DEFERRED ANNUITIES-continued. LVI. To find the present value, in annual payments, of an annuity to continue during the joint existence of any two lives, after a given term. Multiply the present value of £1 at the given rate, for the deferred term, by— The probability of the lives attaining the increased ages; deduct the product from unity; Add this difference to the difference between the present values of an annuity of £1 on the given joint lives, and of an annuity of £1 def. on the joint lives to the incr. ages;— The present value, in a single (LV.) payment, of the given annuity on the deferred joint lives, divided by this last sum, will be the annual payment required. Two ladies, aged twenty-eight and thirty-three respectively, desirous of providing an annuity of £150, to commence when the eldest of the two shall have attained the age of forty, wish to purchase the same for the remainder of their joint lives, by an equal annual premium, payable at the beginning of each year till then; but such annual premium to cease should either life fail in the interim. What annual premium should they pay, reckoning the interest of money at 3 per cent. per ann. ? The present value, by Table III., of £1, due 7 years hence, at 3 per cent., £.813092; 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; Pres. val., by Tab. X., of an ann. of £1, on joint lives, 28 and 33, at 3 per ct.,=£12.474; The probability (XXVIII. 1.) of 28 living 7 yrs. = 45 35; of 33 living 7 yrs. 41609 7 4535 OPERATION. 4010. = 3635 £.813092 × (4010 × 3433)=£.62823, which, ded. from unity, or 1.00000,=£ .37177; Then £.62823 x £11.213 (LV.) the present value, in a single payment, of a deferred annuity of £1 on the given joint lives, at 3 per cent., And £12.474-£7.04433-£5.42967;+£.37177 =£7.04433; =£5.80144: Now £7.04433 × £150=£1056.6495, value in a single payment of def. annuity of £150; Therefore £1056.6495÷£5.80144=£182.1357, or £182 2 9, ann. prem. required. What present value, in annual premiums, payable at the beginning of each year, until the deferred term, but to cease on the failure of either life in the interim, is required to purchase during the remainder, after that period, of the existence of two joint lives, aged 9. 20 and 25, after 7 years, an ann. of £200, at 4 per cent.?—Ans., £241 1 25 and 35, after 10 years, an ann. of £350, at 5 per cent.?-Ans., £186 6 11. 18 and 33, after 9 years, an ann. of £100, at 3 per cent.?-Ans., £ 90 17 6. 31 and 51, after 5 years, an ann. of £ 90, at 4 per cent.?-Ans., £112 11 4. 26 and 44, after 12 years, an ann. of £ 80, at 5 per cent. ?-Ans., £ 25 11 10. 37 and 50, after 8 years, an ann. of £ 30, at 3 per cent.?-Ans., £ 20 5 11. 17 and 23, after 10 years, an ann. of £250, at 4 per cent.?-Ans., £186 10 0. 25 and 27, after 6 years, an ann. of £280, at 5 per cent.?—Ans., £346 12 1. 72 and 76, after 4 years, an ann. of £180, at 3 per cent.?-Ans., £ 58 1 3. 53 and 58, after 8 years, an ann. of £130, at 4 per cent. ?-Ans., £ 49 3 8. What annual premium ought to be paid at the beginning of each year for an annuity of £100, to commence after the next eleven years, and then to continue during the joint lives of two sisters, of the respective ages of fourteen and seventeen, the annual premiums to cease on the death of either lady happening within. the term, reckoning interest at the rate of 4 per cent. per annum? Answer, £69 7 6. TEMPORARY ANNUITIES. LVII. To find the present value of an annuity, to continue a given number of years, at a given rate per cent., upon two given lives jointly surviving the term. Deduct the present value of an annuity of £1, deferred (LV.) on the joint lives the given term, from the present value of an annuity of £1 on the given joint lives; The result, multiplied by the given annuity, will be the present value required. Two ladies, aged twenty-seven and thirty-two respectively, are in the enjoyment of an annuity, payable yearly, of £50, which is to continue for the next ten years, should they both live so long, but to cease in the event of the death of either of them within the term. What is the present value of the annuity, reckoning interest at the rate of 3 per cent. per annum? The present value, by Table III., of £1, due 10 years hence, at 3 per cent.,=£.744094; 10 years, the def. term,+27 years, one of given ages, =37 years, one of increased ages; 10 years, the def. term, +32 years, other given age, 42 years, other increased age; Pres. val., by Tab. X., of an ann. of £1 on joint lives 37 and 42, at 3 per cent., =£10.828; Pres val., by Tab. X., of an ann. of £1 on joint lives 27 and 32, at 3 per cent., =£12.641; The probability (LI.) of 27 and 32 both surviving 10 years=3860 4610 860 3482 4235 Hence £.744094 x £10.828 × (388 × 3483) (LV.)=£5.5467, the present value of an annuity of £1 deferred on the joint lives for 10 years: Therefore (£12.641-£5.5467) × £50-£354.715, or £354 14 3, pres. val. required. What is the present value, during the joint lives of two persons, should they both live so long, aged 19 and 24, for next 7 years, of £140 per ann., at 4 per ct.?-Ans., £ 748 13 9. 23 and 31, for next 4 years, of £270 per ann., at 5 per ct. ?-Ans., £ 882 15 2. 35 and 47, for next 9 years, of £400 per ann., at 3 per ct.?-Ans., £2490 2 9. 54 and 62, for next 5 years, of £ 90 per ann., at 4 per ct.?-Ans., £ 316 15 11. 71 and 74, for next 3 years, of £ 30 per ann., at 5 per ct.?-Ans., £ 68 6 1. 80 and 80, for next 2 years, of £25 per ann., at 3 per ct.?-Ans., £ 31 0 1. Two brothers, aged forty-five and fifty respectively, are entitled for the next ten years, should they both continue to live so long, to the possession of an estate, producing a clear income of £52 10s., payable yearly. What is the present value of their annuity, reckoning the rate of interest at 3 per cent. per annum? Answer, £334 6 10. Two young men, of the respective ages of twenty-seven and (LII.) thirty, purchased an annuity of £105, payable yearly, to be continued during their joint lives until the youngest reached the age of forty, otherwise to cease. What present money did they give for their annuity, calculating interest at the rate of 4 per cent. per annum ? Answer, £838 1 5. What sum ought to be paid down for the lease of an estate, estimated at a clear annual rent of £100, to continue for the next twenty-one years, provided a lady, aged thirty-six, and her brother, aged (LII.) forty-two, both survive so long, allowing the purchaser interest at the rate of 5 per cent. per annum? Answer, £849 2s. LVIII. Two LIVES. Temporary annuities. LVIII. Temporary annuities. Value of leases. Two joint lives. TEMPORARY ANNUITIES-continued. Ten years having lapsed in a lease, granted during the joint existence of two lives, aged thirty and thirty-five respectively, the clear annual rent of which is estimated at £100, what fine ought to be paid for, leave to put in two fresh lives, severally aged fifteen and twenty, in lieu of the original nominees, allowing the tenant interest at the rate of 5 per cent. per annum? Answer, £294 4s. What sum paid down will purchase the lease of an estate for fourteen years, depending on the joint existence of two lives, aged severally fifty-seven and (LII.) sixty, the clear rent of which is £30 per annum, allowing interest at the rate of 3 per cent. ? Answer, £190 12 7. An estate, producing £600 a year, for fifteen years to come, subject to the present occupiers, of the respective ages of forty and forty-five, both living so long, is to be disposed of. What is its present value, supposing the rate of interest at 4 per cent. per annum? Answer, £4691 28. A landlord agreed to grant a lease for fifty years, during the joint lives of two boys, the one aged twelve years and the other (LII.) fourteen, of an estate estimated at £350 per annum, out of which the tenant would be required to pay various charges to the amount of £65 annually. What sum ought to be given for the lease, allowing the purchaser 4 per cent. per annum compound interest for his money? Answer, £3897 15 3. A farmer, aged forty-seven, holds for the term of thirty-three years, contingent, however, on his own life, and that of his son's, aged twenty-nine, both surviving the term, 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, £1249 9 8. What is the present value of the lease of an estate for forty-five years, held on the condition that two children, of the respective ages of nine and fourteen, shall both exist so long, estimated at £167 10s. per annum; but which is charged with the payment of an annuity of £67 10s. for twenty years certain, assuming an allowance of 3 per cent. per annum to the purchaser for the interest of his money? Answer, £1613 5 8. A., aged fifty, and B., aged fifty-five, are entitied for the next twelve years, should they both live so long, but not otherwise, to an income derived from an estate in fee simple, estimated at £150 per annum; after either of which events, C., or his heirs, will enter into possession of it for ever. What is the present value of A. and B.'s annuity, and of C.'s reversion, supposing money to be improved annually at the rate of 5 per cent. interest? Ans., £900 8 10, val. of A. and B.'s ann., £2099 11 2, val. of C.'s rever. Two ladies, one aged thirty-seven, the other forty-three, are entitled for the next seventeen years, or for so many of them as they may both happen to live, to the rents of an estate, yielding £200 per annum; but being desirous to exchange their temporary for a permanent income, to continue during the existence of their joint lives, what equivalent sum ought they to receive yearly, until the death of one of them, reckoning interest at the rate of 4 per cent. per ann.? — Ans., £172 5 9. A gentleman desires to pay down such a present sum as will secure to his two daughters, of the respective ages of twenty-three and (LII.) twenty-five, an annuity of £1200, which is to cease on the youngest life attaining the age of thirty-five, or sooner should either of them fail to live so long. Required the present value of this provision, assuming that money accumulated at the rate of 5 per cent. per ann.? Answer, £8801 16 4. ASSURANCES FOR THE JOINT EXISTENCE OF TWO LIVES. LIX. To find the present value, in a single payment, of an assurance of the reversion of £1, to be received at the end of the year in which either of two given lives may fail. Subtract the present value of an annuity of £1 on the given joint lives, at the given rate,— From the present value of a (XXIV.) perpetuity of £1 at the given rate per cent.; Divide this difference by the value of the perpetuity, increased by unity: The result will be the present value, in a single payment, of the assurance required. Required the premium, in a single payment, to assure £1 on the joint continuance of the lives of two persons, aged ten and fifteen, aged eleven and sixteen, and aged twelve and seventeen, respectively, at the several rates of 3, 4, and 5 per cent. per annum, and according to the Northampton table of mortality? The present value of a (XXIV.) perpetuity of £1, at the several rates of— 3 per cent.=£33.333, which, increased by unity, =£34.333; 4 per cent. £25.000, which, increased by unity, =£26.000; 5 per cent. £20.000, which, increased by unity, =£21.000. = OPERATIONS. Present value of an annuity, by Table X., of £1, on two joint lives, aged 10 and 15, At 3, 4, and 5 per cent.,=15.762, 13.841, and 12.302, respectively; Therefore, at the decease of either of two persons, aged 10 and 15, (£33.333-£15.762)÷£34.333 £.51178, the premium to assure £1, at 3 per cent.; (£25.000 – £13.841)÷£26.000=£.42919, the premium to assure £1, at 4 per cent.; (£20.000 – £12.302)÷£21.000=£.36657, the premium to assure £1, at 5 per cent. Present value of an annuity, by Table X., of £1, on two joint lives, aged 11 and 16, At 3, 4, and 5 per cent.,=15.538, 13.664, and 12.158, respectively; Therefore, at the decease of either of two persons, aged 11 and 16, (£33.333 - £15.538)÷£34.333=£.51831, the premium to assure £1, at 3 per cent.; (£25.000-£13.664)÷£26.000-£.43600, the premium to assure £1, at 4 per cent.; (£20.000-£12.158)÷£21.000-£.37343, the premium to assure £1, at 5 per cent. Present value of an annuity, by Table X., of £1, on two joint lives, aged 12 and 17, At 3, 4, and 5 per cent., 15.308, 13.480, and 12.009, respectively; = Therefore, at the decease of either of two persons, aged 12 and 17, (£33.333-£15.308)÷£34.333 £.52501, the premium to assure £1, at 3 per cent.; (£25,000 – £13.480)÷£26.000=£.44308, the premium to assure £1, at 4 per cent. ; (£20.000–£12.009)÷£21.000=£.38052, the premium to assure £1, at 5 per cent. Continue these operations until the following Table XI. be completely constructed, at the several rates of 3, 4, and 5 per cent. per annum, and with 5 years, 10 years, 15 years, 20 years, and 0, for the differences of the ages. |