XXXV.-TEMPORARY ANNUITIES-continued. Ten years having lapsed in a lease granted during a life aged thirty, the clear annual rent of which is estimated at £100, what fine ought to be paid for leave to put in a fresh life at the original age, in lieu of the present one, allowing the tenant interest at the rate of 5 per cent. per annum? Answer, £123 10s. Required to find what sum paid down will purchase the lease of an estate for fourteen years, depending on a life of sixty-five, the clear rent of which is £30 per annum, allowing interest at the rate of 3 per cent.? Answer, £222 13 3. An estate, producing £600 a year, for fifteen years to come, subject to the present occupier, aged fifty-seven, living so long, is to be disposed of. What is the value of it, supposing interest at 4 per cent. per annum ? Answer, £4944 0 6. What is the present value for the next seven years of an estate yielding £100 yearly, held on a life, aged forty-three, at the rate of 5 per cent. per annum, compound interest? Answer, £528 3 6. What is the lease of an estate for twenty-one years, contingent upon a life of twenty-five, worth, of which the clear annual rent is £140, reckoning interest at 4 per cent. per annum ? Answer, £1666 14s. What sum would be required for the purchase of the lease of an estate, held on a life of thirty-six, for fifteen years, producing a clear annual rental of £250, if 5 per cent. per annum, compound interest, were allowed for the use of the money? Answer, £2237 13 5. A landlord agreed to grant a lease for fifty years, on the life of a boy aged twelve years, 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, £4801 5 6. A farmer, aged forty-seven, holds for the term of thirty-three years, depending on his life, 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, £1494 16 10. What is the present value of the lease of an estate for forty-five years, held on the life of a child ten years of age, estimated at £167 10s. per annum, which is however charged with the payment of an annuity of £67 10s. for twenty years, assuming an allowance of 3 per cent. per annum to the purchaser for the interest of his money? Answer, £2341 18 7. A., aged sixty, is entitled for the next twelve years, should he live so long, to an income derived from an estate in fee simple, estimated at £150 per annum, after either of which events, B., or his heirs, enter into possession of it for ever. What is the present value of A.'s annuity and B.'s reversion, supposing that money improved annually at the rate of 5 per cent. interest? Answer, £1016 19 8, value of A.'s ann.; £1983 0 4, value of B.'s reversion. A lady, forty-one years of age, is entitled for the next seventeen years, or for so many of them as she may happen to live, to the rents of an estate, yielding £200 per annum; but being desirous to exchange her temporary for a permanent income, to continue during her life, what equivalent sum ought she to receive yearly, reckoning interest at the rate of 4 per cent. per annum? Answer, £153 11 3, the equivalent annuity. ASSURANCES FOR THE WHOLE TERM OF LIFE. XXXVI. 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 a given life may fail. Subtract the present value of an annuity of £1 on the given life at the given rate per cent.— Required the premium, in a single payment, to assure £1 at the death of persons aged 15, 16, and 17, respectively, reckoning interest at the several rates of 3, 4, and 5 per cent. per annum, and according to the Northampton table of mortality? The present value (XXIV.) of a perpetuity of £1 at the several rates of 4 per cent.= £25.000, which, increased by unity, = £26.000; = OPERATIONS. The pres. value of an annuity, by Tab. VIII., of £1 on a life of 15, at the several rates Of 3, 4, and 5 per cent. £19.657, £16.791, and £14.588, respectively; Therefore, at the death of a person aged 15, (£33.333-£19.657)÷£34.333= £.398334, the premium to assure £1, at 3 per cent.; (£25.000-£16.791)£26.000-£.315731, the premium to assure £1, at 4 per cent.; (£20.000-£14.588) £21.000-£.257714, the premium to assure £1, at 5 per cent. The pres. value of an annuity, by Tab. VIII., of £1 on a life of 16, at the several rates Of 3, 4, and 5 per cent. £19.435, £16.625, and £14.460, respectively; Therefore, at the death of a person aged 16, per cent.; (£33.333-£19.435) £34.333= £.404800, the premium to assure £1, at 3 (£25.000-£16.625)÷£26.000= £.322115, the premium to assure £1, at 4 per cent.; (£20.000-£14.460)÷£21.000 = £.263810, the premium to assure £1, at 5 per cent. The pres. value of an annuity, by Tab. VIII., of £1 on a life of 17, at the several rates Of 3, 4, and 5 per cent. £19.218, £16.462, and £14.334, respectively; Therefore, at the death of a person aged 17, (£33.333 £19.218)÷£34.333=£.411121, the premium to assure £1, at 3 per cent.; (£25.000-£16.462)÷£26.000=£.328384, the premium to assure £1, at 4 per cent.; (£20.000 – £14.334)÷£21.000=£.269810, the premium to assure £1, at 5 per cent. Continue these operations until the following Table IX. be completely constructed, at the several rates of 3, 4, and 5 per cent. per annum. Showing the value of the reversion, or the single premium for the assurance, of £1, payable at the end of the year in which a given life shall fail, deduced from the Northampton table of mortality, at the several rates of 3, 4, and 5 per cent. per annum. ASSURANCES FOR THE WHOLE TERM OF LIFE-continued. XXXVII. To find, by means of Table IX., what sum, or premium, paid down now, is equal to the value of a given sum, to be received at the end of the year in which a given life may fail, at the several rates of 3, 4, and 5 per cent. per annum, according to the Northampton table of mortality. Multiply the present value of £1, as found by Table IX., due one year after the failure of the given life, at the given rate by the given sum; the result will be the required present value, or premium. What sum, or premium, ought to be paid down as the present value of £1000, to be received by the representatives of a person, aged forty-five, at the end of the year of his decease, allowing interest at the rate of 3 per cent. per annum ? Pres. val. of £1, by Tab. IX., at 3 per cent., on the failure of a life of 45,=£.572073; Therefore £.57207 × £1000=(II. 3.) £572.07, or £572 15, the prem. required. What principal, or premium, paid now, will amount to £121 13 4, at the death of a person aged thirty-three; that is, what is the present value of the reversion of £121 13 4, to be received at the end of the year of the failure of a life of thirty-three, discount being allowed at the rate of 4 per cent. per annum ?—Answer, £49 17s. What premium should be given as the present value of £150 10s., which will become due on the death of a person aged twenty-four, reckoning discount at the rate of 5 per cent. per annum? Answer, £45 9s. A person, requiring present capital, desires to dispose of £3750 three per cent. consols, now valued at £80 for every hundred pounds stock, to which he, or his heirs, are absolutely entitled at the death of his mother, aged fifty-four. What is the present value of his reversion, reckoning discount at the rate of 5 per cent. per annum? Answer, £1490 8 7. What sum laid up now, and improved at compound interest during the life of a lady aged thirty-six, will amount to £500, at the rate of 4 per cent. per annum? Answer, £213 16 11. A party, aged fifty-two, on obtaining a legacy of £1500, appropriated only so much of it to present uses as enabled him, by assuring his life, to secure the full amount of the legacy to his representatives at his decease. What sum did he invest as premium for his policy, and how much did he apply to immediate purposes, assuming that money improved at the rate of 3 per cent. per annum, compound interest? Answer, £935 1 9, the prem.; and £564 18 3, the sum used. A., aged thirty-nine, being indebted, gave a post obit bond for £250, to be paid to B., or his executors. B. immediately disposes of his claim, upon being allowed discount at the rate of 4 per cent. per annuin. What present sum did B. receive upon the bond? Answer, £111 15 7. A certain sum, payable at the death of a person aged sixty, amounted to £175; required to know what that sum was, supposing it to have accumulated at the rate of 5 per cent. per annum, compound interest? Answer, £96 14 8. A widow, aged forty-seven, having two daughters, effected a policy of £360 10s. for the whole term of her life, by a single payment. What premium did she pay, interest being reckoned at the rate of 3 per cent. per ann. ?-Answer, £211 7 4. Required the value of the absolute reversion to one fourth share of £2500 three per cent. reduced annuities, payable on the decease of a lady, aged seventy-two, the stock being valued at £80 per cent., but subject to the payment of a legacy of £250 of the same stock, reckoning the interest of money at 5 per cent. per ann. ? Answer, £310 12 7. ASSURANCES FOR THE WHOLE TERM OF LIFE-continued. XXXVIII.—To find the present value, in annual payments, of the reversion of £1, to be received at the end of the year in which a given life may fail. Divide the present value, in a single payment (XXXVI.), of the reversion of £1, payable at the death of the given life, by the present value of an annuity, increased by unity, of £1 on the given life; the result, multiplied by the given sum, will be the ann. prem. required. Required what premium ought to be paid annually to assure £1000 at the decease of a lady aged forty-five, reckoning interest at 3 per cent.? Pres. val., by Tab. IX., of £1 rever. in single payt., on life of 45, at 3 per cent., =£ .57207; Pres. val., by Tab. VIII., of ann. of £1, incr. by unity, on life of 45, at 3 pr.ct.,=£14.692; Therefore (£.57207÷£14.692) × £1000=£38.9375, or £38 18 9, ann. prem. required. A gentleman, entitled to the absolute reversion of £121 13 4 at the death of his brother, aged thirty-three, desires to exchange his claim for an immediate annuity, to continue during the life of his brother. What annual payment ought he to receive as an equivalent, discount being allowed at the rate of 4 per cent. per annum? Answer, £3 5s. annual premium. What premium, payable (XXXI. 2.) half-yearly, is required to assure the sum of £150 10s. at the death of a person aged twenty-four, reckoning discount at the rate of 5 per cent. per annum ? Answer, £3 3 1. A person entitled, at the death of his uncle, aged fifty-four, to the absolute right to £3750 three per cent. consols, valued at £80 stock, having a large family to support and educate upon very limited means, in his anxiety to increase his income, is willing to dispose of his reversion for an equivalent annuity, payable (XXXI. 3.) quarterly, and to allow the purchaser discount at the rate of 5 per cent. per ann. What annual premium, payable quarterly, ought he to receive in exchange? Answer, £146 4 9. What sum laid up annually, and improved at compound interest during the life of a person aged thirty-six, will amount to £500 at his decease, at the rate of 4 per cent. per annum? Answer, £14 7 5. A party, in the fifty-second year of his age, on obtaining a legacy of £1500, appropriated only so much of it to present uses as enabled him, by assuring his life, to secure the full amount of the legacy to his heirs at his decease. What annual premium did he lay up to accumulate, assuming that money improved at the rate of 3 per cent. per annum, compound interest? Answer, £72 6 5. A., being indebted to B. in the sum of £250, entered into an arrangement to secure the debt by assuring his life for the whole term of its duration for the amount. B. accepted the offer, on condition of A. giving security for the annual payments necessary to keep up the policy. A.'s age was thirty-nine: what was the annual premium for this assurance, reckoning interest at 4 per cent. per annum? Answer, £7 15 6. The executors of a gentleman received £175 from an office, which had been assured upon his life when in his sixtieth year; required to know the annual premium paid, supposing 5 per cent. per annum the rate of interest?—Ans., £10 ̊6s. What principal, in annual payments, will amount to £360 10s. by a policy of assurance, effected on the life of a widow, aged forty-seven, the sum to be paid to her daughters at her decease, discount being allowed at the rate of 3 per cent. per annum ? Answer, £14 17 8. An advocate at Edinburgh, in the forty-second year of his age, wishes to assure his life for £5000 for the benefit of his family, and to pay for the same partly by an immediate sum down of £1000, and the remainder by annual payments. What is the annual premium, supposing the rate of interest 3 per cent. per annum? In questions of this kind, deduct the sum paid down from the whole value of the reversion in a single payment, and then divide by the annuity, increased by unity. Answer, £114 5 2. •K |