Module 2
INTEREST & MONEY – TIME
RELATIONSHIPS
Engr. Gerard Ang
School of EECE
Simple Interest
 Interest – is the return on capital
or cost of using capital. It is the
amount of money paid for the
use of borrowed capital or the
income produced by money,
which has been loaned.
 Simple Interest – is calculated
using the principal only, ignoring
any interest that had been
accrued in preceding period.
𝐈 = 𝐏𝐢𝐧
𝐅=𝐏+𝐈
𝐅 = 𝐏(𝟏 + 𝐢𝐧)
Where:
I = interest
i = rate of interest
n = number of interest period
P = principal or present worth
F = accumulated amount or future worth
Types of Simple Interest
1.
Ordinary Simple Interest – simple interest in which it is
assumed that each month contains 30 days and
consequently each year has 360 days.
1 month = 30 days
1 year = 360 days (banker’s year)
2.
Exact Simple Interest – simple interest in which the
exact number of days per month is used.
1 ordinary year = 365 days
1 leap year = 366 days
Sample Problems on
Simple Interest
1.
2.
3.
4.
5.
A loan of Php50,000 is made for a period of 13 months, from
April 1 to April 30 of the following year, at a simple interest rate
of 20%. What future amount is due at the end of the loan
period?
What is the principal amount if the amount of interest at the
end of 2 ½ years is Php450 for a simple interest rate of 6% per
annum?
Determine the exact simple interest of Php25,000 for the
period of December 27, 2002 to March 23, 2003, if the rate of
interest is 10%.
What is the interest due on a Php1,500 loan for 4 years and
three months if it bears 12% ordinary simple interest?
Determine the exact simple interest of Php4,000 for the period
of Feb. 14, 1984 to November 30, 1984, if the rate of interest
is 18%.
Compound Interest
Compound Interest – the
interest for an interest
period is calculate on the
principal plus total amount
of interest accumulated in
the
previous
period.
Compound interest means
“the interest on top of
interest.”
𝐅 = 𝐏(𝟏
+ 𝐢)𝐧
𝐏 = 𝐅(𝟏 + 𝐢)−𝐧
𝐫
𝐢=
𝐦
𝐧=𝐦×𝐭
Where:
I = interest
i = rate of interest
n = number of interest periods
m = number of compounding periods
t = time
P = present worth
F = future worth
Rates of Interest

Rate of Interest – it is the cost of borrowing money.

Nominal Rate of Interest – it specifies the rate of interest and a
number of interest periods in one year.
Where:
i = interest rate per interest period
r = nominal interest rate
m = number of compounding periods
𝐫
𝐢=
𝐦

Effective Rate of Interest – it is the actual or exact rate of
interest on the principal during one year.
𝐄𝐑 = (𝟏 + 𝐢)𝐦 −𝟏
𝐫
𝐄𝐑 = 𝟏 +
𝐦
𝐦
−𝟏
Where:
ER = effective rate
i = interest rate per interest period
r = nominal interest rate
m = number of compounding periods
Values of “m”







m = 1 for compounded annually (every 12 months)
m = 2 for compounded semi-annually (every 6 months)
m = 4 for compounded quarterly (every 3 months)
m = 6 for compounded bi-monthly (every 2 months)
m = 8 for compounded semi-quarterly (every 1 1/2 months)
m = 12 for compounded monthly (every month)
m = 24 for compounded semi-monthly (every 1/2 month)
F/P and P/F Factors:
Notation and Equations
Find/Given
Standard
Notation
Equation
Equation
with Factor
Formula
Excel
Functions
(F/P,i,n)
Singlepayment
compound
amount
F/P
F = P(F/P,i,n)
F = P(1 + i)n
FV(i%,n,,P)
(P/F,i,n)
Singlepayment
present
worth
P/F
P = F(P/F,i,n)
P = F(1 + i)-n
PV(i%,n,,F)
Factor
Notation
Factor
Name
Sample Problems on
Compound Interest
1.
2.
3.
4.
5.
What rate of interest compounded annually must be received if
an investment of Php5,400 made now will result in a receipt of
Php7,200 5 years hence?
What amount will be accumulated by Php4,100 in 10 years at 6%
compounded annually?
What effective annual interest rate corresponds to the following
situations?
a.nominal interest rate of 10% compounded semi-annually
b.nominal interest rate of 6% compounded monthly
c.nominal interest rate of 8% compounded quarterly
How much should Engr. Cruz deposit now, if after 10 years, this
will amount to Php100,000. Interest rate is 12% compounded
semiannually?
If Php1,000 becomes Php5,734 after 15 years, when invested at
an unknown rate of interest compounded semi-annually,
determine the unknown nominal rate and corresponding effective
rate.
Cash Flow Diagram
 Cash Flow Diagram – depicts the timing and
amount of expenses (negative, downward) and
revenues (positive, upward) for engineering
projects.
Receipt (positive cash
flow or cash inflow
Disbursement (negative
cash flow or cash outflow
Types of Cash Flow Diagrams
P-Pattern
F-Pattern
A-Pattern
G-Pattern
“present”
1
2
3
n
“future”
1
2
3
n
“annual”
1
2
3
n
“gradient”
1
2
3
n
Equation of Value
An equation of value is obtained by setting
the sum of the values on a certain
comparison or local date (or focal date) of
one set of obligations equal to the sum of
the values on the same date of another set
of obligations.
Sample Problems on
Equation of Value
1.
2.
Jay wishes his son, Jason, to receive
Php1,000,000 twenty years from now. What
amount should he invest now, if it will earn interest
of 12% compounded annually during the first five
years and 10% compounded monthly for the
remaining years.
Find the present worth of a future payment of
Php300,000 to be made in 10 years with an
interest rate of 10% compounded annually. What
will be the amount if it will be paid 5 years later (on
the 15th year)?
Discrete Payments
The solution of discrete payments or number of
transactions occurring at different periods is taking each
transaction to the base year and equating each value.
Steps in Solving Discrete Payments:
1. Draw the cash flow diagram.
2. Select any convenient focal date.
3. Years on the left of the focal date have a positive sign
while years on the right of the focal date have a
negative sign.
4. Use the principle: Cash Inflow = Cash Outflow
Sample Problems on
Discrete Payments
1.
2.
3.
Acosta Holdings borrowed Php9,000 from Smith Corporation on
January 1, 1998 and Php12,000 on January 1, 2000. Acosta
Holdings made a partial payment of Php7,000 on January 1,
2001. It was agreed that the balance of the loan would be
amortized by two payments, one on January 1, 2002 and one
on January 1, 2003, the second being 50% larger than the first.
If the interest rate is 12%, what is the amount of each payment?
A contract has been signed to lease a restaurant at Php20,000
per year with annual increase of Php1,500 for 8 years.
Payments are to be made at the end of each year, starting one
year from now. The prevailing rate is 7%. What lump sum paid
today would be equivalent to the 8 year lease program?
Mr. Cruz buys a second hand car worth Php150,000 if paid in
cash. On installment basis, he pays Php50,000 downpayment,
Php30,000 at the end of one year, Php40,000 at the end of two
years and a final payment at the end of four years. Find the final
payment if interest is 14%.
Continuous
Compounding Interest
The solution for interest compounded continuously
can be derived thru differential equations and can be
found as:
𝐝𝐏
= 𝐢𝐏
𝐝𝐭
Where:
i = interest rate compounded continuously
P = present worth
t = time
Sample Problems on
Continuous Compounding Interest
1.
2.
Philip invested $100 on a bank. The bank offers
5% interest compounded continuously in a
savings account. Determine (a) how long will it
require for him to earn $5 (b) the equivalent
simple interest rate for 1 year of the bank.
Which is more advisable to invest Php5,000 for
five (5) years, to bank A that offers 5%
compounded continuously or to bank B that
offers 10% simple interest?
Banker’s Discount
Certain banks lend money in such a way that they
deduct the interest on the money. They actually don’t
lend you money you asked for. This type of computing
money is called banker’s discount. The money received
by the borrower after the discount has been deducted is
called proceeds.
𝟏 − (𝟏 + 𝐧𝐢)−𝐧
𝐝=
𝐧
Where:
i = rate of interest
d = rate of discount
n = number of interest period
Sample Problems on
Banker’s Discount
1.
Ms. Glydel Marquez borrowed money from a bank.
She received from the bank Php1,342 and
promised to repay Php1,500 at the end of 9
months. Determine the following: (a) simple
interest rate (b) discount rate or often referred as
Banker’s discount.