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Annuities - A Series of Equal Payments Occurring at Equal Periods of Time

The document discusses annuities and ordinary annuities. It provides formulas for calculating the present value (P), future value (F), and payment amount (A) of an ordinary annuity given certain variables like interest rate (i), number of periods (n), and payment amount. It also defines terms like uniform series present worth factor, capital recovery factor, sinking fund factor, and ordinary annuity. Examples are provided to demonstrate calculating the annual payment amount to pay off a loan over 5 years at 8% interest and calculating monthly installment payments over 20 months at 12% annual interest on an unpaid balance.

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Marcial Militante
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216 views5 pages

Annuities - A Series of Equal Payments Occurring at Equal Periods of Time

The document discusses annuities and ordinary annuities. It provides formulas for calculating the present value (P), future value (F), and payment amount (A) of an ordinary annuity given certain variables like interest rate (i), number of periods (n), and payment amount. It also defines terms like uniform series present worth factor, capital recovery factor, sinking fund factor, and ordinary annuity. Examples are provided to demonstrate calculating the annual payment amount to pay off a loan over 5 years at 8% interest and calculating monthly installment payments over 20 months at 12% annual interest on an unpaid balance.

Uploaded by

Marcial Militante
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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ANNUITIES

-a series of equal payments occurring at equal periods of time

Kinds of Annuities:

a. Ordinary Annuity
b. Deferred Annuity
c. Annuity Due
d. Perpetuity

ORDINARY ANNUITY

-payments are made at the end of each period

1 2 3 4 n-1 n
______________________________......._________
0

A A A A A A

Where:

P = present value of money


F = future value of money
A = periodic payment
n = number of interest periods
i = interest rate per period

Finding P when A is given:

Write the equation of value...

P = A(1+i)-1 +A(1+i)-2 + A(1+i)-3 +…+A(1+i)-(n-1) + A(1+i)-n (eq. 1)

Multiply both sides by (1+i)…

P(1+i) = A + A(1+i)-1 +A(1+i)-2 + … + A(1+i)-n+2 + A(1+i)-n+1 (eq. 2)


Subtract (eq. 1) from (eq. 2):

P + Pi = A + A(1+i)-1 +A(1+i)-2 + … + A(1+i)-n+2 + A(1+i)-n+1 (eq. 2)

- P = A(1+i)-1 +A(1+i)-2 + A(1+i)-3 +…+A(1+i)-(n-1) + A(1+i)-n (eq. 1)


_____________________________________________________

Pi = A – A (1+i)-n

Pi = A[1 - (1+i)-n]

P = A {[1- (1+i)-n] / i }

Or if the factor (1+i) / (1+i) is multiplied to the formula we have the result:

P = A{[(1+i)n – 1] / [i(1+i)n]}
The factor {[(1+i)n – 1] / [i(1+i)n]} is called the “uniform series present worth factor” and
is sometimes represented as (P/A,i%, n) and is read as “P given A at i percent in n interest
periods.

Finding A when P is given:

A = P{i / [1 – (1+i)-n]}
The factor {i / [1 – (1+i)-n]} is called the “capital recovery factor” and is sometimes
represented as (A/P, i%, n) and is read as “A given P at i percent in n interest periods.”

Finding F when A is given:

1 2 3 4 n-1 F
______________________________......._________ n
0

A A A A A A
F = A(1+i)n-1 +A(1+i)n-2 + A(1+i)n-3 +…+A(1+i)2 + A(1+i) + A (eq. 1)

Multiply both sides by (1+i)…

F + Fi = A(1+i)n +A(1+i)n-1 + A(1+i)n-2 +…+A(1+i)2 + A(1+i) (eq. 2)

Subtract (eq. 1) from (eq. 2):

F + Fi = A(1+i)n +A(1+i)n-1 + A(1+i)n-2 +…+A(1+i)2 + A(1+i) (eq. 2)

F = A(1+i)n-1 +A(1+i)n-2 + A(1+i)n-3 +…+A(1+i)2 + A(1+i) + A (eq. 1)

Fi = A(1+i)n – A
Or

F = A{[(1+i)n – 1] / i}
The factor [(1+i)n – 1] / i is called the “uniform series compound amount factor” and is
sometimes represented as (F/A, i%, n) and is read as “F given A, at i per cent in n interest
periods”.

Finding A when F is given:

A = F{i / [(1+i)n – 1]}


The factor {i / [(1+i)-n - 1]} is called the “sinking fund factor” and is sometimes represented
as (A/F, i%, n) and is read as “A given P, at i percent in n interest periods”.

Relation between Capital Recovery Factor & Sinking Fund Factor:

{ i / [(1+i)n – 1] } + i = [ i + i (1 + i)n - i ] / [(1+ i)n – 1]

= [ i (1 + i)n ] / [(1+ i)n – 1] X [(1 + i)-n / (1 + i)-n ]

{ i / [(1+i)n – 1] } + i = i / [ 1 – (1 + i)-n ]
Sinking fund factor + i = capital recovery factor
Examples:

1. A factory operator bought a generator set for P10,000 and agreed to pay the dealer a
uniform sum at the end of each year for 5 years at 8% interest compounded annually so that
the final payment will cancel the debt for principal and interest. What is the annual payment?

Solution:
P 10,000

1 2 3 4 5
_____________________________________
0

A A A A A
i = 8% compounded annually

A = P {i / [1 – (1+i)-n]} = 10,000 { 0.08 / [1 - (1+0.08)-5}

A = P 2, 504.57 (answer)

2. A man bought a laptop for P21,000 on installment basis at the rate of 12% per year on the
unpaid balance. If he paid a down payment of P6,000 in cash and proposes to pay the balance
in 20 monthly payments, what should these monthly payments be?

Solution:

r = 12% per year

i = r/m = 12% / 12 = 1%

n = 20

P 21,000

1 2 3 4 20
________________________________ ……….._____

P 6,000 A A A A A

P = 21,000 – 6,000 = 15,000

A = P {i / [1 – (1+i)-n]} = 15,000 { 0.01 / [1 - (1+0.01)-20}

A = P831.23 (Answer)
Exercise:

Juan dela Cruz borrowed P2,400 at 1% per month payable in 24 equal payments. How much of
the loan remains unpaid immediately after he has paid the 12 th payment?

Answer: P1,271.60

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