2. ANNUITY – a sequence of payments made at equal (fixed) intervals or periods of time.
◦ Annuities may be classified in different ways, as follows:
ANNUITIES
According to payment
interval and interest
period
Simple Annuity – an annuity
where the payment interval is the
same as the interest period.
General Annuity – an annuity where
the payment interval is not the same
as the interest.
According to time of
payment
Ordinary Annuity or Annuity
Immediate – a type of annuity in
which the payments are made at
the end of each payment interval.
Annuity Due – a type of annuity in
which the payments are made at the
beginning of each payment interval.
According to duration Annuity Certain – an annuity in
which payments begin and end at
definite times.
Contingent Annuity – an annuity in
which the payments extend over an
indefinite (or indeterminate) length
of time.
3. Term of annuity, t – time between the first payment interval and last payment interval
Regular or Periodic payment, R – the amount of each payment
Amount (Future Value) of an annuity, F – sum of future values of all the payments to be made
during the entire term of the annuity
Present value of an annuity, P – sum ofpresent values of all the payments to be made during
the entire term of the annuity
Annuities may be illustrated using a time diagram. The time diagram for an ordinary annuity
(i.e., payments are made at the end of the year is given below.
Time Diagram for an n-Payment Ordinary Annuity
R R R R R . . . R
0 1 2 3 4 5 n
4. Example 1. Suppose Mrs. Remoto would like to save P 3,000.00 every month in a fund that gives 9%
compounded monthly. How much is the amount or future value of her savings after 6 months?
Given: Periodic payment, R = P 3,000.00 Find: amount (future value) at the end of the term, F.
term, t = 6 months
interest rate per annum i(12) = 0.09
number of conversions per year, m = 12
interest rate per period, j = 0.09/12 = 0.0075
Solution:
(1) Illustrate the cash flow in a time diagram
3,000.00 3,0000.00 3,000.00 3,000.00 3,000.00 3,000.00
0 1 2 3 4 5 6
5. (2) Find the future value of all the payments at the end of term (t = 6)
3,000.00 3,000.00 3,000.00 3,000.00 3,000.00 3,000.00
0 1 2 3 4 5 6
3,000.00
3,000.00(1 + 0.0075)
3,000.00(1 + 0.0075)2
3,000.00(1 + 0.0075)3
3,000.00(1 + 0.0075)4
3,000.00(1 + 0.0075)5
◦
6. (3) Add all the future values obtained from the previous steps
3,000.00 = 3,000.00
3,000.00(1 + 0.0075) = 3,022.50
3,000.00(1 + 0.0075)2 = 3,045.169
3,000.00(1 + 0.0075)3 = 3,068.008
3,000.00(1 + 0.0075)4 = 3,091.018
3,000.00(1 + 0.0075)5 = 3,114.20
P 18,340.89
Thus, the amount of this annuity is P 18,340.89
7. The future value F of an ordinary annuity is given by:
Amount (Future Value) of Ordinary Annuity:
(1 + j)n – 1 where: R is the regular payment
F = R j is the interest rate per period
j n is the number of payments
8. Example 2. In order to save for her high school graduation, Marie decided to save P 200.00 at the end of
each month. If the bank pays 0.250% compounded monthly, how much will her money be at the end of 6
years?
Given: R = P 200.00 j = 0.0025/12 = 0.0002083
m = 12 t = 6 years
i(12) = 0.250% = 0.0025 n = mt = (12)(6) = 72 periods
Solution:
F = R {(1 + j)n – 1/j)}
F = P 200.00 {( 1 + 0.0002083 )72 – 1)/0.0002083 )}
F = P 14,507.85
9. Present Value of an Ordinary Annuity:
1 - (1 + j)-n where: R is the regular payment
P = R j is the interest rate per period
j n is the number of payments
10. Example 3. Mr. Ribaya paid P 200,000.00 as down payment for a car. The remaining amount is to be settled
by paying P 16,200.00 at the end of each month for 5 years. If the interest is 10.5% compounded monthly,
what is the cash price of his car?
Given: R = P 200.00 j = 0.105/12 = 0.00875
m = 12 t = 5 years
i(12) = 10.5% = 0.105 n = mt = (12)(5) = 60 periods
Down payment = P 200,000.00
Find: Cash value or cash price of the car
Solution:
P = R {(1 – (1 + j)-n/j)}
P = P 16,200.00 {(1 – ( 1 + 0.00875 )-60)/0.00875)}
P = P 753,702.20
Cash Value = Down payment + Present value
= P 200,000.00 + P 753,702.20
Cash Value = P 953,702.20
11. Periodic payment R of an Annuity:
(1 + j)n - 1 (1 + j)n - 1
F = R R = F/
j j
(1 – (1 + j)-n 1 - (1 + j)-n
P = R R = P/
j j
Where:
R is the regular payment P is the present value of an annuity
F is the future value of an annuity j is the interest rate per period
n is the number of payments
12. Example 4. Paolo borrowed P 100,000.00. He agrees to pay the principal plus interest by paying an equal
amount of money each year for 3 years. What should be his annual payment if interest is 8% compounded
annually?
Given: P = P 100,000.00 j = 0.08
m = 1 t = 3 years
i(1) = 8% = 0.08 n = mt = (1)(3) = 3 periods
Find: Periodic payment R
Solution:
P = R {(1 – (1 + j)-n/j)} then
R = P / {(1 – (1 + j)-n/j)}
R = P 100,000.00 / {(1 – ( 1 + 0.08)-3)/0.08)}
P = P 38,803.35
Thus, the man should pay P 38,803.35 every year for 3 years
13. ACTIVITY 7
Solve the following:
1. Linda started to deposit P 2,000.00 quarterly in a fund that pays 5.5% compounded quarterly. How
much will be in the fund after
2. The buyer of a house and lot pays P 200,000.00 cash and P 10,000.00 every month for 20 years. If
money is 9% compounded monthly, how much is the cash value of the lot?
3. Rebecca borrowed P 150,000.00 payable in 2 years. To repay the loan, she must pay an amount
every month with an interest rate of 6% compounded monthly. How much should he pay every
month?
4. Mr. Sarsonas would like to save P 500,000.00 for his son’s college education. How much should he
deposit in a savings account after 6 months for 12 years if interest is at 1% compounded semi-
annually?
5. A television is for sale at P 17,999.00 in cash or on terms, P 1,600.00 each month for the next 12
months. The money is 9% compounded monthly. Which is lower, the cash price or the present value
of the installment terms?
