EXAMPLES. XVIII. a.

When required the following logarithms may be used.

log 2=3010300, log 3=4771213,

log 7='8450980, log11=10413927.

1. Find the amount of £100 in 50 years, at 5 per cent. compound interest; given log 114.674-2.0594650.

2. At simple interest the interest on a certain sum of money is £90, and the discount on the same sum for the same time and at the same rate is £80; find the sum.

3. In how many years will a sum of money double itself at 5 per cent. compound interest?

4. Find, correct to a farthing, the present value of £10000 due 8 years hence at 5 per cent. compound interest; given

log 67683.94-4.8304856.

5. In how many years will £1000 become £2500 at 10 per cent. compound interest?

6. Shew that at simple interest the discount is half the harmonic mean between the sum due and the interest on it.

7. Shew that money will increase more than a hundredfold in a century at 5 per cent. compound interest.

8. What sum of money at 6 per cent. compound interest will amount to £1000 in 12 years? Given

log 106=2.0253059, log 496974.6963292.

9. A man borrows £600 from a money-lender, and the bill is renewed every half-year at an increase of 18 per cent.: what time will elapse before it reaches £6000? Given log 118=2.071882.

10. What is the amount of a farthing in 200 years at 6 per cent. compound interest? Given log 106=2·0253059, log115·0270=2·0611800.

ANNUITIES.

236. An annuity is a fixed sum paid periodically under certain stated conditions; the payment may be made either once a year or at more frequent intervals. Unless it is otherwise stated we shall suppose the payments annual.

An annuity certain is an annuity payable for a fixed term of years independent of any contingency; a life annuity is an annuity which is payable during the lifetime of a person, or of the survivor of a number of persons.

A deferred annuity, or reversion, is an annuity which does not begin until after the lapse of a certain number of years; and when the annuity is deferred for n years, it is said to commence after n years, and the first payment is made at the end of n + 1 years.

If the annuity is to continue for ever it is called a perpetuity; if it does not commence at once it is called a deferred perpetuity. An annuity left unpaid for a certain number of years is said to be forborne for that number of years.

237. To find the amount of an annuity left unpaid for a given number of years, allowing simple interest.

Let A be the annuity, r the interest of £1 for one year, n the number of years, M the amount.

At the end of the first year A is due, and the amount of this sum in the remaining n 1 years is A+ (n-1) rA; at the end of the second year another A is due, and the amount of this sum in the remaining (n − 2) years is A + (n − 2) rA ; and so on. M is the sum of all these amounts;

Now

.. M={A + (n − 1) rA} + {A + (n − 2) rA} + ...... + (A + rA) + A, the series consisting of n terms;

... `M = nA + (1 + 2 + 3 + +n-1) rA

=nA +

n (n − 1) rA.

2

......

To find the amount of an annuity left unpaid for a

given number of years, allowing compound interest.

Let A be the annuity, R the amount of £1 for one year, n the number of years, M the amount.

At the end of the first year A is due, and the amount of this sum in the remaining n-1 years is AR-1; at the end of the second year another A is due, and the amount of this sum in the remaining n - 2 -2. and so on. is AR-2; years

239. In finding the present value of annuities it is always customary to reckon compound interest; the results obtained when simple interest is reckoned being contradictory and untrustworthy. On this point and for further information on the subject of annuities the reader may consult Jones on the Value · of Annuities and Reversionary Payments, and the article Annuities in the Encyclopædia Britannica.

240. To find the present value of an annuity to continue for a given number of years, allowing compound interest.

Let A be the annuity, R the amount of £1 in one year, n the number of years, V the required present value.

The present value of A due in 1 year is AR-1;

the present value of A due in 2 years is AR ̄*; the present value of A due in 3 years is AR ̄3; and so on. [Art. 235.]

Now payments;

is the sum of the present values of the different

NOTE. This result may also be obtained by dividing the value of M, given in Art. 238, by R". [Art. 232.]

COR. If we make n infinite we obtain for the present value of a perpetuity

V=

A

=

A

241. If mA is the present value of an annuity A, the annuity is said to be worth m years' purchase.

In the case of a perpetual annuity ma

A

hence

==

that is, the number of years' purchase of a perpetual annuity is obtained by dividing 100 by the rate per cent.

As instances of perpetual annuities we may mention the income arising from investments in irredeemable Stocks such as many Government Securities, Corporation Stocks, and Railway Debentures. A good test of the credit of a Government is furnished by the number of years' purchase of its Stocks; thus the 2 p. c. Consols at 961 are worth 35 years' purchase; Egyptian 4 p. c. Stock at 96 is worth 24 years' purchase; while Austrian 5 p. c. Stock at 80 is only worth 16 years' purchase.

242. To find the present value of a deferred annuity to commence at the end of p years and to continue for n years, allowing compound interest.

Let A be the annuity, R the amount of £1 in one year, V the present value.

The first payment is made at the end of (p+1) years. [Art. 236.] Hence the present values of the first, second, third ... payments are respectively

COR.

R-1 R-1'

The present value of a deferred perpetuity to commence after p years is given by the formula

243. A freehold estate is an estate which yields a perpetual annuity called the rent; and thus the value of the estate is equal to the present value of a perpetuity equal to the rent.

It follows from Art. 241 that if we know the number of years' purchase that a tenant pays in order to buy his farm, we obtain the rate per cent. at which interest is reckoned by dividing 100 by the number of years' purchase.

Example. The reversion after 6 years of a freehold estate is bought for £20000; what rent ought the purchaser to receive, reckoning compound interest at 5 per cent.? Given log 1·05-0211893, log 1.340096=1271358.

The rent is equal to the annual value of the perpetuity, deferred for 6 years, which may be purchased for £20000.

Let £A be the value of the annuity; then since R=1.05, we have

244. Suppose that a tenant by paying down a certain sum has obtained a lease of an estate for p+q years, and that when q years have elapsed he wishes to renew the lease for a term p+n years; the sum that he must pay is called the fine for renewing n years of the lease.

Let A be the annual value of the estate; then since the tenant has paid for p of the p+n years, the fine must be equal to the present value of a deferred annuity A, to commence after p years and to continue for n years; that is,

The interest is supposed compound unless the contrary is stated.

1. A person borrows £672 to be repaid in 5 years by annual instalments of £120; find the rate of interest, reckoning simple interest. 2. Find the amount of an annuity of £100 in 20 years, allowing compound interest at 4 per cent. Given

log 1 045 0191163, log 24-117=1.3823260.

=

3. A freehold estate is bought for £2750; at what rent should it be let so that the owner may receive 4 per cent. on the purchase money? 4. A freehold estate worth £120 a year is sold for £4000; find the rate of interest.