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1. What is the amount of an annuity of 401. to con, tinue 5 years, allowing 5 per cent. compound interest ? 1+1105+105i +105° +1'051= 5*52563125

5952563125

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2. If sol. yearly rent, or annuity, be forborn 7 years, what will it amount to, at 4 per cent. per annum, compound interest ?

Ans. 3951

To firid the present Value of Annuities at Compound Interest.

RULE.*

1. Divide the annuity by the ratio, or the amount of 11. for one year, and the quotient will be the present worth of 1 year's annuity.

2. Divide

The reason of this rule is evident from the nature of the question, and what was said upon the same subject in the purchase ing of annuities at simple interest.

Let p = present worth of the annuity, and the other letters as

gta

before, then as the amount =

X11, and as the present worth

Or

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2. Divide the annuity by the square of the ratio, and the.quotient will be the present worth of the annuity for two years.

3. Find, in like manner, the present worth of each by itself, and the sum of all these will be the value of the annuity sought.

year

EXAMPLES

1 X

or principal of this, according to the principles of compound in
terest, is the amount divided by me therefore
rI

r+I-?
=p, and px
piti-pt

ri And from these theorems all the cases, where the purchase of annuities is concerned, may be exhibited in logarithmic terms, as follows :

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I. Log.n+Log:1–- Log.r-1=Log. Þ.

II. Log.p+Log.r-1--Log. 1- I

=Log. n.

=

III.
Log. n-Log. n+p-pr

=t. IV. piti +iXrt+
Log.r

P Let t express the number of half years or quarters, n the half year's or quarter's payment, and r the sum of one pound and or À year's interest, then all the preceding rules are applicable to half-yearly and quarterly payments, the same as to whole years.

a year,

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The amount of an annuity may also be found for years and parts of

thus 1. Find the amount for the whole years as before.

2. Find the interest of that amount for the given parts of a year.

3. Add this interest to the former account, and it will give tlie whole amount required.

The

EXAMPLES.

1. What is the present worth of an annuity of 40l. te continue 5 years, discounting at 5 per cent. per annum, compound interest ?

-3

-4

ratio 1'05)40'00000(38.095=present worth for I year,

do. for 2 years. ratio = 1'1025)40'00000(36*281=

do. for 3 years. ratio =1'157525)40*00000(34°556=

do. for 4 years.' ratio = 1*215506)40*00000(326899=

do. for 5 years. ratio = 1'276278)40*00000(31'342=

173-173 = 1731. 35. 5 d. whole present worth of the annuity required.

5

2. What is the present worth of an annuity of 211. Ios. 9 d. to continue 7 years, at 6 per cent. per annum, compound interest ?

Ans. I2ol. 55.

3. What is gol. per annum, to continue 59 years, worth in present money, at the rate of 5 per cent. per annum ?

Ans. 1321*3021l.

To

The present worth of an annuity for years and parts of a year may

be found, thus : 1. Find the present worth for the whole years as before.

2. Find the present worth of this present worth, discounting for the given parts of a year, and it will be the whole present worth required.

To find the present Worth of a Freehold Estate, or an Annuja

ty to continue for ever, at Compound Interest.

RULE. *

As the rate per cent is to 100l. so is the yearly rent to the value required.

EXAMPLES.

* The reason of this rule is obvious : for since a year's interest of the price, which is given for it, is the annuity, there can neither more nor less be made of that price than of the annuity, whether it be employed at simple or compound interest.

The same thing may be shewn thus : the present worth of an annuity to continue forever is * + ++*, &c. ad infinitum, as has been shewn before ; but the sum of this series, by the rules

n

2

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The following theorems shew all the varieties of this rule.

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n

1.

p. II. r-
I Xp=n. III.

tiar, or -r--I. P

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The price of a freehold estate, or annuity to continue for ever,

I

a simple interest, would be expressed by 17+

1+2r

I

I

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+

&c. ad infinitum ; but the sum of this 'se1+3r 1+ 4r ries is infinite, or greater than any assignable number, which sufficiently shews the absurdity of using simple interest in these

cases.

EXAMPLES.

1. An estate brings in yearly 791. 4s. whať would it sett for, allowing the purchaser 43 per cent. compound interest for his money ? 4:5

79'2

IOO

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2. What is the price of a perpetual annuity of 4ol. discounting at 5 per cent. compound interest? Ans. Sool.

3. What is a freehold estate of 751. a year worth, allowing the buyer 6 per cent. compound interest for his money?

Ans. 1250l.

To find the present Worth of an Annuity, or Freehold Estate,

in Reversion, at Compound Interest.

RULE.*

1. Find the present worth of the annuity, as if it were to be entered on immediately.

2. Find

* This rule is sufficiently evident without a demonstration.

Those, who wish to be acquainted with the manner of computing the values of annuities upon lives, may consult the writings of Mr. DEMOIVRE, Mr. Simpson, and Dr. PRICE, all of whora have handled this subject in a very skilful and masterly manner.

Dr.

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