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REVERSIONARY OR DEFERRED PERPETUITIES.

XXVI. To find the present value of a perpetuity or freehold at a given rate per cent., which is not to be entered upon until the expiration of a given period.

Multiply the present value (XXV. 1.) of the perpetuity by the present value of £1, as found by Table III., for the deferred term, at the given rate per cent.; the result will be the value required.

A freehold estate, producing £60 per annum, is mortgaged for the period of fourteen years; what sum ought to be paid down now as its present value, supposing the interest of money at the rate of 3 per cent. per annum ?

£60.03 (XXV. 1.)

= £2000, value of the perpet.; Pres. val. of £1 for 14 years, at 3 per cent., is, by Table III.= £.661118; Therefore £2000 × £.661118=£1322.236, or £1322 4 9, the sum to be paid down.

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What is the present worth of a perpetual annuity of £200, which is not to be entered upon for the next ten years, allowing interest to the purchaser at the rate of 4 per cent. per annum for the improvement of his money? £200.04 (XXV. 1.) = £5000, value of the perpet.; Pres. val. of £1 for 10 years, at 4 per cent., is, by Table III ̧=£.675564 ; Therefore £5000 × £.675564 =£3377.82, or £3377 16 5, the required value. What is the present value of an estate in fee simple of £75 per annum, the same being let to a tenant, and not available for seven years hence, reckoning at the rate of 5 per cent. per annum for the interest of money? £75÷.05 (XXV. 1.)

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= £1500, val. of the annuity;

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Pres. val. of £1 for 7 years, at 5 per cent., is, by Table III. £.710681;
Therefore £1500 × £.710681 £1066.021, or £1066 0 5, the required value.

A farmer having purchased a lease for twenty years of a freehold estate, estimated at £250 per ann., wishes to obtain perpetual occupancy of the property at the expiration of his term: what sum ought he to pay down to his landlord to entitle him to become its sole owner, assuming he is allowed interest at the rate of 4 per cent. per ann. for his money ? Answer, £2852 8 5.

A. possessed a freehold estate of £500 per ann., £300 of which he sold to B. on lease for eighteen years. Being a fashionable and profligate man, and grossly addicted to the ruinous vice of gambling, he rapidly lost every thing he possessed. His creditors, in his extremity, seized upon his estate and disposed of it entirely to his tenant. What purchase-money did B. give for the ownership, reckoning interest at the rate of 5 per cent. per annum? Answer, £6493 2 6.

This question embraces the present value of a perpetuity of £200 per annum, added to the present value of a perpetuity of £300 per annum after eighteen years.

A lady, entitled, at the expiration of seven years, in her own right to a reversionary freehold of £750 per ann., married a merchant who required additional capital to extend his commercial operations; to raise which, the wife was willing to sacrifice £150 per ann. of her deferred perpetuity. Her aunt, however, who was very rich, considering that an annuity, of whatever kind or amount, should be held sacred, and never, if possible, infringed upon or disposed of, consented, rather than allow her niece to diminish her inheritance, to advance from her own funds so much money on behalf of the husband as was equivalent to £150 of the annuity. What further sum was the merchant enabled by this act of kindness to embark in commerce, reckoning interest at the rate of 4 per cent. per ann.?-Ans., £2849 13 10.

A rapidly improving freehold estate, situate in the high street of Eton, Bucks, consisting of a capital brick-built family residence, with double-fronted shop, and extensive warehouses in the rear, was sold by auction at the Castle Hotel, Windsor. The property was estimated at £80 per annum, and subject to a lease which had twenty-one years to run. What sum did the purchaser give for it, considering that the investment returned him at the rate of 5 per cent. per annum interest for the use of his money? Answer, £574 6 2.

REVERSIONARY OR DEFERRED PERPETUITIES-continued.

XXVII. To find what perpetuity, deferred for a given term, may be purchased for a given sum at a given rate per cent.

Multiply the given sum by the given rate per £, and multiply this product by the amount, as found by Table II., of £1 for the deferred term, at the given rate per cent.; the result will be the perpetuity required.

What perpetuity, to be entered upon fourteen years hence, may be purchased for the sum of £1322 4 9; the rate of interest, to regulate the price, being fixed at 3 per cent. per annum ?

Int. of £1, at 3 per cent., =£.03;

The given sum £1322 4 9=£1322.238;
The amount of £1, by Table II., for 14 years, at 3 per cent. per annum, = £1.512589;
Therefore (£1322.238 × .03) × £1.512589 £60 per annum, the required perpetuity.

Required what perpetual annuity, in reversion after the next ten years, is equivalent to the sum of £3377 16 5 now paid down; the purchaser agreeing to invest his money at the rate of 4 per cent. per annum ?-Ans., £200 perpetual annuity.

The sum of £1066 0 5 was deposited in a public office for seven years, on condition of its being improved at 5 per cent. per annum interest, and at the end of that term finally invested in the purchase of an annuity for ever. What was the amount of the perpetuity? Answer, £75 per annum.

A perpetual annuity, deferred for twenty years, cost £2852 8 5, by which the purchaser improved his money at the rate of 4 per cent. per annum; what was the amount of the deferred annuity? or, supposing the purchaser disposed to exchange. it, what annuity, to be entered upon at once, and to continue for ever, ought he to receive as an equivalent ?—Answer, £250, the perpet. ann., deferred for 20 years: Now, £25 being the value (XXIV.) of £1 perpetuity; it follows, that£2852 8 5÷£25; or, which is the same thing, (XXV. 2.) £2852 8 6×.04

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£114 1 11, the equivalent immediate perpetuity.

The celebrated George Robins advertised for sale a magnificent freehold estate. The property, according to the glowing statement of the auctioneer, was a very place of paradise; abounding in hill and dale; in flood and field; in every thing, in short, which poetry could depict or imagination create. The only drawback was, that it would remain in the possession of another for eighteen years to come. An opulent merchant of the city, casually entering the room, and charmed by the description of its beauties, resolved to become its proprietor, and at once offered £4155 4 2 as the purchase money. The price was accepted; he became its possessor; and realised at the rate of 5 per cent. per annum interest by the investWhat was the amount of his reversionary perpetuity?

ment.

Answer, £500 per annum.

A. being entitled, at the expiration of seven years, to the possession of a freehold, valued at a present sum of £2849 13 10, desires to exchange it for a perpetual annuity to commence at once. Now B., being a rich and calculating man, has no objection to give an immediate perpetuity of equivalent value to A. for his reversionary interest, provided he can by the transaction be certain of improving his money at the rate of 4 per cent. per annum. What is the estimated income of A.'s reversionary freehold, and what equivalent immediate perpetuity ought B. to give in exchange for it?

Answer, £150 0 0 per aun., A.'s reversionary freehold ;

And, £113 19 9 per ann., the immediate equiv. perpetuity to be given by B.

The true value of money, contingent only upon its accumulation at a given rate of interest for a given period, having been fully illustrated in the preceding chapter, the interesting calculation of its value, dependant upon the duration of human existence, whether for one or more lives, remains now to be considered. There is, however, no difference in the application of principles in either case. The appreciation of annuities certain is, indeed, more simple than that of life annuities, because the term of the former is ascertainable, on all occasions, to the minutest fractional exactness; whereas no earthly knowledge or subtlety can, except by probability and approximation, determine in any one instance the term or duration of the latter. With a view of arriving at a correct estimate of individual existence, observations, extending over a long series of years, and embracing a large mass of persons of all ages and of both sexes, have been made at different places in various countries by many eminent philosophers and mathematicians. London, Northampton, Carlisle, France, Sweden, and many other towns and kingdoms, have been selected as places suitable for conducting their investigations and establishing their conclusions. We shall, however, confine ourselves exclusively, in this treatise, to the table of mortality constructed by Dr. Price, on the mortality of the population of Northampton. The same theorems, rules, and calculations, which apply to one, apply equally to every table and system of mortality.

The pupil will readily perceive, as he proceeds, that the operations connected with this interesting subject are as simple as they are valuable; and well does this branch of education deserve the attention of every reflecting person. The parent-the guardian-and testator; the heir— the executor-and lawyer; the man of present property-and the man of property in anticipation; whoever has any possessions to bequeath--whoever has any legacy to expect, ought each and all, in every case depending upon human mortality, whether the claim be immediate, deferred, or in absolute reversion at death, to acquire the all-important knowledge of the value of life annuities. This is the only mode to enable them to appreciate correctly, and to secure infallibly, their own and their representatives' real interests.

ANNUITIES AND ASSURANCES,

IMMEDIATE,

DEFERRED,

AND

TEMPORARY,

IN

SINGLE AND ANNUAL PREMIUMS,

ON

SINGLE LIVES.

Showing the number of persons living and dying at every age, according to the observations made at Northampton.

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