What is Annuity?

Is a series of equal payments

(made in each period) for a given
total period of time and subject to a
fixed compound interest rate.
Types of Annuity

Ordinary Annuity – annuity in which

the periodic payment is made at the end
of each payment interval.

Annuity Due – an annuity in which the

periodic payment is made at the
beginning of each payment interval.
Types of Annuity

Deferred Annuity – the periodic

payment is not made at the beginning
nor at the end of each payment interval,
but some later date.

Perpetuity – the type of annuity similar

to ordinary annuity except that the
payments continue infinitely.
Future Value – is the total accumulation
of the payments and interest.

Present Value – is the principal that

must be invested today to provide the
regular payments of an annuity.
Annuity?
m

Annually 1
F = 𝑃(1 + 𝑖) 𝑛
Semi
2
Annually

Quarterly 4

𝐹 Monthly 12
P=
(1+𝑖)𝑛 Daily 360
 Ordinary Annuity

(1+𝑖)𝑛 −1
0 1 2 3 4 n
F=𝐴 [ ]
𝑖

A A A A A
P F

(1+𝑖)𝑛 −1
P=𝐴 [ 𝑛 ]
1+𝑖 (𝑖)
 (1+𝑖)𝑛 −1 (1+𝑖)𝑛 −1
F=𝐴 [
𝑖
] Ordinary Annuity P=𝐴 [ 𝑛
1+𝑖 (𝑖)
]

What is the accumulated amount of five-year annuity paying P 6000 at the end of
each year, with interest at 15% compounded annually?

px F =
A[CI in 1
+ -

-6166
F 3
=
F :
6000
[CI
F = 40 , 454 29
.
 (1+𝑖)𝑛 −1 (1+𝑖)𝑛 −1
F=𝐴 [
𝑖
] Ordinary Annuity P=𝐴 [ 𝑛
1+𝑖 (𝑖)
]

A man decided to deposit an amount every end of the year in order to buy a farm lot
on his retirement in 10 years. How much must his yearly deposits be to earn P
4,000,000 at his retirement if i = 5%.
[ i) 15
C+ -

F A =

PT 1
23 410

-A _
A
-AA
F :M
4m = A
[CI +05 y
A =
318 018 30
, .
 (1+𝑖)𝑛 −1 (1+𝑖)𝑛 −1
F=𝐴 [
𝑖
] Ordinary Annuity P=𝐴 [ 𝑛
1+𝑖 (𝑖)
]

Mr. Cruz plans to deposit for the education of his 5 years old son, P 500 at the end of
each month for 10 years at 12% annual interest compounded monthly. The amount
that will be available in two years is:

PT 1 2345 at
F =
A[-1]
↓to 1500/500 1000 100 1500
F=?
F =

500 [CI2C
F =
13 486 73
,
.
 (1+𝑖)𝑛 −1 (1+𝑖)𝑛 −1
F=𝐴 [
𝑖
] Ordinary Annuity P=𝐴 [ 𝑛
1+𝑖 (𝑖)
]
F =
p(1 + i)n

How many quarterly payments of P 40,000 at 10% annually compounded quarterly

must be made in order to pay a car costing P 680,000 by installments instead of
paying it at once?
Fo80k Fook Fol =

x680k
A[(I i) 1)
+
p P(1 i)"
-

I +
234 U =

YOK401YOK 40K
-

680000(1 + 0 025)"
40000
-

Fyok
.

i = = 25
.
or 0 025
.

n = 22 41 v 23
.
 (1+𝑖)𝑛 −1 (1+𝑖)𝑛 −1
F=𝐴 [
𝑖
] Ordinary Annuity P=𝐴 [ 𝑛
1+𝑖 (𝑖)
]
F =
p(1 + i)n

How many quarterly payments of P 40,000 at 10% annually compounded quarterly

must be made in order to pay a car costing P 680,000 by installments instead of
paying it at once? 680000(1 + 0
025)"
40000
.

7680K
x680k
p I 680000(1 + 0 025" .

1,600 ,
000/D + 0 025" 1)
234
=
U .
-

680 ,000 (1 025)" 1600000 (1 025)"

YOK401YOK
.
=
40K . -

1600, 000
- -

Fy0K 680 ,000 (1 025)"

.
=
1600000(1 .
025)" = -
1 600 , 000
,

i = = 25
.
or 0 025
.
920 ,000 (1
027" .
= 160 , 000
Log (920 000) + Log (1 025)"
, .
=
Gog (1 600, 000)
,

Log(920000 + nLog (1 025) .

=
Log (1600000)
nLog (1 025) .
=
Log (1600000) -

Log (920,000)
rog(1 025) .

Log(1 .
025)
n = 22 41223
.
 Annuity Due
n-1
0 1 2 3 4

A A A A A
P F

1+𝑖 𝑛 −1 1+𝑖 𝑛+1 −1

F=𝐴[ ](1+i) or F = 𝐴 [ − 1]
𝑖 𝑖

(1+𝑖)𝑛 −1
P=𝐴[ 𝑛−1 ]
1+𝑖 (𝑖)
 Annuity Due F =
p(1 + ijn
1+𝑖 𝑛 −1 1+𝑖 𝑛+1 −1 (1+𝑖)𝑛 −1
F=𝐴[ ](1+i) or F = 𝐴 [ − 1] P=𝐴[ ]
𝑖 𝑖 1+𝑖 𝑛−1 (𝑖)

How much money can you deposit now that will pay your next 12 months of lease
amounting to P 3,000 each paid every beginning of the month if money is worth 7%
annually compounded monthly? Fz() Fp =

Fp
A - 1f (i)
↑P , 2345 1 P(ti)" =

3K313K3k3k
00503) [C1 0008- (
↓

Es 0(1 + 0 3000
+

0005
.
=

i =
107 = 0 .
00583
P = 34 , 874 24.
 F = p(1 + in
Annuity Due
1+𝑖 𝑛 −1 1+𝑖 𝑛+1 −1 (1+𝑖)𝑛 −1
F=𝐴[ ](1+i) or F = 𝐴 [ − 1] P=𝐴[ ]
𝑖 𝑖 1+𝑖 𝑛−1 (𝑖)

A man deposits P 50,000 every year for 8 years start on the day his son was born so
that his son can withdraw the amount on his 18th birthday. How much will this
amount be if money is worth 4% annually?
F +
A[x i) 1](1
-

=
479: 139 77 + 1)
.

X
o
500CIO
1s + 50
:

KOSK5OKF
-
=

sokOK

479
,
139 77 .
 F = p(1 + in
Annuity Due
1+𝑖 𝑛 −1 1+𝑖 𝑛+1 −1 (1+𝑖)𝑛 −1
F=𝐴[ ](1+i) or F = 𝐴 [ − 1] P=𝐴[ ]
𝑖 𝑖 1+𝑖 𝑛−1 (𝑖)

A man deposits P 50,000 every year for 8 years start on the day his son was born so
that his son can withdraw the amount on his 18th birthday. How much will this
amount be if money is worth 4% annually?
479: 139 77
.

PCI +is" = w
X
1 a 3456
!
- 18 479 379
,
.

77(1 + 04)1 0 .
=
w

SokokSK50
-

243 9/
09
.

,
 Annuity Due
1+𝑖 𝑛 −1 1+𝑖 𝑛+1 −1 (1+𝑖)𝑛 −1
F=𝐴[ ](1+i) or F = 𝐴 [ − 1] P=𝐴[ ]
𝑖 𝑖 1+𝑖 𝑛−1 (𝑖)

A farmer bought tractor costing P 25,000 payable in 10 semi-annual payments, each

installment payable at the beginning of each period. If the rate of interest is 26%
compounded semi-annually. Determine the amount of each installment.

"
P 25,
=,

-I
00
p =A
[]
25, 000
= a
=]
A =

13)
[ (1
1

I
+0 -

(1 + 0 (3)10 1(0-13)
=

A =
4077 . 0
 Annuity Due
1+𝑖 𝑛 −1 1+𝑖 𝑛+1 −1 (1+𝑖)𝑛 −1
F=𝐴[ ](1+i) or F = 𝐴 [ − 1] P=𝐴[ ]
𝑖 𝑖 1+𝑖 𝑛−1 (𝑖)

If P 200 is deposited in a savings account at the beginning of each 15 years and the
account draws interest at 7% per year, compounded annually, the value of the
account at the end of 15 years will be most nearly

112 15
F =

ATH-1]( i) +

CoolCIOOFIO
200 200 200 200

F
-
- - - -
-

F =
5377 61
.
Types of Annuity

Deferred Annuity – the periodic

payment is not made at the beginning
nor at the end of each payment interval,
but some later date.

Perpetuity – the type of annuity similar

to ordinary annuity except that the
payments continue infinitely.
 Deferred Annuity
P

0 1 2 3 4 n

A A A

1+𝑖 𝑛 −1 1− 1+𝑖 −𝑛 −𝑘
F=𝐴[ ] P=𝐴 [ ](1 + 𝑖)
𝑖 𝑖
 (1+𝑖)𝑛 −1
P=𝐴 [ ]
𝑖 1+𝑖 𝑚+𝑛
1− 1+𝑖 −𝑛
F=𝐴[
1+𝑖 𝑛 −1
𝑖
] Deferred Annuity P=𝐴 [
𝑖
](1 + 𝑖)−𝑘

Mr. Lee wants to receive a consistent amount from a bank that offers 7% interest to
be compounded annually. If he will delay his withdrawal for the first 5 years, how
much should he deposit now to receive P 100,000 every year for the next 10 years
after the period of deferral?
A[ (11) "J(1
-

=
P
-

5
+ i)
=

-M
#
4544 i 100000 11+00] (1 007)
-

P =
+

look 100K DOK

p = 500 , 771 .
66
 (1+𝑖)𝑛 −1
P=𝐴 [ ]
𝑖 1+𝑖 𝑚+𝑛
1− 1+𝑖 −𝑛
F=𝐴[
1+𝑖 𝑛 −1
𝑖
] Deferred Annuity P=𝐴 [
𝑖
](1 + 𝑖)−𝑘

Determine the present worth of a deferred annuity, consisting of 10 semi-annual

payments, each P 1000 the first at the end of the third year. Money worth 14%
compounded semi-annually.

Af ( nJ(1
i) k
+ i)
-

P = +

↑P (k 1k ( 14)
P =

1000/- 11 j
1k

p =
5007 72 .
 (1+𝑖)𝑛 −1
P=𝐴 [ ]
𝑖 1+𝑖 𝑚+𝑛
1− 1+𝑖 −𝑛
F=𝐴[
1+𝑖 𝑛 −1
𝑖
] Deferred Annuity P=𝐴 [
𝑖
](1 + 𝑖)−𝑘

A man loans P 187,400 from a bank with interest at 5% compounded annually. He

agrees to pay his obligations by paying 8 equal payments, the first being due at the
end of 10 years. Find the annual payments.

A[1 + "j(1
-

( k
X
0 =
+ 2)
-

!
N 187400 =

AsCH005)
*

A
A AAA

A = 44980 57 .
 (1+𝑖)𝑛 −1
P=𝐴 [ ]
𝑖 1+𝑖 𝑚+𝑛
1− 1+𝑖 −𝑛
F=𝐴[
1+𝑖 𝑛 −1
𝑖
] Deferred Annuity P=𝐴 [
𝑖
](1 + 𝑖)−𝑘

A parent on the day a child is born wishes to determine what lump sum would have
to be paid into an account bearing interest at 5% compounded annually, in order to
withdraw P 20,000 each on the child’s 18th, 19th , 20th and 21st birthday. How much is
the lump sum amount?
At + ](
"
(1 is k
P
-

= + i)
1231819902

"It's [1 11+00554]It0
=

p = 20000

201 201 20k 201

p =
30941 73 :
 (1+𝑖)𝑛 −1
P=𝐴 [ ]
𝑖 1+𝑖 𝑚+𝑛

Perpetuity

0 1 2 3 4 ∞
𝐴
P=
A A A A A 𝑖
 𝐴
Perpetuity P=
𝑖
A man, pondering of retirement, determined that P 100,000 per year will be
sufficient for him to live without the need to work. Upon his retirement, how much
must he invest in a bank that offers 5% interest rate to receive the amount every
year continuously?

P =
A =
100 =
2000, 0
 𝐴
Perpetuity P=
𝑖
On average, it needs P 250,000 per year and an additional P 600,000 every 4 years
for major repairs to maintain a certain three-storey office building. How much must
be paid now to take care of these payments on a bank that offers 8% per year?

412345678
I 1111111
Prsok =
25000 Plook =200

↓
↓↓t ↓↓t
250k 250K 250k 2501 250k 250k Prok = 3 , 125
, 000
Po = 1 , 666 606 67
, .

600 600
P =
Perok + Poo
effective rate :

in =
(1 + 1)
m
-
1 =
(1 + 008)4 + = 0 36
.
p = 3125 ,00 + 1 666 666 67
. ,
·

p = 4 791 , 666 67
,
.
 𝐴
Perpetuity P=
𝑖
The city wishes to set up a retirement fund. At 10% interest rate compounded
continuously, how large a fund in millions of pesos is required to guarantee an
average annual retirement income of P 20,000.00? 3000 retirees are expected to
participate.

p :-20

200 , 000 X 3000 = 600 , 000 ,000