Example. Let the age of the life in possession be 40 years, and the ages of the two lives in expectation be 20 and 65 years ; in this case, the value of the three lives being 15,902, and that of the life in possession 11,837, the answer will be 4,065 years purchase : so that, if the an nuity was to he 5001. the value of the re version would be 20321. 108.
Problem 4. To find the value of an annuity certain for a given term after the extinction of any given life or lives.
Subtract the value of the life or lives from the perpetuity, and reserve the re mainder: then say as the perpetuity, is to the present value of the annuity certain, so is the said reserved remainder to a fourth proportional, which will be the number of years purchase required.
Example. Suppose A and his heirs are entitled to an annuity certain for 14 years, to commence at the death of B, aged 25.—What is the present value of A's interest in this annuity ?—The value of the life of B, is 13,567, which subtract ed from 20 (the perpetuity) leaves 6,433 for the remainder : therefore, as 20 is to 9,198, the value of an annuity certain for 14 years, so is 6,433 to 3,183, the number of years purchase required.
Problem 5. B, who is of a given age, will, if he lives till the decease of A, whose age is also given, become possessed of an estate of a given value ; what is the worth of his expectation in present mo ney ? Find the vahie of an annuity on two equal joint lives, whose common age is equal to the age of the oldest of the two proposed lives, which value subtract from the per petuity, and take half the remainder; then say, as the expectation of duration of the younger of the too lives is to that of the older, so is the said half remainder to a fourth proportional ; which will be the number of years purchase required when the life of B in expectation i the older of the two ; but if B be the younger, then add the value so found'to that of the joint lives A and B, and let the sum be subtracted from the perpetuity, which gives the answer in this case.
Example 1. Suppose the age of A to be 20, and that of B 30 years ; and the annual value of the estate 50/. Then the value of two equal joint lives ,aged 30 being 10,255, and the perpetuity 20, the difference will be 9,745, the half of which is 4,872. Therefore, as 33,43, the expecta tion of A, is to 28,27 the expectation, of B, so is 4,872 to 4,119 years purchase, which being multiplied by 50, the given annual •value, we have 205/. 19s. for the required value of 13's expectation.
Example 2. Let the age of A be 30, that of E 20 years ; and the rest as in the preceding example. Then, the value of the joint lives is 10,707, which being add ed to 4,119 found above, the sum is 14,826; and this subtracted from 20,the perpetuity, and multiplied by 50, gives 258/. 14s. for
the value in this case.
Probiem 6 To find the value of a given estate at the death of B, provided that should happen after the death of A.
Find the value of an annuity upon the longest of two equal lives, whose common age is that of the older of the two lives, A and B, which value subtract from the perpetuity and take. half the remainder. Then, as the expectation of duration of the younger of the lives is to that of the older, so is the said half remainder to the number of years purchase required, when is the older of the two. But if B be the younger, then to the number of years pur chase thus found add the value of an an nuity on the longest of the lives, A and 13, and subtract the sum from the perpetuity for the answer in this case.
Example 1. Let the age of A be 30, and that of B 60 years ; the given estate 120/. per annum. Then the value of an an nuity on the longest of two lives aged 60 each will be found to be 10,896, which taken from 20, the perpetuity, leaves 9,104 for the remainder. Therefore it will be as 28,27, the expectation of A, is to 13,21, the expectation of B, so is 4,552 the half remainder, to 2,127, the number of years riu•chase required, which, being multi plied by 120, gives 255/. 4s. 9d. for the present value.
Example 2. Let the age of A be 60 and that of B 30 years ; then, to the number of years purchase found in the preceding ex ample, add 14,172, the value of an annuity on the longest of the two lives, the sum is 16,299, and this subtracted from 20, the perpetuity, and multiplied by 120, gives 444/. 28. 4d. for the value in this case.
The solutions of the two last problems comprehend all the cases of survivorship between two lives for their whole dura tion; but an expectation dependent on survivorship is sometimes restricted to a term of years less than the whole dura tion of the lives. Those who have occa sion for the rules for resolving questions of this description, or of the various cases which may arise when three or more lives are concerned, are referred to Mr. T. Simpson's Doctrine of Annuities, Dr. Price's Treatise on Reversionary Pay ments, or Mr W. Morgan's Treatise on Annuities and Assurances.
Reversionary interests being a species of property of which purchasers are not always readily found, those who have oc casion to dispose of an interest of this kind generally sell it by public auction, M which mode it very seldom happens that more than two thirds of the true cal culated value is obtained.