Page:Supplement to the fourth, fifth, and sixth editions of the Encyclopaedia Britannica - with preliminary dissertations on the history of the sciences - illustrated by engravings (IA gri 33125011196181).pdf/646

 age of 55, is $$\frac{4073}{4727} \times 0{\cdot}613913$$; therefore, by No. 55, the required value is $$\frac{4073 \times 0{\cdot}613913 \times 10{\cdot}347}{4727} = 5{\cdot}473$$ years purchase; so that if the annuity were L. 200, its present value would be L. 1094, 12s.

70. Ex. 5. Required the present value of an annuity to be received for the next ten years, provided that a person now 45 years of age, shall so long live.

Solution.

The present value of an annuity on a life of 45, to

is the required number of years purchase. And, if the annuity were L. 200, its present value would be L. 1435.

71. Ex. 6. An annuity on a life of 45, deferred 10 years, was shown in No. 69, to be worth 5·473 years purchase in present money; let it be required to determine the equivalent annual payment for the same, to be made at the end of each of the next 10 years, but subject to failure upon the life failing in the term.

Solution.

The present value of L. 1 per annum on the proposed life for the next 10 years, has just been shown to be L. 7·175, and this, multiplied by the required annual payment, must produce L. 5·473; that payment must, therefore, be $$\frac{5{\cdot}473}{7{\cdot}175} = 0{\cdot}76279$$. And, since the annual payment for the deferred annuity of L. 1 per annum would be L. 0·76279, that for an annuity of L. 200 must be L. 152, 11s. 2d.

72. Ex. 7. Let the present value be required of an annuity on a life now 40 years of age, to be payable only while that life survives another now of the age of 50 years.

years purchase is the required value (60).

Therefore, if the annuity were L. 100, it would be worth L. 340, 12s. in present money.

73. If the annuity in the last example were to be paid for by a constant annual premium at the end of each year while both the lives survived; by reasoning as in No. 71, it will be found, that such annual premium for an annuity of L. 1 should be $$\frac{3{\cdot}406}{9{\cdot}984} = \text{L. } 0{\cdot}341146$$; for an annuity of L. 100 it should therefore be L. 34, 2s. 3½d.

74. But if one of the equal premiums for this annuity is to be paid down now, and another at the end of each year while both the lives survive; the number of years purchase the whole of these premiums are worth, will evidently be increased by unity, on account of the payment made now, it will, therefore, be 10·984; and each premium for an annuity of L. 1 must, in this case, be $$\frac{3{\cdot}406}{10{\cdot}984} = \text{L. } 0{\cdot}310087$$; for an annuity of L. 100 it should, therefore, be L. 31, 0s, 2d.

75. Ex. 8. Let it be required to determine the present value of the reversion of a perpetual annuity after the failure of a life now 50 years of age.

Solution.

The value of the perpetuity is 20 years purchase (8.)

purchase, the required value of the reversion (63.)

So that if the annuity were L. 300, its present value would be L. 2502.

76. In the same manner it will be found, by the 68th number and those referred to in the last example, that the reversion of a perpetuity, after the failure of the last survivor of three lives, now aged 50, 55, and 60 years respectively, is worth 5·999 years purchase in present money; therefore, if it were L. 100 per annum, its present value would be L. 599, 18s.

III. OF ASSURANCES ON LIVES.

77. An assurance upon a life, or lives, is a contract by which the Office or Underwriter, in consideration of a stipulated premium, engages to pay a certain sum upon such life or lives failing within the term for which the assurance is effected.

78. If the term of the assurance be the whole duration of the life or lives assured, the sum must necessarily be paid whenever the failure happens; and, in what follows, that payment is always supposed to be made at the end of the year in which the event assured against takes place. The anniversary of the assurance, or the day of the date of the policy, being accounted the beginning of each year.

79. At the end of the year in which any proposed life or lives may fail, the proprietor of the reversion of a perpetual annuity of L. 1 after their failure, will receive the pound then due, and will, at the same time, enter upon the perpetuity; therefore, the present value of the reversion is the same as that of L. 1 added to the money a perpetual annuity of L. 1 would cost, supposing this sum not to be receivable until the expiration of the year in which the failure of the life or lives might happen.

80. Hence we have this proportion. As the value of a perpetuity increased by unity is to L. 1, so is the present value of the reversion of a perpetual annuity of L. 1, after the failure of any life or lives, to the present value of L. 1, receivable at the end of the year in which such failure shall take place.

81. Therefore, if the value of an annuity of one pound on any life or lives, be subtracted from that of the perpetuity, and the remainder be divided by the value of the perpetuity increased by unity; the