Chapter 1 - No.02 Formula For Interest and Equivalents - Engineering Economy

EASTERN INTERNATIONAL
UNIVERSITY
School of Engineering
ENGINEERING ECONOMY
Lecturer: Thach Dung Chinh

Email: chinh.thach@eiu.edu.vn
MB: 0978104193
Chapter 1. Formula for Interest and Equivalents
Why?
Why do people say that one dollar today is worth more than one
dollar in the future?
Why do people believe that when they have money, businesses
should invest immediately? Because investing early helps
minimize the impact of inflation on the currency's value. Early
investment is the magical integration of money?
Objectives:
- Basic Concepts of Currency: Interest, Simple
Interest, and Compound Interest
- The Time Value of Money
- Applications of the Time Value of Money in
Practice.
Objectives:
1. Concept of Interest Rate.
2. Simple interest, compound interest.
3. Cash flow diagram.
4. Equivalent formulas for cash flow.
Time value of money
In financial terms, money at different points in time has different values due to:
 Opportunity cost of holding money;
 Inflation;
 Risk.
Using the time value of money to:
 Convert to equivalent value;
 Enable comparisons.
The time value of money is specifically characterized by two fundamental
concepts:
 Present value;
 Future value.
Time value of money
Economic equivalence
 Different amounts of money at different times
can be equal in economic value.
 At an interest rate of 10% per year, 1 million
today is equivalent to 1.1 million a year later.
 If you deposit P $ in a savings account today for
n periods at an interest rate of i, you will have F
(> P) $ at the end of period n."
Time value of money
 Interest Income: The amount of money that an
individual receives after a specific period from
the initial principal amount invested according
to a particular method, such as through
lending, is referred to as "Interest Income.“
 Interest Rate: It is the ratio of the interest
earned in one unit of time to the principal
amount during that period.
Interest Income
Interest Rate = x100%
Initial capital
Time value of money
Simple Interest: It is the amount of interest determined based on the
principal amount (initial investment) at a fixed interest rate. Calculating
interest in this manner is known as the simple interest method.
Simple interest is determined by the following formula:
I=P×i×n
Where:
I: Simple Interest.
P: Initial capital.
i: Interest rate.
n: Number of interest periods.
Time value of money
Example 1:
A savings deposit of 100 USD is made in a bank for 5 years at an interest
rate of 6% per year. Calculate the simple interest.
Solution:
Annual interest = 100 x 0.06 = $6

Time value of money
Present Future
Interest
Value
The value of 100 USD at the end of the fifth year is 130 USD
Time value of money
Compound Interest:
 It is the amount of interest determined based on the cumulative interest
from previous periods, which is added to the principal to serve as the
basis for calculating interest for the next period. Calculating interest in
this manner is known as the compound interest method.
 The total amount of principal and interest after n periods is:
P(1+i)n
Time value of money
Example 2:
Present Future
Interest
Value
106 = 100 + 100x6%

= 100(1+6%)
Time value of money
Calculate compound interest:
Present Future
Interest
Value
Time value of money
Present Future
Interest
Value
The value at the end of the fifth year = $133.82

Time value of money
Present Future
Interest
Value
The value at the end of the fifth year = $133.82

Real interest rate and nominal interest rate.
 Usually, the interest rate is used to calculate interest for a period of 1
year, also known as the interest accrual period of 1 year. In practice, the
interest accrual period may be less than 1 year.
 Example: Interest rate of 12% per year, compounded quarterly, with
interest paid once every 6 months.
 Interest accrual period: 1 year
 Compounding period: 1 quarter
 Interest payment period (calculation period): 6 months
 When the interest accrual period aligns with the compounding period, it
is the real interest rate (effective interest rate). If the interest accrual
period is different from the compounding period, it is the nominal
interest rate.
 Calculate the conversion of nominal interest rates for different periods
(months, quarters, years)
Example 3:
 Interest rate of 3% per quarter By default, the effective interest rate
per quarter is 3% (compounded quarterly).
 Nominal interest rate of 3% per quarter The nominal interest rate per
year is 3% * 4 = 12% per year.
 Interest rate of 20% per year, compounded quarterly Nominal interest
rate per year, compounded quarterly Nominal interest rate per quarter
= Effective interest rate per quarter = 5% per quarter.
 Calculate the conversion from nominal interest rate to effective interest
rate:
 Step 1: Convert from nominal interest rate to effective interest rate
during the compounding period.
 Step 2: Convert from effective interest rate during the compounding
period to effective interest rate during the calculation period.
Cash Flow
Concept of Cash Flow Diagram
 Cash Flow of the project (CF): Inflows and outflows
 Convention: Inflows and outflows occur at the end of each period.
 At each period: Net Cash Flow = Inflow - Outflow

Cash Flow
Symbols on the Cash Flow Diagram:
: Positive cash flow, income
: Negative cash flow, expenses
 P (Present Value): Present value, conventionally at some milestone (often
at the end of year 0, beginning of year 1 of the project)
 F (Future): Future value at some designated milestone (different from
point 0).
 A (Annual/Uniform value): Sequence of cash flows with equal values,
occurring at the end and continuously for a number of periods.
 n (Number): Number of periods (e.g., years, months, quarters, ...)
 i% (Interest): Interest or discount rate
Cash Flow
F (Future value)
F (Cash Inflow )
P (Present value) F (Cash outflow )
F (Future value)
A (Cash Inflow )
P (Present value) A (Cash outflow )

Future Value of Simple Interest
 Future Value: It is the value that can be
received at a specific point in the future,
including the principal amount and all the
interest accrued up to that time.
F=P(1+i n)
Where:
 F: Future value of the amount at the end of
the nth period.
 P: Principal amount (initial investment).
 i: Interest rate per period.
 n: Number of interest periods.
Future Value of Simple Interest
Example 4:
Mr. A lends 100 million VND. How much will Mr. A receive after one quarter?
(assuming an interest rate of 12% per annum, calculated using simple
interest).
Solution:
P = 100 million VND, i = 12%/ year, n = 1 quarter
F=P(1+i.n)=100(1+12%. )=103 million VND.
Future Value of Compound Interest
It is the value that can be received at a specific
point in the future, including the principal amount
and the entire compound interest (interest
reinvested into the principal for the next period)
calculated up to that time. F=P(1+i)n
Where:
P,i,n: As explained above.
(1+i)n : Represents the future value of 1 unit
after n periods with interest compounded each
period at the rate i. It is called the compound
interest factor and is denoted as (F/P,i,n).
So: F=P.(F/P,i,n)
Example 5:
A person deposits 100 million VND in a savings account for a term of 1 year, with
an interest rate of 10% per year. After 5 years, the person withdraws the principal
and interest. How much money will the person receive after 5 years?
Solution:
The amount the person can receive at the end of the 5th year is:
F5 = 100 x (1 + 10%)5 = 100 x (F/P,10%,5) = 100 x 1,611 = 161,1 (million VND)
Calculated of simple interest:
Link:
F5 = 100 x (1 + 10% x 5) = 150 (million VND)
The difference between compound interest and simple interest is:
161,1 – 150 = 11,1 (million VND)
Present Value is the initial value of the investment (initial capital).
P = F(1+i.)-n
Example 6:
A company discounts a promissory note worth 200 million VND at a bank with an
interest rate of 8% per year. This promissory note will mature in 4 years. Determine
the present value of the promissory note.
Solution:
F=200 million VND , i=8%/year, n=4 years
P = F(1+i.)-n = 200(1+8%)-4 = 147 million VND
Compound Interest
Compound Interest Multiple Times per Year
Interest is compounded m times per year. The amount an investor receives after n
years is given by:
Px(1 + )n.m
The amount of interest the investor receives is:
Px(1 + )n.m - P
P: Principal amount (initial investment).
r: Annual interest rate.
m: Number of times interest is compounded per year.
n: Number of years.
Compound Interest
Example 7:
A customer deposits 700 million VND in a bank for a period of 5 years. Determine
the amount the customer will receive after 5 years, knowing that interest is
compounded:
a. Monthly with an interest rate of 6% per year.
b. Quarterly with an interest rate of 6.5% per year.
c. Annually with an interest rate of 6.8% per year.
Compound Interest
Solution:
a. 700x(1+ )
% 5.12
= 944,195 million VND
b. 700x(1+ )
, % 5.4
c. 700x(1+ )
, % 5.1
Compound Interest
Continuous Compounding: The number of compounding periods per year
approaches infinity.
F = A.er.n
with e = 2,71828
Example 8: A person buys a 5-year bond with A=250 million VND, i=3% per year,
and interest is paid once at maturity (calculated with continuous compounding).
What is F?
Solution:
F = 250.e3%.5 = 290,459 million VND
Thank
you!

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EASTERN INTERNATIONAL

Lecturer: Thach Dung Chinh

Annual interest = 100 x 0.06 = $6

106 = 100 + 100x6%

The value at the end of the fifth year = $133.82

The value at the end of the fifth year = $133.82

 At each period: Net Cash Flow = Inflow - Outflow

P (Present value) F (Cash outflow )

P (Present value) A (Cash outflow )

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