CHAPTER V.

COMPOUND INTEREST.

(Art. 153.) Logarithms are of great utility in resolving some questions in relation to compound interest and annuities; but for a full understanding of the subject, the pupil must pass through the following investigation:

Let p represent any principal, and r the interest of a unit of this principal for one year. Then 1tr would be the amount of $1, or £1. Put A=1+r.

Now as two dollars will amount to twice as much as one dollar, three dollars to three times as much as one dollar, &c. Therefore, 1: A :: A : A2 = the amount in 2 years, And 1: A :: A2: A3 = the amount in 3 years, &c. &c.

Therefore, A is the amount of one dollar or one unit of the principal in n years, and p times this sum will be the amount for p dollars. Let a represent this amount; then we have this general equation

pA=a.

In questions where n, the number of years, is an unknown term, or very large, the aid of logarithms is very essential to a quick and easy solution.

For example, what time is required for any sum of money to double itself, at three per cent. compound interest? Here a=2p, and A=(1.03), and the general equation becomes p(1.03)=2p

Or

n log. (1.03)=log.2, or years nearly.

(1.03)=2. Taking the logarithms log.2 .30103 -23.45

n=

(log. 1.03) .012837

2. A bottle of wine that originally cost 20 cents was put away for two hundred years; what would it be worth at the end of that time, allowing 5 per cent. compound interest?

This question makes the general equation stand thus:

(20 cts. being of a dollar)

(1.05)200= a

Therefore,

(1.05)200=5a

Taking the logarithms 200 log. (1.05)= log. 5+log. a Hence, log. a=200 log. (1.05)-log. 5. Ans. $3458.10. 3. A capital of $5000 stands at 4 per cent. compound interest; what will it amount to in 40 years?

Ans. $24005.10. 4. In what time will $5 amount to $9, at 5 per cent.compound interest? Ans. 12.04 years. 5. A capital of $1000 in 6 years, at compound interest, amounted to $1800; what was the rate per cent?

Ans. log. (1+r)= log. 1.8

6

Ans. 10% nearly.

6. A certain sum of money at compound interest, at 4 per cent. for four years, amounted to $350.95; what was the sum? Ans. $300.

7. How long must $3600 remain, at 5 per cent. compound interest, to amount to as much as $5000, at 4 per cent. for 12 years. Ans. 16 years, nearly.

ANNUITIES.

(Art. 154.) An annuity is a sum of money payable periodically, for some specified time, or during the life of the receiver. If the payments are not made, the annuity is said to be in arrear, and the receiver is entitled to interest on the several payments in arrear.

The worth of an annuity in arrear, is the sum of the several payments, together with compound interest on every payment after it became due.

On this definition we proceed to investigate a formula to be applied to calculations respecting annuities.

Let p represent the annual principal or annuity to be

paid, and 1+r=A, the amount of unity of principal for one year at the given rate r.

Let n represent the number of years, and put A' to represent the entire amount of the annuity in arrear.

It is evident that on the last payment due no interest could accrue, and therefore the sum will be p. The preceding payment will have one year's interest; it will therefore be pa; the payment preceding that will have two years' compound interest, and of course will be represented by PA2 (Art. 149.) Hence, the whole amount A' will be A'=p+pA+pA2+pA3 &c. to pa

This is a geometrical series, and its sum (Art. 120.) is

A_PA-P_p[(1+r)-1]

This general equation contains four quantities, A', p, r, and n; any three of them being given in any question the other can be found.

/

EXAMPLES.

1. An annuity of $50 has remained unpaid for 6 years, at compound interest on the sums due, at 6 per cent. what sum is now due?

_50[ 50[(1.06)-1]

By the general equation, A'=

Taking the logarithm, we have

.06

log. A'= log. 50+ log. [(1.06)-1]-log. .06

The value of (1.06), as found by log. is 1.41852, from which subtract 1, as indicated, and take the log. of the decimal number .41852. Then

log. A'=1.69897+(-1.62172) - (-2.778151)=2.54218, From which we find A'=$348.56, ans.

2. In what time will an annuity of $20 amount to $1000 at 4 per cent. compound interest?

The equation applied, we have

1000_20[(1.04)—1]

.04

Dividing by 20, and multiplying by 0.4, we have

2=(1.04)-1 or (1.04)=3.

3. What will an annuity of $50 amount to, if suffered to remain unpaid for twenty years, at 3 per cent. compound interest? Ans. $1413.98.

4. What is the present value of an annuity or rental of $50 a year, to continue 20 years, discounting at the rate of 3 per cent. compound interest?

N. B. By question 3d, we find that if the annuity be not paid until the end of 20 years, the amount then due would be $1413.98. If paid now, such a sum must be paid as, put out at compound interest for the given rate and time, will amount to $1413.98.

Now if we had the amount of $1 at compound interest for 20 years, at 3 per cent., that sum would be to $1 as $1413.98 is to the required sum, $713.50.

(Art. (155.) To be more general, let us represent the present worth of an annuity by P. By (Art. 153.) the amount of one dollar for any given rate and time is A; A being 1+r and n the number of years. By (Art. 150.) the value of any annuity p remaining unpaid for any given time, n years, at any rate of compound interest r, is PA-P

or A'.

T

Now by the preceding explanation we may have this pro

portion:

A: 1 :: A': P, or P=

A'

(1)

A

Hence, to find the present worth of an annuity, we have this RULE. Divide the amount of the annuity supposed unpaid for the given number of years, by the amount of one dollar for the same number of years.

If in equation (1) we put the value of A', we shall have PA-PA-P

Divide both members by An and we have

This last equation will apply to the following problems: 5. The annual rent of a freehold estate is p pounds or dollars, to continue forever. What is the present value of the estate, money being worth 5 per cent. compound interest? Here, as n is infinite, the term, 2 becomes 0, and equa

An

tion (2) becomes P=P=P20p; that is, the present

T

.05

value of the estate is worth 20 years' rent.

6. The rent of an estate is $3000 a year; what sum could purchase such an estate, money being worth 3 per cent. compound interest? Ans. $100000.

7. What is the present value of an annuity of $350, Ans. $2356.46.

assigned for 8 years, at 4 per cent?

8. A debt due at this time, amounting to $1200, is to be discharged in seven annual and equal payments; what is the amount of these payments, if interest be computed at 4 per cent.? Ans. $200, nearly.

The rent of a farm is $250 per year, with a perpetual lease. How much ready money will purchase said farm, money being worth 7 per cent. per annum?

Ans. $35714.