PAMANTASAN NG LUNGSOD NG VALENZUELA
Tongco Street, Maysan, Valenzuela City
COLLEGE OF ENGINEERING AND INFORMATION TECHNOLOGY
Civil Engineering Department
CASH FLOW DIAGRAM
Types of Annuities
- a graphical representation of cash flows drawn on a
time scale.
1. Ordinary Annuity
- equal payments are made at the end of each
payment period starting from the first period.
- horizontal line divided into period time units.
- vertical arrows represent cash flows.
- Net cash flow = Inflows – Outflows.
2. Deferred Annuity
- payment of the first amount is deferred a certain
number of periods after the first.
CASH FLOW DIAGRAM RULES:
1. Cash flows cannot be added/subtracted unless
they occur at the same point in time.
2. Cash flows occurring during a period are counted
at the end of the period.
3. In comparing values, every cashflow must be in the
same time period.
3. Annuity Due
- payments are made at the start of each period
beginning from the first period.
Notations for cash flow diagram:
𝑷 = 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑠𝑢𝑚 𝑜𝑓 𝑚𝑜𝑛𝑒𝑦.
𝑭 = 𝑓𝑢𝑡𝑢𝑟𝑒
𝑠𝑢𝑚
𝑜𝑓
𝑚𝑜𝑛𝑒𝑦.
𝑨 = 𝑒𝑛𝑑 − 𝑜𝑓 − 𝑝𝑒𝑟𝑖𝑜𝑑 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤𝑠 𝑖𝑛 𝑎 𝑢𝑛𝑖𝑓𝑜𝑟𝑚
𝑠𝑒𝑟𝑖𝑒𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑛𝑔 𝑓𝑜𝑟 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓
𝑝𝑒𝑟𝑖𝑜𝑑𝑠.
𝑮 = 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑎𝑚𝑜𝑢𝑛𝑡𝑠.
𝑰 = 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡.
𝒊 = 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑒𝑟𝑖𝑜𝑑.
𝒏 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑒𝑟𝑖𝑜𝑑.
4. Perpetuity
- Periodic payments continue indefinitely.
ANNUITY
- A series of equal payments occurring at equal
interval of time.
′
(𝟏 + 𝒊)𝒏 − 𝟏
𝑭 = 𝑨[
]
𝒊
′
(𝟏 + 𝒊)𝒏 − 𝟏
𝑷 = 𝑨[
]
𝒊(𝟏 + 𝒊)𝒏
𝑤ℎ𝑒𝑟𝑒:
𝐹 = 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑒𝑑 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑟 𝐹𝑢𝑡𝑢𝑟𝑒 𝑊𝑜𝑟𝑡ℎ
𝑃 = 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑟 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑤𝑜𝑟𝑡ℎ
𝐴 = 𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑎𝑚𝑜𝑢𝑛𝑡, 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑜𝑟 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝑖 = 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
𝑛′ = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠
𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑒𝑟𝑖𝑜𝑑
Prepared by: Engr. Karl Megan S. Garcia
ENGINEERING ECONOMICS: EM5
𝑷=
𝑨
𝒊
∗ 𝑛𝑜𝑡𝑒 ∗
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑤𝑜𝑟𝑡ℎ 𝑜𝑓 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 𝑎𝑝𝑝𝑒𝑎𝑟𝑠 𝑎 𝑦𝑒𝑎𝑟
𝑏𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
PAMANTASAN NG LUNGSOD NG VALENZUELA
Tongco Street, Maysan, Valenzuela City
COLLEGE OF ENGINEERING AND INFORMATION TECHNOLOGY
Civil Engineering Department
Sample Problems
1. A student deposits ₱1,500.00 in a 9% account
today. He intends to deposit another ₱3,000.00 at the
end of 2 years. He plans to purchase in five years his
favorite shoes with ₱5,000.00. Calculate the money
that will be left in his account one year after the
purchase.
2. A man bought a lot worth ₱1,000,000.00 if paid in
cash. On the installment basis, he paid a down
payment of ₱200,000.00; ₱300,000.00 at the end of
one year; ₱400,000.00 at the end of three years and
a final payment at the end of five years. What was the
final payment if interest was 20% compounded
annually?
3. If ₱500.00 is invested at the end of each year for 6
years at an annual interest rate of 7%, what is the total
peso amount upon the deposit of the sixth payment?
4. A manufacturing firm wishes to give each 80
employee a holiday bonus. How much is needed to
invest monthly for a year at 12% nominal interest rate
compounded monthly so that each employee will
receive a ₱2,000.00 bonus?
5. If ₱500.00 is deposited in an account at the
beginning of each year for 6 years at an annual
interest rate of 7%, how much can be withdrawn after
6 years?
6. How much can you deposit now that will pay your
next 12 months of rent amounting to ₱3.000.00 each
if money is worth 7% compounded monthly?
7. A Civil Engineer loans ₱200,000.00 from a bank
with interest at 5% compounded annually. He agrees
to pay the obligations by paying 8 equal annual
payments, the first being due at the end of 10 years.
Find the annual payments.
8. Find the value after 20 years in pesos of an annuity
₱20,000.00 payable annually for 8 years, with the first
payment at the end of 2 years if money is worth 5%.
9. What is the present value of a perpetuity of ₱500.00
paid annually discounted back to the present at 8
percent?
10. A fund is to provide an annual scholarship at
₱4,000.00 for the first 5 years; ₱6,000.00 for the next
5 years and ₱9,000.00 thereafter. The fund will be
established 1 year before the first scholarship is
awarded. If the fund earns 12% interest, what sum
must be deposited?
Prepared by: Engr. Karl Megan S. Garcia
ENGINEERING ECONOMICS: EM5