IMMEDIATE ANNUITIES-Continued. XXXII.-1. To find, by means of Table VIII., what annuity a given sum will purchase during the whole term of a given life, according to the Northampton table of mortality, 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 VIII., of an annuity of £1 on the given life, at the given rate per cent.; the result will be the annuity required. The trustees of a widow lady, aged forty-five, purchased from a public company an annuity, payable yearly, during the remainder of her life, for the sum of £684 12s. What was the amount of the ann., allowing int. at the rate of 3 per cent. per ann.? Pres. val., as found by Tab. VIII., of an ann. of £1, on a life of 45, at 3 per ct.,=£13.692; Therefore £684 12s., or £684.6,÷13.692=£50, the annuity required. Legacy duty was paid at Somerset House on the sum of £989 15 6, the value of an annuity, payable yearly, during life, which had been bequeathed to a person aged forty. 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, £75 per annum. A merchant, desirous in his prosperity of securing a permanent provision to his wife, now in the twenty-seventh year of her age, purchases an annuity, to be payable to her half-yearly, during the remainder of life, for the sum of £1362 14s. What amount of annuity, payable half-yearly, ought the lady to receive, supposing the merchant invested his money at the rate of 5 per cent. per annum interest? Increase the annuity on the given age by .25, because payable (XXXI. 2.) half-yearly. Answer, £100 per annum, payable half-yearly. A gentleman, deriving a large income of eighty thousand pounds sterling per annum, terminable with his life, having three daughters, aged severally twenty-two, twenty-five, and twenty-seven, considered himself bound as a parent to sink a present sum of £20000 upon each of them, in the purchase of annuities, to continue during the whole term of their respective lives. What annuity, payable quarterly, did he thus secure to each of them, assuming the rate of int. at 4 per cent. per ann. ? Increase the annuity, as found by Table VIII, of £1, upon each age, at the given rate, by .375, because payable (XXXI. 3.) quarterly. Answer, to the age of 22, an annuity of £1236 14 1; to the age of 25, an annuity of £1264 15 8; and to the age of 27, an ann. of £1285 8 7; each payable quarterly. The aged sister of the celebrated Sir Sidney Smith, the hero of Acre, having been reduced by a long series of misfortunes to poverty and destitution, a few generous friends undertook to raise a subscription for her relief. Amongst others, the accomplished Jane Porter pleaded urgently and successfully in her favor. Her most gracious Majesty Queen Victoria, her Royal Highness the Duchess of Kent, and many of the nobility and gentry, contributed liberally to this benevolent fund, which ultimately amounted to £370 9s. This sum was immediately invested in the purchase of an annuity, payable quarterly, to this unfortunate lady, who was in the sixty-sixth year of her age, during the remainder of her life. What amount of annuity was thus obtained, reckoning at the rate of 5 per cent. per annum for the interest of money? Answer, £50 per annum, payable quarterly. A., who is fifty-four years of age, being in possession of a freehold estate, worth, in present money, the sum of £2000, is willing to exchange the same for an equivalent annuity, payable yearly, during the remainder of his life, with B. What ann. ought to be granted to A., and to what immediate perpetuity does B. become entitled, supposing the interest of money in both cases to be at the rate of 4 per cent. per ann.? Answer, £191 18 5, annuity, payable yearly to A. during life; and £80, immediate annuity, payable (XXV. 2.) for ever to B. XXXII.-1. IMMEDIATE ANNUITIES-continued. A lady, aged thirty-five, sank £2000 in the purchase of an irredeemable annuity, payable half-yearly, during the whole period of her existence. To what annuity was she entitled, allowing that money constantly accumulated at the rate of 3 per cent. per annum? Answer, £123 11s. per ann., payable half-yearly. 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 of a person in the forty-first year of his age, reckoning interest at the rate of 5 per cent. per annum? Pres. val., by Tab. V., of £1 per ann. long ann., for 30 yrs., at 5 per ct.,=£15.372451; And 15.372451 × 100 (II. 3.)=£1537.2451, pres. val. of £100 per ann. long annuities; The pres. val., by Tab. VIII., of an ann. of £1, on a life of 41, at 5 per cent., =£11.695; Therefore £1537.2451-11.695 £131 8 11, the equivalent annuity. A nobleman, in consideration of the faithful services of his late steward, who died prematurely, ordered £1000 to be laid out in the purchase of an annuity, to be paid quarterly, for his orphan daughter, now in the seventeenth year of her age. Required the annuity granted by his lordship's kindness, allowing 5 per cent. per annum for the interest of money?-Ans., £67 19 9 per ann., payable quarterly. A., aged sixty-three, relinquished to B. his claim to £5000 consols, when this stock was at the price of 90, for an equivalent annuity, payable half-yearly, during the remainder of his life. What annuity should B. grant to A. in exchange, reckoning the interest of money at the rate of 3 per cent. per annum ? (£5000 × 90)÷100-£4500, the present value of £5000 consols, at 90; And, by Table VIII., an annuity of £1, payable half-yearly, on a life of 63,=£9.16 ; Therefore £4500-9.16 £491 5 4, the equivalent annuity, payable half-yearly. 2. To find the age at which an annuity commences. Divide the given sum by the given annuity; the result will be the present value, as found by Table VIII., of an annuity of £1 at the given rate, at the age required. An annuity of £50, payable yearly, was granted to a maiden lady, during her life, for the sum of £684 12s. Required the age at which the annuity commenced, assuming 3 per cent. per annum the rate of interest? £684 128., or £684.6,÷50=£13.692, the present value of an annuity of £1, at per cent., at the age required; And in Table VIII., under 3 per cent., £13.692 corresponds to 45, the required age. A landed proprietor, for the sum of £3983 8s., encumbered his estate with the payment of £300 a year, during the existence of a certain life. At what age was the annuity granted, supposing that the mortgagee was allowed interest at the rate of 5 per cent. per annum for the use of his money? Answer, at the age of 28. Required the age of a party to whom an annuity, to continue during his life, of £39 17 6 was granted for the sum of £477 13 3; the purchaser having been allowed, in this transaction, interest at the rate of 5 per cent. per annum ? Answer, 39 years. DEFERRED ANNUITIES. XXXIII. To find the present value of an annuity to continue during the remainder of a given life after a given term. Multiply the pres. val., as found by Tab. III., of £1, at the given rate, for the deferr. term, byThe present value of an annuity of £1 on the given life at the increased age; and this byThe probability of the given life attaining the given age: The result will be the present value required. A lady, aged forty-five, will be entitled to an annuity, payable yearly, during the remainder of her life, after the age of fifty-five. What is her reversion worth in present money, supposing it to be £50 per annum, and the rate of interest to be 3 per cent.? 10 years, the deferred term,+45 years, the given age,=55 years, the increased age; 3248 A young man, in the twenty-seventh year of his age, being of a contemplative and provident disposition, paid down such a sum of money as would secure him an annuity, payable half-yearly, of £75, at the age of forty, should he live so long, and then to continue during his life. What present money did he give, calculating interest at the rate of 4 per cent. per annum ?-Answer, £477 11 10. What premium ought to be paid down for an annuity of £100, to commence after the next twenty-one years, and then to continue during the life of a lady, now aged twenty-seven years, allowing the purchaser interest at the rate of 5 per cent. per annum ? Answer, £249 2 8. A lady, aged twenty-two, will be entitled, after the next eleven years, to an annuity of £1236 14 1 during the remainder of her life: what is the present value of her reversion, reckoning the interest of money at 4 per cent. per annum? Answer, £9618 1 7. A gentleman, possessing a large income, terminable at his death, desires to pay down such a present sum as will secure to his only daughter, now in her twentyfifth year, an annuity during life of £1264 15 8, upon her attaining the age of thirty-five. Required the present value of this deferred annuity, assuming that money accumulated at the rate of 5 per cent. per annum ?-Answer, £8177 17 3. A young nobleman, now aged twenty-seven, will be entitled to an annuity of £1285 8 7, during life, provided he shall attain the age of forty. Having, however, been profligate and extravagant, his importunate creditors compel him to dispose of his claim, in order to enable him to meet his pecuniary engagements. The present value of his title cancelled his obligations exactly. How much was he indebted, allowing interest at the rate of 4 per cent. per ann.?—Ans., £8033 5 9. What premium ought to be given now to secure an annuity of £67 19 9, payable (XXXI. 2.) half-yearly, during the life of a young lady, now aged seventeen, after the next thirteen years, reckoning interest at 4 per cent. per annum? Answer, £505 17 5. A certain sum was the purchase-money of an annuity of £191 18 5, payable (XXXI. 3.) quarterly, to be enjoyed for life by a person, aged fifty-four, if he attained the age of sixty. Required the price of the annuity, interest being allowed at the rate of 5 per cent. per annum ? Answer, £1011 7 11. XXXIII.-DEFERRED ANNUITIES-continued. What sum ought to be given for the remainder of the lease of an estate for twenty-one years, after the death of a person aged forty; allowing interest to the purchaser of 4 per cent. per annum, and the clear annual rent being £70? Answer, £203 12 7. A freehold estate (XXV. 1.), producing £60 per annum, is mortgaged during the life of its present occupant, aged sixty-four; what is the present value of the reversion, supposing the interest of money at the rate of 3 per cent. per annum? Answer, £1483 6 9. A perpetual annuity of £200 is to be entered upon, after the next ten years, by a lady, aged twenty-five, should she live so long. What is the present value of her title, allowing interest at the rate of 4 per cent. per annum for the improvement of money? Answer, £2845 12s. A., or his heirs, is entitled to an estate in fee simple of £75 per annum, after the next seven years, or sooner should the present tenant, aged seventy-two, not survive the term. What is the present value of A.'s inheritance, reckoning at the rate of 5 per cent. per annum for the interest of money? Answer, £1187 9 2. What sum ought to be given for the reversion of a freehold estate, after the extinction of a life aged thirty-nine, the estate being estimated at the clear annual rent of £250, and assuming the interest of money at 4 per cent. per annum? Answer, £2906 5s. What sum ought to be given for the perpetual tithes of a rectory, producing a clear rent of £300 per annum, after the death of the present incumbent, aged fifty-nine, so as to allow the purchaser at the rate of 5 per cent. interest for his money? Answer, £3420 6s. A person, aged sixty-eight, having purchased the lease of an estate for fifty years, the clear annual rent of which is £460, desires to receive so much present money as it may be worth after his decease. What sum ought to be paid to him, in order that the purchaser may be allowed interest at the rate of 3 per cent. per annum for the use of his money? Answer, £8446 17 5. What sum ought to be given for the remainder of a lease of forty years, after the death of the present tenant, aged seventy-six, the estate being estimated at £65 per annum, allowing interest at the rate of 4 per cent. per annum? Answer, £980 7 8. A., being entitled on the death of his father, aged sixty-four, to the possession of a freehold, producing £600 per annum, desires to exchange it for a perpetual income, to commence at once. B. is willing to grant an immediate perpetuity of equivalent value to A. for his reversionary title, on being allowed interest at the rate of 4 per cent. per annum for the use of his money. What is the value of A.'s reversionary freehold, and what equivalent immediate perpetuity ought B. to give in exchange Answer, £10182 0 0, pres. val. of A.'s reversionary freehold; And £407 5 7, the equiv. perpet. to be given by B. for it? DEFERRED ANNUITIES-continued. XXXIV. To find the present value, in annual payments, of an annuity, to continue during the remainder of a given life, after a given term. Multiply the present value of £1 at the given rate for the deferred term by The probab. of the given life attaining the increased age; deduct the product from unity; Add this difference to the difference between the present values of an annuity of £1 at the given age, and of an annuity of £1 on the given age attaining the deferred age:The present value of the given deferred annuity, divided by this last sum, will be the annual payment required. A lady, aged forty-five, desirous of providing an annuity of £50, to commence when she shall attain the age of fifty-five, wishes to purchase the same by an equal annual premium at the beginning of each year till then; but such annual premium to cease with her life, should that fail in the interim. What annual premium should she pay, reckoning the interest of money at 3 per cent. per annum? 10 years, the deferred term,+45 years, the present age,=55 years, the increased age; The present value, by Table III., of £1 for 10 years, at 3 per cent.,= £.744094; The probability of a life of 45 attaining (XXVIII. 1.) the age of 55= Pres. val., by Tab. VIII., of an ann. of £1 on given life, 45, at 3 per ct.,=£13.692; Pres. value, by Tab. VIII., of an ann. of £1 on the increased age, 55,=£11.150. OPERATION. = 2448 3243 £.439180; £6.253148; £7.878032; £.744094 × 3+8=£.56082, which, deducted from unity, or 1.00000,= Then £.56082 x £11.150 (XXXIII.), the present value of a deferred annuity of £1 on the given life, And £13.692- £6.253148=£7.438852;+.43918 And £6.253148 × £50, gives the pres. val. of deferred annuity of £50=£312.6574; Therefore £312.6574-£7.878032=£39.6872, or £39 13 9, the ann. prem. required. = A young man, who has just completed the twenty-seventh year of his age, wishes to pay immediately, and at the beginning of each successive year, such a premium as will secure him a deferred annuity of £75 at the age of forty; but such annual premium to cease, should he not survive the term; that is, he desires to make thirteen annual payments to entitle him, on the day of his attaining the forty-first year of his age, to an annuity of £75, which shall continue afterwards every year during the remainder of his life. What sum per annum ought he to pay, supposing the rate of interest at 4 per cent. per annum? £.600574; 13 years, the deferred term,+27 years, the present age, =40 years, the increased age; OPERATION. £.600574 x 3833-£.473555, which, deduct. from unity, or 1.000000,= Then £.473555 x £13.197 (XXXIII.), the present value of a deferred annuity of £1 on the given life, And £15.184-£6.2495-£8.9345;+.526445 = 3635 4610 £.526445; £6.24950; = £9.460945; And £6.2495 × 75, gives the pres. val. of a deferred annuity of £75-£468.7125; Therefore £468.7125÷£9.460945=£49.5418, or £49 10 10, the ann. prem. required. |