The Time Value of
Money
Interest: The Cost of Money
 Interest
is defined as the amount of money paid for
the use of borrowed capital for a certain
period of time.
 Capital
refers to wealth in the form of money or
property that can be used to produce more
wealth
Concept of Interest:
 Lender
= income for derived for letting other use
his capital or resources
 Borrower
= expenses for using someone’s capital or
resources
Elements of Transactions involving
Interest
 Principal
(P)
amount of money in transactions involving
debts or investments
 Interest
Rate (i)
measure the cost or price of money
expressed as percentage per period of time
continuation…
 Interest
period
specified length of time that marks the
duration of the transactions
 No.
of interest period
determines how frequently interest is
calculated
continuation..
 Plan
for Receipts/ Disbursements (A)
cash flow pattern over a specified length
of time to pay off debts or investments
 Future
Amount of Money (F)
cumulative effect of interest rate over the
number of interest period
equal to the sum of principal and the total
interest earned.
Simple Interest
 directly
proportional to the length of time
the amount or principal is borrowed
I = Pin
F = P+I
= P + Pin
= P(1 + in)
I = total interest earned
by the Principal
P = amount of principal
i = interest rate
n = no. of interest
periods
F = total amount to be
paid after the interest
period
Types of Simple Interest
 Ordinary
Simple Interest
computed based on one banker’s year (360
days; 12months each consist of 30 days)
 Exact
Simple Interest
computed based on the exact numbers of
days in a year (365 days for regular year
and 366 days for leap year)
d
Ordinary simple = Pi
360
d
Exact Simple = Pi
(regular year)
365
d
= Pi
(leap year)
366
Sample Problem 3.1
 Determine
the ordinary simple interest on
Php10,000 for 9 months and 10 days if the
rate of interest is 12%.
Problem Details
P = Php10,000
i = 12% ordinary
simple
n = 9 mos. & 10days
Ans. I = Php933.33
Sample Problem 3.2
A
bank charges 12% simple interest on a
Php300,000 loan. How much will be repaid
if the loan is paid back in one lump-sum
amount after three years?
Sample Problem 3.2
A
bank charges 12% simple interest on a
Php300,000 loan. How much will be repaid
if the loan is paid back in one lump-sum
amount after three years?
Ans. F3= Php408,000
Sample Problem 3.3
 If
you borrowed money from your friend
with simple interest of 12%, find the
present worth of Php50,000 which is due at
the end of 7 months
Sample Problem 3.3
 If
you borrowed money from your friend
with simple interest of 12%, find the
present worth of Php50,000 which is due at
the end of 7 months
Ans. P0 = Php46,728.971….
Sample Problem 3.4
A
man borrows Php10,000 from a loan firm.
The rate of simple interest is 15%, but the
interest is to be deducted from the loan at
the time the money is borrowed. At the
end of one year, he has to pay back
Php10,000. What is the actual rate of
interest?
Sample Problem 3.4
A
man borrows Php10,000 from a loan firm.
The rate of simple interest is 15%, but the
interest is to be deducted from the loan at
the time the money is borrowed. At the
end of one year, he has to pay back
Php10,000. What is the actual rate of
interest?
Ans. i = 17.6470…..%
Compound Interest
 the
interest earned by the principal is not
paid at the end of each interest period, but
is considered as added to the principal, and
therefore will also earn interest for the
succeeding periods
F = P (1 + i )
n
where: (1+i)n = Single Payment Compound Amount Factor
(SPCAF)
Sample Problem 3.5
 The
amount of Php50,000 was deposited in
the bank earning an interest of 7.5% per
annum. Determine the total amount at the
end of 5 years, if the principal and interest
were not withdrawn during the period
Sample Problem 3.5
 The
amount of Php50,000 was deposited in
the bank earning an interest of 7.5% per
annum. Determine the total amount at the
end of 5 years, if the principal and interest
were not withdrawn during the period
Ans. F5 = Php71,781.47
Sample Problem 3.6
 Find
the present worth of a payment of Php30,000
to be made in five years with an interest rate of
8% per annum.
Sample Problem 3.6
 Find
the present worth of a payment of
Php30,000 to be made in five years with an
interest rate of 8% per annum
Ans. P0 = Php20,417.495…..
Sample Problem 3.7
A
sum of Php1,000 is invested now and left
for eight years, at which time the principal
is withdrawn. The interest that has
accrued is left for another eight years. If
the effective annual interest rate is 5%,
what will be the withdrawal amount at the
end of the 16th year?
Sample Problem 3.7
A
sum of Php1,000 is invested now and left
for eight years, at which time the principal
is withdrawn. The interest that has
accrued is left for another eight years. If
the effective annual interest rate is 5%,
what will be the withdrawal amount at the
end of the 16th year?
Ans. F16 = Php705.42
Sample Problem 3.8
A
student plan to deposit Php1,500 in the
bank now and another Php3,000 for the
next two years. If he plans to withdraw
Php5,000 three years after his last deposit
for the purpose of buying shoes, what will
be the amount of money left in the bank
after one year of his withdrawal? Effective
annual interest rate is 10%.
Sample Problem 3.8
A
student plan to deposit Php1,500 in the
bank now and another Php3,000 for the
next two years. If he plans to withdraw
Php5,000 three years after his last deposit
for the purpose of buying shoes, what will
be the amount of money left in the bank
after one year of his withdrawal? Effective
annual interest rate is 10%.
Ans. F6 = Php1,549.6415…
Sample Problem 3.9
 How
long will it take money to triple itself
if invested at 8% compounded annually?
Sample Problem 3.9
 How
long will it take money to triple itself
if invested at 8% compounded annually?
Ans. n = 14.27 years
Nominal Rate of Interest vs. Effective Rate
of Interest
 nominal
rate of interest
statement of rate of interest and the number
of interest periods per year.
 effective
rate of interest
actual rate of interest on the principal for one
year
Relationship between Nominal & Effective
Rate of Interest
M
r 

i = 1 +  − 1
 M
where: M = is the number of compounding periods per
year
Sample Problem 3.10
 Calculate
the effective rate corresponding
to each of the following rates:
a.
10% compounded semi-annually
b.
10% compounded quarterly
c.
10% compounded bi-monthly
d.
10% compounded monthly
Sample Problem 3.10
 Calculate
the effective rate corresponding
to each of the following rates:
a.
10% compounded semi-annually
b.
10% compounded quarterly
c.
10% compounded bi-monthly
d.
10% compounded monthly
Ans. a.) i = 10.25%
b) i = 10.38%
c.) i = 10.43%
d.) i = 10.47%
Sample Problem 3.11
 Find
the future worth of an investment
worth Php30,500 after 10 years with an
interest rate of 12% compounded quarterly
Sample Problem 3.11
 Find
the future worth of an investment
worth Php30,500 after 10 years with an
interest rate of 12% compounded quarterly
Ans. F10 = Php99,492.15266
Sample Problem 3.12
 How
many years is required for Php2,000 to
increase by Php3,000 if the interest rate is
12% compounded semi-annually?
Sample Problem 3.12
 How
many years is required for Php2,000 to
increase by Php3,000 if the interest rate is
12% compounded semi-annually?
Ans. n = 7.862…..years