1. What is the interest of 3787. 14s. Od., from the 9th of November to the 16th of March, at 5 per cent.?

Suppose the rate had been at 41 per cent.

Subsect. IV.-COMPOUND INTEREST.

If a man lend 1007. for one year, and exact the payment of principal and interest when due, he will receive 1057.; if he lends out this money to the first holder, or any other, for another year precisely, he will receive back 1107. 5s., and in three years it would be 1157. 15s. 3d., &c. Hence arises compound interest, which, though it is prohibited by the laws, every banking company exact in effect, as they take particular care that no cash account shall remain unsettled beyond a year. This section also admits of four varieties, which we shall not illustrate with many examples, as all questions which occur in this rule are briefly answered by the Tables to be found at the end of the work.

Multiply the amount of 17. for a year so often into itself as are years proposed, abating one, and the last product, multiplied by the principal, gives the amount; from which deduct the principal, for the interest.

If there are also days beyond complete years, add to the amount, or principal and compound interest formerly found, the simple interest of that amount for them.

1. What is the interest and amount of 7001. for 3 years, at 5 per cent. per annum?

ANNUITIES are periodical payments to persons for a term of years, for life, or for ever. Thus, the dividends on stock in the public funds are annuities for unlimited terms, except otherwise expressed, as the Government " Long Annuities," which terminate in 1860. The rents of freehold estates are also annuities. The reversions of leases, after the expiration of under-leases, are deferred annuities. So, also, are the reversions to freeholds, after the expiration of terminable leases. Hence we have immediate annuities, perpetual annuities, deferred annuities, and deferred perpetuities.

Annuities certain are of two kinds; first, those which are forborne, or in arrears; secondly, those which hereafter become payable at some future time. Respecting those in arrear or said to be forborne, we regard only the amount.

When they hereafter become payable, we contemplate their present worth. The former exceeds the sum of the several payments by the interest which has accumulated: the latter falls short of that sum by the discount to be deducted. Both, however, depend, not only on the intervals at which the payments successively become due, but also on the periods at which the interest is convertible into principal. Calculations of this sort are easily managed by means of the Tables we give at the end of the work, the construction of which we shall here illustrate by examples.

An annuity is in arrear 6 years; i.e. the first payment became due 5 years ago; the second 4 years ago; the third 3; the fourth 2; the fifth 1; and the sixth is just due.

The interest is 4 per cent.; and the several payments, as shown by Table IV., are, upon 17., as follow:

Payment just due, owes no interest, and is £1.00000
The payment due 1 year back

Their sum

The payment due 2 years ago

The sum of the three payments is -
The payment due 3 years ago

1.04000

Therefore this total is called the years' purchase, and is 67. 12s. 8d. Hence the amount of 17. per annum for 2 years is 2.047.; for 3 years it is 3.12167.; for 4 years it is 4.24647.; for 5 years it is 5-41637.; and for 6 years it is 6·6327., or 67. 12s. 8d.

When the annuity is made payable at the beginning of the year, each payment, and consequently their aggregate, is increased by

one year's interest; so that the amount of

the last payment would be

Of that preceding it

Of the one preceding this

Of the fourth preceding

Of the fifth preceding

Making a total of

- £1.04000

1.08160

1.12486

1.16985

1.21665

- £5.63296

In which case we see that the annuity payable at the beginning of the year is less by unity than an annuity for 6 years payable at the end of the year.

Ex. Let the annuity be 20l., forborne for 20 years at 4 per cent. compound interest. Then by the Table the amount of 11. for 20 years being, at 4 per cent., 29.778087, we multiply this by 201., the amount of the annuity, and the product, 5957. 11s. 3d., is the amount of the annuity.

Present worth of Annuities. Here we consider, in addition to the above, the period when the annuity is to commence*, and the term for which it is to continue; as explained in the beginning of this article.

Thus, for the present value of an annuity of one pound due one year hence at 4 per cent., we have

The present value of 17. or 1-038462 dis

count=

-£0.961538

To which adding that of 17. due 2 years hence 0.924556 Their sum is the present value of 17. per

annum for 2 years

To which adding the value of 17. due 3 years hence

years

And we have the present value of 17. per
annum for 3
Adding to this sum the value of 17. due 4
years hence

The total shows the value of 17. per annum

for 4 years

* An annuity payable yearly is said to commence, or be entered upon, one year before the first payment becomes due; and an annuity payable by half-yearly instalments is said to commence half a year before the first instalment becomes due; and so on.

Hence the construction of Table V. of present values of 17. annuity due for any number of years from 1 year up to 50 years.

If the annuity be payable at the beginning of the year, the present value of the first payment would evidently be

That of the second, discounted for 1 year, at 4 per cent., would be

That of the third, discounted for 2 years, at 4 per cent., would be

£1.00000

0.961538

0.924556

0.888996

That of the fourth, discounted for 3 years, at 4 per cent., would be

So that the total would be

Showing the present worth of 17. annuity payable at the beginning of the year for 4 years, which is simply unity added to the present value of a like annuity payable at the end of the year for 3 years. The same principle applies to

any other period, and to any other rate of interest: i. e. adding unity to the tabular number opposite 1 year less than the given term.

Ex. 1. Thus, if the improved rent of a farm held under a lease for 21 years be 100l., reckoning interest at 5 per cent. (Table V., opposite 21 years, 5 per cent. column), the present value of the lease is 12.821153 × 100=12827. 3s 1d.

Ex. 2. And the present value of a perpetual annuity of 301., reckoning interest at 4 per cent., is 25.00000 × 30= 7507. Here 25·0000=100÷÷4: see page 42.

Ex. 3. In like manner an estate in fee-simple, yielding a net annual rent of 500l., is, at 4 per cent., worth 25 × 500, or 12,500l.; for 100÷4=25: see page 42.

Ex. 4. Two persons, A. and B., divide an annuity of 1007. for the next 30 years between them, so that A. and his heirs enjoy it for the next 10 years, and B. and his heirs for the remaining 20 years. Required the present value of B.'s share or deferred annuity, reckoning interest at 3 per cent.?