2. Annuity
- is a sequence of payments (usually of
equal size) made at equal intervals of
time. Examples of which are:
- monthly house rent payment
- Annual premiums of life insurance policy
- Installment payments in purchasing a house
- monthly retirement benefits (pension plan)
3. Simple Annuity
- An annuity in which the payment period is same
as the interest period (conversion period)
Payment period – time between successive period
of annuity.
Term – time from the beginning of the first
payment period to the last payment period.
Periodic Payment (R) – size of each annuity
payment
4. Ordinary Annuity
- An annuity in which the payments are
made at the end of each payment period.
Amount of Ordinary Annuity (S) – sum of
accumulated values of the payments at
the end of its terms.
Present Value of O. Annuity (A) – sum of the
present values of the payments.
5. Amount and Present Value of
Ordinary Annuity
Amount Present Value
j
i
m
= n tm=and
,
1 (1 )
nli
n
nli
A Ra
where
i
a
i
−
=
− +
=
,
(1 ) 1
nli
n
nli
S Rs
where
i
s
i
=
+ −
=
6. Problem Set 3.3
1. Find the Present Value and the amount of
an P8,000 ordinary annuity payable
quarterly for 10 years if the money is worth
12% converted quarterly.
7. Problem Set 3.3
8. The purchaser of a portable DVD player
with USB will pay P1,500 cash and
P565.80 at the end of each month for 12
months to discharge all principal and
interest at 9% compounded monthly. Find
the cash price of the DVD player.
8. Problem Set 3.3
11. Linda deposits P10,000 every 3 months
for 7 years in an account paying 4%
compounded quarterly. How much will she
save in her account at the end of 7 years
assuming no withdrawals were made?
9. Problem Set 3.3
17. Find the amount of a 4- year ordinary
annuity whose present value is P8,300 if
the money is worth 7% m = 4.
10. Problem Set 3.3
18. Find the present value of a 7- year
ordinary annuity whose amount is P72,900
if the money is worth 6% m = 2.
12. Problem Set 3.4
1. How much should be invested in a fund at
the end of each year in order to
accumulate to P150,000 at the end of 10
years if the fund is earns 6.5% effective?
13. Problem Set 3.4
1. The borrower of a P500,000 loan plans to
repay the loan by making equal payments
at the end of each three months for 10
years. If the rate of the interest is 17%
compounded quarterly, find the quarterlt
payment.
14. Problem Set 3.4
6. A video camera is worth P35,950.
Nancee bought one by paying P5,000 and
an equal payment at the end of each
month for 18 months. Find the equal
payments if the interest rate is 18% m =
12.
15. Problem Set 3.4
9. The Citizen Cottage Industry has a high-
speed sewing machine that will retire in 5
years. How much must be set aside each
3 months in order to by a new sewing
machine that costs P720,000 to replace
the old one if the fund is invested at 8%
compounded quarterly?
16. Finding the nominal rate using (A)
2 2
2
( 1) 6( 1) 12 1 0
4
2
nR
n i n i
A
b b ac
i
a
− + + + − = ÷
− ± −
=
2 2
2
6( 1) (6( 1)) 4( 1)(12(1 ))
2( 1)
nR
n n n
Ai
n
j im
n tm
− + ± + − − −
=
−
=
=
a b c
Given A
17. Finding the nominal rate using (S)
2 2
2
( 1) 6( 1) 12 1 0
4
2
nR
n i n i
S
b b ac
i
a
− − − + − = ÷
− ± −
=
2 2
2
6( 1) ( 6( 1)) 4( 1)(12(1 ))
2( 1)
nR
n n n
Si
n
j im
n tm
− ± − − − − −
=
−
=
=
Given S
a b c
18. Problem Set 3.5
1. At what nominal rate compound
semiannually is P5,000 the present value
of P1,000 ordinary annuity payable
semiannually for 3 years?
A = 5000
R = 1000
m = 2 (semiannually)
t = 3 years
n = (3)(2) = 6
19. 2 2
2
1
2
2
1
( 1) 6 1 35
6( 1) 6(6 1) 42
6(1000)
12 1 12 1 2.4
5000
42 42 4(35)( 2.4)
0.05465...
2(42)
42 42 4(35)( 2.4)
1.2547
2(42)
(2) 0.1093 10.93%
a n
b n
nR
c
A
i
i
j im i
= − = − =
= + = + =
= − = − =− ÷ ÷
− + − −
= =
− − − −
= =−
= = = ≈
20. Problem Set 3.5
4. A man invests P10,000 at the end of every 3
months. If he has P390,000 in 7 years, at what
rate compounded quarterly did his investment
earn interest?
S = 390000
R = 10000
m = 4 (quarterly)
t = 7 years
n = (7)(4) = 28
21. 2 2
2
1
2
2
1 1
( 1) 28 1 783
6( 1) 6(28 1) 162
28(10000)
12 1 12 1 3.3846
390000
( 162) ( 162) 4(783)(3.3846)
0.18331...
2(783)
( 162) ( 162) 4(783)(3.3846)
0.02358...
2(783)
(4)
a n
b n
nR
c
S
i
i
j im i
= − = − =
= − − = − − = −
= − = − = ÷ ÷
− − + − −
= =
− − + − −
= =
= =
2 2
0.73324 13.32%
(4) 0.09432 9.43%j im i
= ≈
= = = ≈
22. Problem Set 3.5
6. A 31-inch LCD television costs P61,990.
Stanley bought one by making a down
payment of P7,000 and paying P5,040.75
at the end of every 3 months for 3 years.
At what rate converted quarterly was the
interest charged?
23. Problem Set 3.5
10. Marian invests P11,600 at the end of
each year in a fund. If she wants to have
P246,500 in the fund in 14 years, at what
rate compounded annually should the
money be invested?
