per

petuity, and reserve the remainder. Then say, as the petuity is to the present value of the annuity certain, so is the reserved remainder, to the number of years purchase required.

Ex. 1. What is the value of an annuity certain for 14 years, to commence at the death of a person aged 35, allowing 5 per cent.?

The value of a life of 35 (Table, p. 204) 12.502; this subtracted from 20, the perpetuity, leaves 7.498 reserved remainder. Then, as 20 9.898† 7.498: 3.7107 number of years purchase.

Ex. 2. A and his heirs are entitled to an annuity certain for 25 years, to commence at the death of a cousin aged 45 years; what can A sell his interest in this annuity for?

NOTES.

* PERPETUITY, is the number of years purchase to be given for an annuity which is to continue for ever; and it is found by dividing 100l. by the rate of interest; thus, allowing 5 per cent., the

These are the number of years purchase to be given for a perpetual annuity, on the supposition that it is receivable yearly: but, as annuities are more commonly received half-yearly, and the interest of money likewise paid half-yearly; in this case the perpetuity will be somewhat greater or less than the above, as the periods at which the annuity is payable are more or less frequent than those at which the rate of interest is here supposed payable.

+ The value of an annuity certain for 14 years. Table, p. 196.

CASE III. To find the value of an annuity for a term certain; and also for what may happen to remain of a given life after the expiration of this term.

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 certain is proposed. Multiply this value by 11. payable at the end of the given term, and also by the probability that the given life will continue so long. Add the product to the value of the annuity certain for the given term, and the sum will be the answer.

Ex. 1. What is the value of an annuity of 607. for 14 years, and also for the remainder of a life now aged 35, after the expiration of that term? 35+ 14 = 49.

The value of a life aged 49 (Table 1. p. 204) The value of 17. payable at the end of 14 years (Table, p. 209.)

The probability that the life will exist so long, (Table, p. 200 and 201)

Therefore, 10,443 X .505068 X

2936

4010

10.443

.505068

2936

4010

3.861; this added to 9.898,

the value of an annuity certain for 14 years, (see Table, p. 195) = 13.759, the number of years purchase; and 13.759 X 60 = 8251. 10s. 9 d.

Ex. 2. What is the value of an annuity of 75l. for 10 years, and also the remainder of a life now aged 24, after the expiration of that term?

CASE IV. To find what annuity can be purchased for a given sum, during the joint lives of two persons of given ages, and also during the life of the survivor, on condition that the annuity shall be reduced one-half at the extinction of the joint lives.

RULE. Divide twice the given sum by the sum of the value of the two single lives, and the quotient will give the annuity to be paid during the joint lives, one-half of which is therefore the annuity to be paid during the remainder of the surviving life.

Ex. 1. A man and his wife, aged 35 and 27, are desirous of sinking 20007., in order to receive an annuity during their joint lives, and also another annuity of half the value during the remainder of the serviving life: what annuities ought to be granted them?

25.879

annuity during their

joint lives and 77l. 5s. 7d. annuity during the life of the survivor.

Ex. 2. A single man, aged 60, possessed of 1500l. is desirous of purchasing with it an annuity for himself and his sister, aged 40, during their joint lives, with one of half the value, during the remainder of the life of the survivor, at the death of either; what will be the value of the annuities?

Ex: 3. A man, possessed of 1000l., which he will sink in the same way, and for the same purposes, during the joint lives of himself and father; the age of the one is 55, of the other 80: what annuities can be given for it?

V. To find the value of the expectation of a perpetual annuity, provided one person of a given age survives ano- › ther of a given age.

(1). IF THE EXPECTANT BE THE ELDER.

RULE. Find the value of an annuity on two equal joint lives, whose common age is equal to the age of the oldest of the two proposed lives; subtract this value from the perpetuity, and take half the remainder: then say,

As the expectation of the duration of life of the younger, Is to that of the elder;

So is the half remainder to a fourth proportional: which will be the number of years purchase, if the expectant is the older.

(2). IF THE EXPECTANT BE THE YOUNGER.

Add the value found, as above, to that of the joint lives, and let the sum be subtracted from the perpetuity, and the remainder is the answer.

Ex. 1. What is the value of B's expectation, (aged 30), of an estate of 507. per annum, provided he survive A, aged 20?

Value of two joint lives, aged 30, (Table H. p. 206) 10.255, the difference between which and 20, (the perpetuity), is 9.745, the half of which is 4.872: therefore,

Ex. 2. What is the value as above, when B is 20, and A 30?

Then, to 4.119, just found, add

10.707, value of the joint lives (Table II. p. 206.) 14.826; this subtracted from 20, the perpetuity, and

the remainder, 5.174 × 50258l. 14s. is the true answer.

EXAMPLES FOR PRACTICE.

Ex. 1. What is the additional value of an annuity certain for the term of 28 years, if extended to the longest of 3 lives that may survive that term, each now 30 years of age?

Ex. 2. What is the additional value of an annuity certain for the term of 40 years, for the longest of 3 lives that may survive that term, each now 32 years of age?

Ex. 3. What is the difference in the value of an annuity of 201. certain for 30 years, and an annuity of the same amount on the longest of two lives, aged 25 and 40?

Ex. 4. What is the value of an estate of 150l. per annum, held on the longest of two lives, aged 40 and 50, subject to the payment of an annuity of 147. tca life of 62, and another annuity of 187. to a life of 65?

Ex. 5. What is the present worth of 2000l. to be received at the decease of a person aged 65 ?

Ex. 6. What is the present value of 361. a-year, being the third part of a farm in Essex, after the death of a person aged 54 years?

Ex. 7. What is the present value of a reversionary annuity of 2591. 3s. 8d. during the life of a person aged 24, in case he survives his brother, aged 34?

Ex. 8. What should be the consideration to be paid at the death of a person aged 80, for 1000l. now advanced to a person aged 25, in case the latter survives the former?

Ex. 9. What is the reversion worth of 2000l. 3 per cent. consols, if a life of 37 survives one of 53 ?

Ex. 10. What is the present worth of the reversion of 30751. 3 per cent. consols, if a life of 37 survives two lives, one aged 58 years, and the other 30?

Ex. 11. What is the value of the reversion of 911. per annum for ever, after the death of a person aged 53 ?

Ex. 12. A person aged 52, is entitled to 800l. at the death of another aged 76, provided the former survives the latter what is its present worth?

:

Ex. 13. What is the present value of an annuity on the longest of two lives, now aged 25 and 30, the annuity not to commence till 14 years hence?

Ex. 14. What is the value of an annuity of 1351., secured by, and payable out of, the dividends on 3 per cent. Reduced Bank Annuities, to which the purchaser will become entitled on the decease of a lady in her 72d year, and to enjoy the same during the life of a healthy gentleman, now in his 40th year, if he is the surviver?

LEASES.

A LEASE is a conveyance of any lands and tenements, made, in consideration of rent, or of a present sum of money, for life, or for a term of years.

The purchaser of a Lease may be considered as the purchaser of an annuity equal to the rack-rent of the estate; its value must therefore be calculated on the same principles as that of an annuity.

The sum paid down for the grant of a lease is so much, as being put out to interest will enable the landlord to repay himself the rack-rent of the estate, or the yearly value of his interest therein.

The value of the lease depends on the length of the term, and the rate of interest which the landlord can make of his money.

The value of leases at 5 per cent. compound interest my be found from Table, p. 196.

1

Thus, the value of a lease for 14 years, of a farm worth 150l. per anr.. is, by that table 9.898641 X 150 1484l. 15s. 11d.

Ex. 1. What ought to be given for a lease of 26 years of an estate of 187. per annum clear annual rent, * in order that a purchaser may make 5 per cent. of his money?

Ex. 2. A friend has just purchased the lease of a house for 54 years, for which he gave 5507., and he is to pay a ground-rent of 17. per annum: how much ought the house to let for, allowing 5 per cent. interest only?

Leases are generally calculated at a higher rate of interest; we shall therefore insert the following

TABLE, †

Shewing the Number of Years Purchase that ought to be given for a Lease, for any Number of Years not exceeding 100, at 6, 7, and 8 per cent. interest.

NOTES.

* That is, the next surplus rent, after deducting the reserved rent, if any, and all taxes and other annual charges.

Leases.

See Gregory's Dictionary of Arts and Sciences: also Baily on

L 2