2. 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.
3. EXAMPLES OF GENERAL ANNUITY:
1. Monthly installment payment of a car, lot, or house with an interest rate
that is compounded monthly.
2. Paying a debt semi-annually when the interest is compounded
monthly.
4. FUTURE AND PRESENT VALUE OF A GENERAL ORDINARY
ANNUITY
The future value F of General
Ordinary Annuity
𝐹 = 𝑅
1 + 𝑗 𝑛
− 1
𝑗
The present value F of General
Ordinary Annuity
𝑃 = 𝑅
1 − 1 + 𝑗 −𝑛
𝑗
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
5. Note: The formulas for F and P are same as those in simple annuity.
The extra step occurs in finding j: the given interest rate per period
must be converted to an equivalent rate per payment interval.
6. 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 t=15 𝑟4 = 6% 𝑚1 = 12 𝑚2 = 4
𝐧 = 𝑚1𝑡 = 12 15 = 180 payments
Find: F
Solution:
Step 1: Convert 6% compounded quarterly to its equivalent interest rate for monthly payment interval.
𝐹1 = 𝐹2
1 + 𝑟12
12 = 1 +
𝑟4
4
4
1 + 𝑟12
12 = 1 +
0.06
4
4
1 + 𝑟12
12
= 1 + 0.015 4
1 + 𝑟12
12
= 1.015 4
1 + 𝑟12 = 1.015 4
1
12
𝑟12 = 1.015
1
3 − 1
𝑟12 = 1.00498 − 1
𝑟12 = 0.00498
7. Step 2: Apply the formula in finding the future value of an ordinary annuity using the computed equivalent rate.
Thus, the interest rate per monthly payment interval is 0.00498 or 0.498%.
𝐹 = 𝑅
1 + 𝑗 𝑛 − 1
𝑗
𝐹 = 1000
1 + 0.00498 180 − 1
0.00498
𝐹 = 1000
1.00498 180 − 1
0.00498
𝐹 = 1000
2.44532 − 1
0.00498
𝐹 = 1000
1.44532
0.00498
𝐹 = 1000 290.22490
𝑭 = 𝟐𝟗𝟎, 𝟐𝟐𝟒. 𝟗𝟎
Thus, Cris will have P290,224.90 in the fund after 15
years.
8. 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?
Find: F
Solution:
Step 1: Convert 0.25% compounded monthly to its equivalent interest rate for each semi-annual payment interval.
𝐹1 = 𝐹2
1 + 𝑟2
2
= 1 +
𝑟12
12
12
1 + 𝑟2
2 = 1 +
0.0025
12
12
1 + 𝑟2
2
= 1 + 0.00021 12
1 + 𝑟2
2 = 1.00021 12
1 + 𝑟2 = 1.00021 12
1
2
𝑟2 = 1.00021 6
− 1
𝑟2 = 1.00126 − 1
𝑟2 = 0.00126
Given: R = 5,000 t=10 𝑟12 = 0.25% 𝑚1 = 2 𝑚2 = 12
𝐧 = 𝑚1𝑡 = 2 10 = 20 payments
9. Step 2: Apply the formula in finding the future value of an ordinary annuity using the computed equivalent rate.
Thus, the interest rate per semi-annual payment interval is 0.00126 or 0.126%.
𝐹 = 𝑅
1 + 𝑗 𝑛 − 1
𝑗
𝐹 = 5000
1 + 0.00126 20 − 1
0.00126
𝐹 = 5000
1.00126 20 − 1
0.00126
𝐹 = 5000
1.02550 − 1
0.00126
𝐹 = 5000
0.02550
0.00126
𝐹 = 5000 20.23810
𝑭 = 𝟏𝟎𝟏, 𝟏𝟗𝟎. 𝟓𝟎
Thus, a teacher will have P101,190.50 in the fund after
10 years.
10. Example 3: Ken borrowed an amount of money from Kat. He agrees to pay the principal plus
interest by paying P38,973.76 each year for 3 years. How much money did he borrow if interest is
8% compounded quarterly?
Find: P
Solution:
Step 1: Convert 8% compounded quarterly to its equivalent interest rate for each payment interval.
𝐹1 = 𝐹2
1 + 𝑟1
1
= 1 +
𝑟4
4
4
1 + 𝑟1
1 = 1 +
0.08
4
4
1 + 𝑟1
1
= 1 + 0.02 4
1 + 𝑟1
1 = 1.02 4
1 + 𝑟1 = 1.02 4
1
1
𝑟1 = 1.02 4
− 1
𝑟1 = 1.08243 − 1
𝑟1 = 0.08243
Given: R = 38,973.76 t=3 𝑟4 = 8% 𝑚1 = 1 𝑚2 = 4
𝐧 = 𝑚1𝑡 = 1 3 = 3 payments
11. Step 2: Apply the formula in finding the present value of an ordinary annuity using the computed equivalent rate.
Thus, the interest rate per payment interval is 0.08243 or 8.243%.
𝑃 = 𝑅
1 − 1 + 𝑗 −𝑛
𝑗
𝑃 = 38,973.76
1 − 1 + 0.08243 −3
0.08243
𝑃 = 38,973.76
1 − 1.08243 −3
0.08243
𝑃 = 38,973.76
1 − 0.78849
0.08243
𝑃 = 38,973.76
0.21151
0.08243
𝑃 = 38,973.76 2.56593
𝑭 = 𝟏𝟎𝟎, 𝟎𝟎𝟑. 𝟗𝟒
Hence, Ken borrowed P100,003.94 from Kat.
12. 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?
Find: P
Solution:
Step 1: Convert 9% compounded semi-annually to its equivalent interest rate for each monthly payment interval.
𝐹1 = 𝐹2
1 + 𝑟12
12
= 1 +
𝑟2
2
2
1 + 𝑟12
12 = 1 +
0.09
2
2
1 + 𝑟12
12
= 1 + 0.045 2
1 + 𝑟12
12 = 1.045 2
1 + 𝑟12 = 1.045 2
1
12
𝑟12 = 1.045
1
6 − 1
𝑟12 = 1.00736 − 1
𝑟12 = 0.00736
Given: R = 3,000 t=0.5 𝑟2 = 9% 𝑚1 = 12 𝑚2 = 2
𝐧 = 𝑚1𝑡 = 12 0.5 = 6 payments
13. Step 2: Apply the formula in finding the present value of an ordinary annuity using the computed equivalent rate.
Thus, the interest rate per month payment interval is 0.00736 or 0.736%.
𝑃 = 𝑅
1 − 1 + 𝑗 −𝑛
𝑗
𝑃 = 3,000
1 − 1 + 0.00736 −6
0.00736
𝑃 = 3,000
1 − 1.00736 −6
0.00736
𝑃 = 3,000
1 − 0.95694
0.00736
𝑃 = 3,000
0.04306
0.00736
𝑃 = 3,000 5.85054
𝑭 = 𝟏𝟕, 𝟓𝟓𝟏. 𝟔𝟐
Thus, the cost of the TV set is P17,551.62
14. ANOTHER EXAMPLES:
1. ABC Bank pays interest at the rate of 2% compounded quarterly. How
much will Ken have in the bank at the end of 5 years if he deposits
P3,000 every month?
2. A sala set is for sale at P16,000 in cash or on monthly installment of
P2,950 for 6 months at 12% compounded semi-annually. Which is
lower: the cash price or the present value of the installment?
15. FAIR MARKET VALUE
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.
FAIR MARKET 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.
