4. General Annuity
an annuity where the length of the payment
interval is not the same as the length of the
interest compounding period.
General Ordinary Annuity –
a general annuity in which
the periodic payment is
made at the end of the
payment interval.
5. Examples of General annuity:
1. Monthly installment payment of a
car, lot, or house with an interest
rate that is compounded annually
2. Paying a debt semi-annually
when the interest is
compounded monthly
6. Future and Present Value of a General
Ordinary Annuity
The future value F and present value P of a
general ordinary annuity is given by:
F = R
(1+𝑗)𝑛
−1
𝑗
and
P = R
1− (1+𝑗)−𝑛
𝑗
7. where
R is the regular payment;
j is the equivalent interest rate per payment
interval converted from the interest rate per
period; and
n is the number of payments.
8. Example 1. Cris started to deposit P1,000
monthly in a fund that pays 6%
compounded quarterly. How much will be
in the fund after 15 years?
Given: R = 1,000
n = 12(15)
= 180 payments i(4)
= 0.06m
= 4
Find: F
9. Solution:
The cash flow for this problem is shown in the
diagram below.
F
1000 1000 1000 …. 1000 1000
0 1 2 3 179 180
15. Example 2. A teacher saves P5,000 every 6
months in a bank that pays 0.25% compounded
monthly. How much will be her savings after 10
years?
Given: R = 5000
n = 2(10) = 20 payments
i(12) = 0.25% = 0.0025
m = 12
Find: F
16. Solution
The cash flow for this problem is shown in the
diagram below.
5000 5000 5000 …5000 5000
0 1 2 3 … 19 20
17. (1) Convert 0.25% compounded monthly to its
equivalent interest rate for each semi-annual
payment interval.
F1 = F2
P(1 +
𝑡2
2
)(2)t = P(1 +
𝑡2
2
)(12)t
(1 +
𝑡2
2
)2 = (1 +
0.0025
2
)12
(1 +
𝑡2
2
)2 = (1.00020833)12
18. (1 +
𝑡2
2
)2 = [ (1.00020833)12](1/2)
𝑡2
2
= (1.00020833)6 – 1
𝑡2
2
= 0.00125063 = j
Thus, the interest rate per semi-annual
payment interval is 0.00125063 or
0.125%.
19. (2) Apply the formula in finding the future value of
an ordinary annuity using the computed equivalent
rate
F = R
(1+0.00125063)20
−1
𝑗
F = 5000
(1+0.00125063)20
−1
0.00125063
F = 101,197.06
Thus, the teacher will be able to
save P101,197.06 after 10 years.
20. Example 4.
Mrs. Remoto would like to buy a television (TV) set
payable for 6 months starting at the end of the
month. How much is the cost of the TV set if her
monthly payment is P3,000 and interest is 9%
compounded semi-annually?
Given:
R = 3,000 m = 2 i(2) = 0.09 n = 6 payments
Find: cost (present value) at the
beginning of the term P
22. 𝑖(12)
12
= 0.00736312 = j
Thus, the interest rate
per monthly payment
interval is 0.00736312
or 0.736312%.
23. (2) Apply the formula in finding the present value
of an ordinary annuity using the computed
equivalent rate j = 0.00736312.
P = R
1−(1+𝑗)−𝑛
𝑗
P = 3000
1−(1+0.00736312)−6
0.00736312
P = 17,545.08
Thus, the cost of the TV set is
P17,545.08.
24. A cash flow is a term that refers to
payments received (cash inflows) or
payments or deposits made (cash
outflows). Cash inflows can be
represented by positive numbers and
cash outflows can be
represented by negative
numbers.
25. The fair market value or economic
value of a cash flow (payment stream)
on a particular date refers to a single
amount that is equivalent to the value
of the payment stream at that date.
This particular date is called
the focal date.
26. Example 5. Mr. Ribaya received two offers on
a lot that he wants to sell. Mr. Ocampo has
offered P50,000 and a P1 million lump sum
payment 5 years from now. Mr. Cruz has
offered P50,000 plus P40,000 every quarter
for five years. Compare the fair market values
of the two offers if money can earn 5%
compounded annually. Which
offer has a higher market
value?
27. Given:
Mr. Ocampo’s offer Mr. Cruz’s offer
P50,000 down payment P50,000 down payment
P1,000,000 after 5 years P40,000 every quarter
for 5 years
Find: fair market value of each offer
28. Solution.
We illustrate the cash flows of the two offers using time
diagrams.
Mr. Ocampo’s offer:
50,000 1 million
0 1 2 3 4 5
Mr. Cruz’s offer:
50,000
40,000 40,000 40,000 …. 40,000
0 1 2 3 … 20
29. Choose a focal date and determine the
values of the two offers at that focal
date. For example, the focal date can be
the date at the start of the term.
Since the focal date is at t = 0, compute
for the present value of each offer.
30. Mr. Ocampo’s offer: Since P50,000 is offered
today, then its present value is still
PhP 50 000.The present value of P1,000,000
offered 5 years from now is
P = F(1 + j)-n
P = 1,000,000 (1+0.05)-5
P = P783,526.20
Fair Market Value (FMV) = Down payment +
Present Value
= 50,000 + 783,526.20
FMV = P833,526.20
31. Mr. Cruz’s offer: We first compute for the present
value of a general annuity with quarterly payments
but with annual compounding at 5%.
Solve the equivalent rate, compounded quarterly, of
5% compounded annually.
F1 = F2
P(1 +
𝑖(4)
4
)(4)(5) = P(1 +
𝑖(4)
4
)(1)(5)
(1 +
𝑖(4)
4
)20 = (1 +
0.05
1
)5
1 +
𝑖(4)
4
= (1.05)(1/4)
32. 𝑖(4)
4
= (1.05)(1/4) - 1
𝑖(4)
4
= 0.012272
The present value of an annuity is given by
P = R
1−(1−𝑗)−𝑛
𝑗
P = 40000
1−(1−0.012272)−20
0.012272
P = PhP 705,572.70
Fair Market Value = Downpayment
+ present value
= 50,000 + 705,572.70
Fair Market Value = P755,572.70
33. Hence, Mr. Ocampo’s offer has a higher
market value. The difference between the
market values of the two offers at the
start of the term is
833 526.20 - 755 572.70 = P77,953.50
Alternate Solution (Focal
date at the end of the term):
34. Mr. Ocampo’s offer:
The future value of P1,000,000 at the end of
the term at 5% compounded annually is given
by
F = P(1 + j)n
F = 50,000(1+0.05)5
F = 63,814.08
The fair market value of this
offer at the end of the term is
63,814.08 plus 1million pesos
amounting to P,1,063 814.08.
35. Mr. Cruz’s offer:
The future value of this ordinary general
annuity is given by:
F = R
(1+𝑗)𝑛
−1
𝑗
F = 40000
(1+0.012272)20
−1
0.012272
F = 900,509.40
36. The future of P50,000 at the end of the term is
P63,814.08, which was already determined earlier.
Fair Market Value = 900,509.40 + 63,814.08 =
P964,323.48
As expected, Mr. Ocampo’s offer still has a higher
market value, even if the focal date was chosen to
be at the end of the term. The difference between
the market values of the two offers at the end of
the term is
1,063,814.08 –964,323.48 = P99,490.60.
37. You can also check that the present value
of the difference is the same as the
difference computed when the focal date
was the start of the term:
P = 99,490.60(1 + 0.05)-5
= P77,953.49.
