242 17. Find the value of a life-annuity of £560, secured upon a freehold property, receivable after the termination of a lease of which 7 years have to run; the age of the person being 25, and the interest of money reckoned at 4 per cent. Ans. £6623, 4s. 6d. 18. What is the value of an annuity of £85, 17s. 6d, receivable during the joint continuance of two lives of 21 and 25, but not to commence until 10 years have expired, interest at 5 per cent. ? Ans. £533, 14s. 44d. 19. What is the present value of an annuity of £30 for the next 15 years, dependent upon the existence of a life whose age Ans. £316, 1d. nearly. is 19, interest at 4 per cent.? Present value = (a19a34 x 104 × -15 5417 X 30. VI. TO FIND THE PRESENT VALUE OF A PERPETUITY TO BE ENTERED UPON AFTER THE FAILURE OF A GIVEN LIFE OR LIVES. RULE. From the present value of the perpetuity subtract the present value of an annuity on the given life or lives; the remainder will be the value of the perpetuity in reversion. THE REASON of the rule is obvious. Exercises. 20. What is the value of a freehold estate of £1721, 10s. yearly rent, to be entered upon after the decease of the present possessor, Ans. £28244, 13s. whose age is 64, interest at 4 per cent.? 21. What is the value of a freehold estate of £1721, 10s. yearly rent, to be entered upon after the failure of one of two lives, whose ages are 35 and 40, interest at 3 per cent.? Ans. £33199, 14s. 01d. OF A GIVEN SUM VII. TO FIND THE PRESENT VALUE PAYABLE ON THE FAILURE OF A GIVEN LIFE OR LIVES. RULE.-Multiply the value of an annuity of £1 on the given life or lives by the interest of £1 for a year, and subtract the product from 1; multiply the remainder by the present value of £1 for a year, and then by the given sum; the result will be the value in a single payment. The annual payment is found by dividing the single payment by the value of an annuity on the given life or lives increased by 1. Or, Subtract the present value of £1 for 1 year from 1, and multiply the remainder by the value of an annuity of £1 on the given life or lives; subtract this product from the present value of £1 for 1 year, and the remainder, multiplied by the given sum, will be the value in a single payment. This is called an assurance on the given life or lives. The annual premiums are in all cases supposed to be payable in advance. Example.-What ought a gentleman, whose age is 36, to pay just now to secure to his heirs £700 payable at his death ?-and what ought the annual payments to be, during his life, to assure the same, interest at 5 per cent.? REASON OF THE RULE.-This Rule may be made to depend upon Rule VI., by converting the sum assured into a perpetual annuity, the amount of which will manifestly be the interest of the sum assured for 1 year. In the case of a perpetual annuity, however, the first payment is supposed to be made at that term next succeeding the death of the present occupant; but in the case of a sum assured, payable at the same term, one year would require to elapse before the first year's interest could be drawn hence the present value of a reversionary perpetuity, equal to the interest of the sum assured, is more valuable than the present value of the reversionary sum assured by one year's purchase. It follows, then, that the result obtained by last rule, multiplied by the present value of £1 for 1 year, will be the present value of the sum assured in a single payment. Thus, in the Example, suppose the sum assured is £1: since the rate per cent. is 5, .. the present value of a perpetuity of £1 is ; hence, by Rule VI., 1 *05 the present value of a reversionary perpetuity of £1 = Now, if 1 ⚫05 1 *05 a36 be multiplied by '05, the interest of £1 for a --1 year, we get 1-'05 × α36· For the reason above stated, this requires to be multiplied by 1.05, to obtain the present value of £1 assured on a life aged 36; that is, present value = (1 —·05 × a„) × 1·051. Hence the first Rule: the second is only a modification of this. Exercises. 22. What single premium, or what annual premium, would be required to secure £739, 7s. 6d., payable at the end of the year in which the existence of a person now aged 29 shall fail, interest at 3 per cent.? Ans. Single premium, £292, 5s. 7ąd.; annual premium, £14, 1s. 6d. 23. What is the value of £2300 payable on the failure of two joint lives, aged 55 and 60, interest at 4 per cent. ? Ans. £1541, 10s. 74d. paid during the joint to secure the payment 24. What annual premium must be existence of two lives aged 25 and 30, of £869, 13s. 6d. after the extinction of the longer of the two lives, interest at 4 per cent.? Ans. £10, 11s. 24d. 25. What ought a person aged 51 to pay annually to assure £1000 on his life, interest at 5 per cent.? Ans. £32, 19s. 2ąd. 26. What ought a person aged 30 to pay annually for an assurance of £350 on his life, interest at 3 per cent.? Ans. £6, 16s. 73d. Miscellaneous Exercises. 1. A father is desirous of providing a dowry of £1560 for his daughter on her arriving at the age of 21; what sum should he pay to secure it, interest at 4 per cent., his daughter's age at present being 12? Ans. £1035, 11s. 72d. 2. What is the annual premium during the joint existence of two lives aged 45 and 50, for an assurance of £7000, payable on the death of the last survivor, interest at 3 per cent.? Ans. £239, 0s. 2 d. 3. Required the single payment for an annuity of £120 to a lady aged 37, after the decease of her husband, aged 40, in the event of her surviving him, interest at 5 per cent. Ans. £316, 8s. 91d. 4. Required the annual premium payable during marriage, for an annuity of £400 on the life of a lady aged 30, in the event of her surviving her husband, whose age is 35, interest at 4 per cent. Ans. £92, 15s. 5ąd. 5. What is the price of an annuity of £250 during the joint lives of a gentleman aged 70 and a lady aged 55, interest at 3 per cent.? Ans. £1504, 15s. 6. What is the present value of the next presentation to a living, the age of the present incumbent being 86, the rate of interest 4 per cent.; assuming that the age of the presentee is 30, and that the annual value of the living is £420? Ans. £5873, 14s. 7. A gentleman assures his life for £2000; what annual premium ought he to pay, supposing that he pays £500 just now, that his age is 47, and that the rate of interest is 4 per cent. ? Ans. £25, 7s. 73d. 8. What is the value of a perpetual annuity of £827, to be entered upon after the decease of a lady aged 61, interest at 5 per cent. ? Ans. £9335, 3s. 6d. 9. A party proposed to lay out £400 in the purchase of an annuity, to be entered on at the end of 9 years, and to continue so long as a life, now aged 32, shall survive that time; what sum per annum will he be entitled to, interest at 3 per cent.? Ans. £34, 2s. Old. 10. A society of married men pay each £10 of entry, besides an annual contribution during marriage, depending upon their ages at entering; what ought a man aged 43 to contribute annually to secure an annuity of £75 for his wife, aged 30, in the event of her surviving him, interest at 4 per cent.? Ans. £23, 4s. 11 d. 11. A person aged 40 is desirous of purchasing an annuity of £200 upon his life, to commence on his arriving at the age of 65, and to be paid for by an annual premium until that time; what is the annual premium, the rate of interest being 4 per cent.? Ans. £26, 1s. 2 d. 12. A man and his wife, aged 35 and 27, wish to sink £2000 upon two annuities; one during their joint lives, and another, of half the value, during the remainder of the surviving life: what annuities ought to be granted, interest at 5 per cent.? Ans. During their joint lives, one for £137, 0s. 51d.; during the surviving life, one for £68, 10s. 2ad. EXCHANGE. EXCHANGE is the rule by which sums in the money of one country are converted into sums of equivalent value, in the money of another.* PAR OF EXCHANGE means that sum in the currency of one country which, in intrinsic or real value, is equal to a given sum estimated in the currency of another. Thus, according to the mint regulations of Great Britain and France, £1 sterling is equal to 25 fr. 22 cents: this is called the par of exchange. THE COURSE OF EXCHANGE, at any date, means that sum in the currency of one country which at that date is equal to a given sum estimated in the currency of another. The course of exchange is seldom at par, but is sometimes above, and sometimes below par, according to circumstances. In the calculation of exchanges, a variable sum in one country is allowed for a certain sum in another. Thus, London gives France £1 sterling for a variable number of francs, and Portugal a variable number of pence for 1 milrea. The calculations in Exchange are performed by Simple Proportion and Practice. The pupil will find no difficulty in the solution of any of the exercises. MEASURES OF FOREIGN MONEY. = AUSTRIA.-4 pfennings 1 creutzer, 60 creutzers = 1 florin. The money of account and exchange is the florin, which is equal to 2s. Old. sterling nearly; the par of exchange with London is 9 fl. 50 cr. per £1 sterling. AUSTRIAN ITALY. -100 centisimi = 1 lira Austriacha. The money of account and exchange is the lira Austriacha, which is equal to 8 d. sterling nearly; the par of exchange with London is 6 lire Austriache per 483d. sterling. CANADA.-100 cents 1 dollar. The dollar is equal to about 4s. 1 d. sterling. In exchange, the dollar is estimated at 4s. 6d. sterling, and bills on London sell at a premium of 9 per cent. Accounts were formerly kept in pounds, shillings, and pence currency: £100 currency were equal to £90 sterling, giving a premium for bills on London. FRANCE.-100 centimes 1 franc. The money of account and exchange is the franc, which is equal to 94d. sterling nearly; the par of exchange with London is 25 fr. 22 ct. per £1 sterling. For full information, see Chambers's Commercial Tables, pp. 242 to 253. |