the several rates of 3, 4, and 5 per cent. per annum. Multiply the present value, as found by Table X., of an annuity of £1 on the joint existence of the given lives, at the given rate per cent., by the given annuity; the result will be the present value required. What is the present value of an annuity of £50, payable yearly, during the joint existence of two persons, aged forty-five and fifty respectively, reckoning the interest of money at 3 per cent. per annum? Pres. val., by Tab. X., of an ann. of £1 on two joint lives, 45 and 50, at 3 per ct.,=£9.204; Therefore £9.204 × £50=£460.200, or £460 4 0, the present value required. Government regulates the legacy duty, chargeable upon annuities, by the Northampton table of mortality, at 4 per cent. interest: upon what present sum, therefore, ought two persons, of the respective ages of thirty and forty, to be charged, to whom an annuity of £75 is bequeathed during the existence of their joint lives? Answer, £786 15s. A merchant and his wife, severally aged forty-two and twenty-seven, possessing a freehold estate, estimated at £100 per annum, desire to convert it into present money, for the purposes of commerce: what sum ought they to receive for it, supposing they are willing to allow the purchaser interest at the rate of 5 per cent. per annum, and averaging the duration of human life by the Northampton table of mortality? Answer, £952 4s. A., aged thirty-four, and B., aged fifty-four, deriving from freehold property an annuity, payable yearly, of £200, during their joint lives, wish to transfer it for an equal annuity to their friend C., aged thirty-two: what sum ought A. and B. to pay to equalise the values, reckoning the interest of money at 4 per cent. per annum ? Answer, £1725 16s. What is the present value of an annuity of £60, payable (XXXI. 2.) half-yearly, during the joint existence of two persons, aged thirty-six and (LII.) thirty-nine respectively, reckoning interest at 3 per cent. per annum?-Answer, £688 13 7. What is the present value of the lease of an estate of the clear rent of £600 per annum, payable (XXXI. 3.) quarterly, during the joint lives of two persons, aged severally thirty-eight and (LII.) forty-five, reckoning the interest of money at 4 per cent. per annum? Answer, £5931 12s. Two brothers, of the respective ages of forty-four and (LII.) fifty-six, holding the lease of an estate during their joint lives, of the clear annual rental of £80, being suddenly called abroad, are desirous to dispose of the same immediately, and are willing to allow the purchaser interest at the rate of 5 per cent. per ann.: what sum ought they to receive as the present value of their title? Ans., £587 12s. What is the present value of an annuity of £500, payable yearly, during the joint lives of two persons, of the several ages of twenty and (LII.) thirty-seven, reckoning interest at 4 per cent, per annum? Answer, £5634 18s. A farmer, aged forty-nine, holds, during the joint lives of himself and his nephew, aged thirty-three, an estate by lease of the full value of £290 per annum, for which he pays an annual rent of £110, and also other charges for tithes, taxes, and similar expenses, to the amount of £40 annually: what sum ought he to receive for the disposal of his lease, supposing 5 per cent. per annum the rate of interest allowed? Answer, £1204 2 3. IMMEDIATE ANNUITIES-continued. LIV. To find, by means of Table X., what annuity a given sum will purchase during the joint existence of any two lives, at the several rates of 3, 4, and 5 per cent. per annum. Divide the given sum by the present value, as found by Table X., of an annuity of £1 during the existence of the joint lives, at the given rate per cent.; the result will be the annuity required. The trustees of a widow lady, aged forty-five, and of her sister, aged fifty, purchased from a public company an annuity, payable yearly, during their joint lives, for the sum of £460 4s. What was the amount of the annuity, allowing interest at the rate of 3 per cent. per annum? Pres. val., by Tab. X., of an ann. of £1 during two joint lives, 45 & 50, at 3 per ct.,=£9.204; Therefore £460 4 0, or £460.200÷£9.204=£50, the annuity required. What annuity, payable yearly, may be purchased during the joint lives of two persons, aged 6 4. 9 3. 5 4. 9 6. Eighteen and twenty-five, . for £2000, at 3 per cent.? . . Ans., £143 10 2. Twenty-four and thirty-seven, for £1000, at 4 per cent.? Ans., £ 90 Thirty-two and thirty-four,. . for £ 500, at 5 per cent.?.. Ans., £ 50 Sixty-nine and seventy-five,. . for £ 100, at 4 per cent.? . . Ans., £ 28 Seventy-eight and eighty,. . . for £ 250, at 3 per cent.? . . Ans., £109 Fifty and fifty, for £910, at 5 per cent.?.. Ans., £120 19 7. Legacy duty was paid at Somerset-house on the sum of £786 15s., being the value of an annuity, payable yearly, which had been bequeathed to two persons, during their joint lives, aged twenty-seven and (LII.) thirty-six respectively. Now the government charge for annuities by legacy is always regulated by the Northampton table of mortality, at 4 per cent. interest: what, therefore, was the amount of the annuity bequeathed? Answer, £71 7 5. Two ladies, the one aged thirty-three, the other (LII.) thirty-five, sunk £1500 in the purchase of an irredeemable annuity, payable (XXXI. 2.) half-yearly, during their joint lives. To what annuity were they entitled, allowing that money constantly accumulated at the rate of 5 per cent. per annum ? Answer, £149 8 2. What annuity, payable yearly, is equivalent to £100 per annum in the government long annuities, which have thirty years to run, and now in the enjoyment, during their joint lives, of two persons, each aged sixty-one, reckoning interest at the rate of 4 per cent. per annum? Answer, £286 15 4. A., aged forty-nine, relinquished to B. his claim to £5000 consols, when this stock was at the price of £90, for an equivalent annuity, payable (XXXI. 2.) halfyearly, during the joint lives of himself and his wife, aged thirty-one. What annuity should B. grant to A. and his wife in exchange, reckoning the interest of money at 3 per cent. per annum ? Answer, £427 17 1. A merchant sank the sum of £1500 in the purchase of an annuity, payable (XXXI. 3.) quarterly, during the joint lives of his son and daughter, respectively aged thirty-five and thirty. What was the amount of the annuity, supposing the merchant invested his money at the rate of 5 per cent. per annum interest? Answer, £145 4 5. Two persons, aged severally fifty-three and fifty-seven, hold the lease, during their joint lives, of an estate of the estimated value of £1850. Being desirous to convert the same into an immediate equivalent annuity, payable yearly, how much per annum ought they to receive in exchange for their lease, allowing interest at the rate of 4 per cent. per annum? Answer, £259 1 4. DEFERRED ANNUITIES. LV.-To find the present value, in a single payment, of an annuity, to continue during the joint existence of any two lives after a given term. Multiply the pres. val., as found by Tab. III., of £1, at the given rate, for the def. term, byThe present value of an ann. of £1 on the joint lives at the increased ages; and this byThe probability of the given lives attaining the given ages; The result, multiplied by the given annuity, will be the present value required. Two ladies, of the respective ages of twenty-eight and thirty-three, will be entitled, after the next seven years, to an annuity, payable yearly, during the remainder of the existence of their joint lives. What is their reversion worth in present money, supposing it to be £150 per annum, and the rate of interest to be 3 per cent.? The present value, by Table III., of £1, due 7 years hence, at 3 per cent.,=£.813092; 7 years, the def. term,+28 years, one of given ages,=35 years, one of increased ages; 7 years, the def. term,+33 years, other given age, 40 years, other increased age; Pres. val., by Tab. X., of an ann, of £1 on joint lives, 35 and 40, at 3 per ct.,=£11.213; The probability (XXVIII. 1.) of 28 living 7 of 33 living 7 yrs. yrs. 45359 Theref. (£.813092 × £11.213) × (1936) £150 £1056.645, or £1056 13 0, the present value required. = 4010 · = 3635 41609 What is the present value, during the joint lives of two persons, aged20 and 25, after 7 years, of £200 per ann., at 4 per cent. ?-Ans., £1379 14 10. 25 and 35, after 10 years, of £350 per ann., at 5 per cent.?—Ans., £1307 19 4. 18 and 33, after 9 years, of £100 per ann., at 3 per cent.?-Ans., £ 645 10 8. 31 and 51, after 5 years, of £ 90 per ann., at 4 per cent.?-Ans., £ 475 13 11. 26 and 44, after 12 years, of £ 80 per ann., at 5 per cent.?-Ans., £ 194 16 4. 37 and 50, after 8 years, of £ 30 per ann., at 3 per cent.?—Ans., £ 124 6 11. What sum ought to be given for the reversion of an annuity of £250 per ann., held during the joint existence of two persons, aged respectively seventeen and (LII.) twenty-three, to be entered upon ten years hence; assuming the interest of money at 4 per cent. per annum? Answer, £1398 6 1. Two brothers, of the respective ages of twenty-five and (LII.) twenty-seven, will, after the next six years, be entitled, during their joint lives, to the lease of an estate, producing a clear income of £280 per ann. An eligible opportunity to commence business having presented itself, they are willing to dispose of their reversion for an equivalent sum down; what present money ought they to receive, reckoning interest at the rate of 5 per cent. per annum ? Answer, £1711 5 3. What premium ought to be paid down for an annuity of £100, to commence after the next eleven years, and then to continue during the joint lives of two sisters, severally aged fourteen and seventeen, allowing the purchasers interest at the rate of 4 per cent. per annum? Answer, £565 16 3. A., or his heirs, will be entitled to an estate in fee simple of £180 10s. per ann., after the next four years, provided the present tenants, aged respectively seventytwo and (LII.) seventy-six, jointly survive the term. What is the present value of A.'s inheritance, reckoning interest at 3 per cent. per ann. ?— Ans., £171 11 3. Two ladies, aged fifty-three and fifty-eight respectively, will, after the next eight years, be entitled, during the existence of their joint lives, to the enjoyment of an annuity of £130; but being limited in their means, and anxious to protect themselves against the painful uncertainty contingent on the death of either of them in the interim, are willing to dispose of their reversionary annuity for its present value. What equivalent sum ought they to receive, allowing the purchaser interest at the rate of 4 per cent. per annum ? Answer, £273 1 9. |