Sidebilder
PDF
ePub

is worth 14.78 years purchase: and an annuity at 5 per cent. for a London life, is worth 11.6 years purchase; and for a Northampton life, at the fame rate of intereft, 13.07 years purchase. This table also shows the value of an annuity of 11. for a single life, at all the above-mentioned rates of intereft: thus, an annuity of 1l. per ann. for a single life, at 30 years of age, according to the London tables, at 3 per cent. is worth 151.; and the fame, according to the Northampton tables, is worth 161. and upwards of 18s. &c.

This table is esteemed the best of any extant, and preferable to any other of a different form. But those who fell annuities have generally a table of 2 years more value than the lives in this table, for purchafers who are upwards of twenty years of age.

Definitions.

1. The probability of life is the chance that any person or perfons have of living to any certain time, and is denoted by a fraction, whofe numerator is the chance of living, and denominator that of living and dying. Thus, fuppose it were required to find the probability of a person of the age of 20 attaining to the age of 37, according to Mr. Simpson's table. Here it must be obferved, that of 360 perfons living at the age of 20, only 252 survive to the age of 37; therefore, 108 perfons have died between the two ages. Thus, 252 is the chance of the faid perfon's living to the age of 37, and 360 the chance of the said person's dying before he attains the age of 37, and the probability of life of that person is expressed by the fraction or; therefore, the odds in that person's favour, or the chance that he fhall live to that age, is 7 to 3.

2. The probability of dying is expressed by a fraction, which is the difference between the former fraction and unity. Thus, the probability that the aforefaid person shall die before the age of 37 is expreffed by 10% or fo, which

[merged small][ocr errors][merged small]

fhows that the chance of that perfon's dying before the faid age is as 3 to 7.

3. The extremity of life is the period beyond which there is no probability of furviving. In the Northampton tables this is 96 years.

4. The complement of life is the number of years which any perfon's age wants of the full extremity of life; this in the Northampton table, for a life aged 80, is 17 years.

5. The expectation of life is the number of years due to the life of a perfon of a certain age, upon an equality of chance. And it is the number of years purchase, which an annuity for life is worth in ready money, without allowing any intereft. And in fingle lives it is always equal to the fum of all the probabilities of furviving to the extremity of life.

6. The number of years purchase of annuities, at any rate of intereft, is that number which, if multiplied by the annuity, is equal to the prefent value thereof, according to fuch rate of intereft; therefore, it is the present value of an an nuity of 11. according to a given rate of intereft, as seen in Table V.

7. The reverfion of a life annuity is where two or more lives are in joint poffeffion, and the expectation depends upon the probability of one particular life surviving the rest.

PROBLEM I.

To find the Value of an Annuity for the Life of any Perfon, at a given Rate of Intereft.

RULE. Seek the age of the perfon in the first column of Table V. and against it, under the proper intereft, is the number of years purchase for either the London or Northampton life; or, which is the fame thing, the prefent value of an annuity of 17. during fuch life. Multiply this by the annuity, and the product is the answer,

EXAMPLE.

EXAMPLE. Suppofe the given age be 49, the rate of intereft 3 per cent. and the annuity 201. Here again 49, and under 3 per cent, ftands 11.6 according to the London bills, which, multiplied by 207. gives 232/. for the value of a London life. And in the next column stands 12.69, the value according to the Northampton bills, which, multiplied by 201. as before, produces 2537. 16s. for the value of a Northampton life.

SECT. II.

OF THE VALUE OF AN ANNUITY DURING JOINT LIVES.

PROBLEM I."

To find the Value of an Annuity for the joint Continuance of two Lives; that is, one Life failing, the Annuity to cease.

CASE I.

When both Perfons are of the fame Age.

RULE. Find the value of any one of the lives from Table V. Multiply this value by the intereft of 17. for a year at the given rate; fubtract the product from 2, divide the aforefaid value by this remainder, and the quotient will be the value of 17. annuity, or the number of years purchase.

EXAMPLE. What is the value of 100l. annuity, for the joint lives of two perfons, aged 40 years each, according to the London tables, reckoning intereft at 5 per cent.? Here, by the table, one life for 40 years is,'

[blocks in formation]
[blocks in formation]

And 1.485)10.3 (6.9 value of 14 annuity, which multiplied by 100 is 690l. the value of the annuity fought.

[ocr errors]

CASE II.

When the two Perfons are of different Ages.

RULE. Find the values of the two lives in Table V. Multiply them one into the other, and call the refult the first product; then multiply the faid firft product by the interest of il. for a year, at the given rate, calling the result the second product: add the values of the two lives together, and from the fum fubtract the second product; divide the first product by the remainder, and the quotient will be the value of 14 annuity, or the number of years purchase.

EXAMPLE. What is the value of 50l. annuity, for the joint lives of two perfons, whereof one is 20, and the other 30 years of age, according to the Northampton tables, interest at 4 per cent.?

[blocks in formation]

And 21.333064)236.9234(11.1 value of 17. annuity.

50

555.0 Value required.

PROBLEM

[blocks in formation]

To find the Value of an Annuity during the Life of the longest Survivor of two Lives; that is, as long as either of the two Parties live.

RULE. From the fum of the values of the fingle lives, fubtract the value of the joint lives, and the remainder will be the value fought.

EXAMPLE. What is the value of an annuity of 17. to continue during the longeft of two lives: the one perfon being 30, and the other 40 years of age; interest at 4 per cent. the life of 30 years of age valued according to the London bills, and that of 40 years of age according to the Northampton bills?

By the table, the value of 30 years is
The value of 40 years is

The value of their joint lives, by Problem II.

Cafe 2, is

The value fought

13.1

13.20

26.30

8.9

17.4

If the annuity be any other than 17. multiply the above found value, by the given annuity; and if the two perfons be of equal ages, the value of their joint lives must be found by Cafe 1, of Problem II.

PROBLEM IV.

To find the Value of an Annuity for the joint Continuance of three Lives; that is, one Life failing, the Annuity to cease.

RULE. Multiply the value of the three fingle lives continually into each other, calling the refult the product of the three lives; multiply that product by the intereft of 17. and that product again by 2, calling the refult the double product ; then from the fum of the feveral products of the faid lives,

taken

« ForrigeFortsett »