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 any sum of annuity for a given term and rate of interest, we have only to multiply the corresponding tabular value by the sum or the annuity proposed; the product will be the amount or the value sought, according as the case may be.

16. Example 1. To what sum will L. 100 amount, when improved at compound interest during 20 years? the rate of interest being 4 ''per cent. per annum''.

By Table III., it appears, that L. 1 so improved, would, at the expiration of the term, amount to L. 2·191123, therefore L. 100 would amount to 100 times as much, that is, to L. 219·1123, or L. 219, 2s. 3d.

17. Ex. 2. What is the present value of L. 400, which is not to be received until the expiration of 14 years, when the rate of interest is 5 per cent.?

The present value of L. 1 to be received then, will be found by Table I. to be L. 0·505068: L. 400 to be received at the same time, will therefore be worth, in present money, 400 times as much, or L. 202·0272, that is, L. 202, 0s. 6½d.

18. Ex. 3. Required the present value of an annuity of L. 50 for 21 years, when the rate of interest is 5 per cent.

Table II. shows the value of an annuity of L. 1 for the same term to be L. 12·8212; the required value must therefore be 50 times as much, or L. 641·06, that is, L. 641, 1s, 2½d.

19. Ex. 4. What will an annuity of L. 10, 10s. or L. 10·5, for thirty years, amount to, when each payment is put out as it becomes due, and improved at compound interest until the end of the term? The rate of interest being 4 per cent.

The amount of an annuity of L. 1 so improved, would be L. 56·084938, as appears by Table IV., the amount required will therefore be 10·5 times this, or L. 588·89185, that is, L. 588, 17s. 10d.

20. When the interval between the time of the purchase of an annuity and the first payment thereof, exceeds that which is interposed between each two immediately successive payments; such annuity is said to be deferred for a time equal to that excess, and to be entered upon at the expiration of that time.

21. If two persons, A and B, purchase an annuity between them, which A is to enter upon immediately, and to enjoy during a certain part of the term, and B, or his heirs, or assigns, for the remainder of it; the present value of B’s interest will evidently be, the excess of the value of the annuity for the whole of the term from this time, above the value of the interest of A.

So that when the entrance on an annuity ts deferred for a certain term, its present value will be the excess of the value of the annuity for the term of delay and continuance together, above the value of an equal annuity for the term of delay only.

22. Example 1. Required the value of a perpetual annuity of L. 120, which is not to be entered upon until the expiration of 14 years from this time, reckoning interest at 3 per cent.

The perpetuity, with immediate possession, would be worth 33⅓ years’ purchase (8); and an annuity for the term of delay is worth 11·2961 (Table II.)

23. Ex. 2. Allowing interest at 5 per cent. what sum should be paid down now for the renewal of 14 years lapsed, in a lease for 21 years of an estate producing L. 300 per annum, clear of all deductions?

This is the price of an annuity for 14 years, to be entered upon 7 years hence; the term of delay, therefore, is 7 years, and that of the delay and continuance together 21 years.

By Table Il. it appears, that the present value of an annuity

24. Hitherto we have proceeded upon the supposition of the annuity being payable, and the interest convertible into principal, which shall reproduce interest, only once a-year.

But annuities are generally payable half-yearly, and sometimes quarterly; and the same circumstances that render it desirable for an annuitant to receive his annual sum in equal half-yearly or quarterly portions, also give occasion to the interest of money being paid in the same manner.

But whatever has been advanced above, concerning the present value or the amount of an annuity, when both that and the interest of money were only payable once a-year, will evidently be true when applied to half the annuity, and half the interest paid twice as often, on the supposition of half-yearly payments; or to a quarter of the annuity, and a quarter of the interest, paid four times as often, when the payments are made quarterly.

25. Half-yearly payments are, however, by far the most common, and these four tables will also enable us to answer the most useful questions concerning them.

For we have only to extract the present value, or the amount, from the table, against twice the number of years in the term, at half the annual rate of interest, and, in the case of an annuity, to multiply the number so extracted, by half the annuity proposed.

26. Ex. 1. To what sum will L. 100 amount in 20 years, when the interest at the rate of 4 ''per cent. per annum'', is convertible into principal half-yearly?

This being the amount in 40 half years at 2 per cent. interest for every half year, will be the same as the amount in 40 years at 2 ''per cent. per annum'', which, by Table III. will be found to be 220·804, or L. 220, 16s. 1d.; and is only L. 1, 13s. 10d. more than it would amount to if the interest were not convertible more than once a-year (16).