Ex. 3. A merchant marries a lady aged 28, whose fortune for life is 300l. per annum, being desirous of converting the same into money, what ought he to have for it, allowing interest 5 per cent. ? Ex. 4. What is the value of an annuity of 2001. during the life of a person aged 25 years? Ex. 5. What is the value of 50l. per annum, payable during the life of a person aged 41 years? Ex. 6. What is the value of a clear annuity of 75l. during the life of an old man aged 76? Ex. 7. What is the value of a landed estate during the life of a person aged 38, producing nett 30l. 9s. per annum? Ex. 8. What is the life interest of a person aged 53, in 1250l. 3 per cent. Consols worth? Ex. 9. A gentleman aged 60, who receives an annuity of 150l. per annum, for life, out of a freehold estate, wishes to exchange his life for that of his wife, aged 32: what ought to be required of him for so doing? Ex. 10. A person having an annuity of 100l. during a life of 37 years, agrees to exchange it for an equivalent annuity during a life of 45; what annuity should be granted him? Ex. 11. What annuity will 100l. purchase during the life of a person aged 28 ? Ex. 12. A parish means to raise a sum of money for building a workhouse, by life annuities; at what ages should they grant 7, 8, and 9 per cent. ?* Ex. 13. What is the difference in value between an annuity of 40%. during a life of 36, and an annuity certain for 20 years?+ Ex. 14. A person aged 27 is possessed of 60l. per annum in the government long annuities, which have 51 years to run, and which he is willing to relinquish for an annuity during his life; what should the equivalent annuity be? Ex. 15. What annuity should be granted to a person aged 57 during his life, for 2,000l. five per cent. stock, which is now at 991⁄2? NOTES. * Questions of this sort are answered by dividing 100l. by the rates per cent., and opposite to the numbers in the table that are nearest the quotient, are the required ages: thus, to find at what age a life annuity of 9 per cent. should be granted, 100 9 11.111, the nearest number in the table is 11.105, by the side of which is 45, hence, to ages of 45, an annuity of 9 per cent. may be granted, + See Tables, p. 204 and 196. TABLE II. Shewing the Value of an Annuity during the joint continuance of Two Lives, according to the probabilities of Life at Northampton, reckoning interest at 5 per cent. Ages. Vaiue. Ages. Value. Ages. Value. Ages. 'Value. 5-5 5-10 5-15 5-20 11.561 15-50 5-25 11.281 15-55 8:403 5-30 10.959 15-60 7.622 5-35 11.984 15-35 10.555 CASE 1. To find the value of an annuity on the longest of two single lives. RULE. From the sum of the values of the single lives : subtract the value of their joint continuance, and the remainder will give the value of the longest of the lices. Ex. 1. What is the value of the longest of two lives 10 and 15 = 12.302 Table II. The value of the joint continuance of two > lives of 17.425 Value of the longest of the two lives Therefore an annuity of 100l. a year upon the longest of two lives, one 10 and the other 15, would be worth nearly 17 years and a half purchase, or more accurately, 1742l. 10s. Ex. 2. What is the value of an annuity on the longest of two lives whose ages are thirty and forty. CASE II. To find the value of an annuity on three joint lives. RULE. Take the value of the two elder, and find the age of a single life equal to that; then find the value of the joint lives of this now found, and the youngest. Ex. 1. Let the three lives be 20, 30, and 40. The value of the joint continuance of the two eldest; viz of 30 and 40 (by Table II.) is equal to 9.576, which answers to a single life (by Table 1.) of 54. Now, the value of the joint lives of 20 and 54 by Table II., or the ages which come nearest, viz. 20 and 55, is 8.216* for the value sought: hence an annuity of 401. on three joint lives would be worth about 328/. 12s. Ex. 2. To find the value of 3 joint lives of the ages 15, 30, and 45. Ex. 3. What is the value of an annuity of 150l, on the joint-continuance of three lives of the ages 50, 60, and 70? CASE HI. To find the value of the longest of any three lives. RULE. From the sum of the values of all the single lives, subtract the sum of the values of all the joint lives, combined two and two. To the remainder add the value of the three joint lives, and the sum will be the value of the longest of the three lives. Ex. 1. What is the value of the longest of three lives, whose ages are 20, 30, and 40? Table I. Value of a life-of-20: 14.007 * The numbers 9.576 and 8.216, are not quite accurate, because the limits of this book do not admit of a table giving the combinations of all ages. 8.6968.216 (the value of the joint lives found in Ex. 1. Case II.) 16.912 the value of the longest of the three lives. Ex. 2. What is the value of the longest of three lives, whose ages are 15, 30, and 45 ? Ex. 3. What is the value of an annuity on the longest of three lives, whose ages are 50, 60, and 70? EXAMPLES FOR PRACTICE. Ex. 1. What is the present value of an annuity of 50l., on the joint lives of two persons, each 30 years of age? Ex. 2. What is the present value of an annuity of 657., during the joint lives and the life of the survivor, of a man aged 45, and his wife aged 35 ? Ex. 3. What is the value of a lease producing 27l. 13s. per annum, on the longest of two lives aged 60 and 45? Ex. 4. What is the value of an annuity of 40l. on two joint lives of 70 and 5 years? Ex. 5. What is the value of an annuity of 50%. on the longest of two lives of 70 and 5 years? CASE IV. To find the value of an annuity on a given life for any number of years. RULE. Find the value of a life as many years older than the given life as are equal to the term for which the annuity is proposed. Multiply this value by 11. payable at the end of this term, and also by the probability that the life will continue so long. Subtract the product from the present value of the given life, and the remainder multiplied by the annuity will be the answer. Ex. 1. What is the value of an annuity of 501. per ann. for 14 years, on a life of 35? 35 +14 The value of a life of 49 (14 years older than the given life, by Table I.) The value of 11. payable at the end of 14 years (Table, p. 209) The probability that a life of 35 will continue 14 years (Table, p. 200, and the 2d Case in p. 201.) ) 10.443 X .505068 X :(2996). 4010. 49. .7322 3.861, which, subtracted from 12.502, the value of a life of 35, Table I. gives 8.641; and 8.641 X 50 432l. 1s. Ex. 2. What is the value of an annuity of sol. per annum for 20 years, provided a person aged 45 live so long? TABLE, Shewing the present Value of 17. to be received at the end of any number of years, not exceeding 100; discounting at 5 per Cent. Compound Interest. In order to find the present worth of any sum which is to be received at the end of a certain number of years-Multiply the number in the table opposite to the term of years, by the sum, and the product will be the answer. Ex. 1. What is the present value of 750l., to be received at the expiration of 9 years? The number in the table even with 9 years is .644609, which is to be multiplied by 750. .644609 |