Time Value of Money
Module 7
Time Value of Money
• Founded on the principle that money has
earning capacity.
• The peso that you receive today has a
greater value compared to the peso you
will receive in the future.
Illustration
• Student A receives Php 500.00 today.
Student A invests money in a time deposit with an earning capacity of 5% per
annum. Thus, by the end of 2019, Student A earns Php 25.00 from his
investment.
By the time, Student B receives his Php 500.00, Student A already earned Php
525.00 from his investment.
• Student B receives Php 500.00 in 2019.
Time line
• A graphical representation of the timing of
cash flows.
Period
0
Cash PV = Php 500.00
Where: PV = Present Value
FV = Future Value
1
2
3
FV = ?
Time line
• A graphical representation of the timing of
cash flows.
0
Beginning of
Period 1
1
Cash PV = Php 500.00
Where: PV = Present Value
FV = Future Value
2
3
FV = ?
Time line
• A graphical representation of the timing of
cash flows.
End of Period 1
0
Cash PV = Php 500.00
Where: PV = Present Value
FV = Future Value
1
2
3
FV = ?
Time line
• A graphical representation of the timing of
cash flows.
0
Cash PV = Php 500.00
Where: PV = Present Value
FV = Future Value
1
Beginning of
Period 2
2
3
FV = ?
Time line
• A graphical representation of the timing of
cash flows.
End of Period 2
0
Cash PV = Php 500.00
Where: PV = Present Value
FV = Future Value
1
2
3
FV = ?
Simple vs Compound Interest
• Interest – amount earned when money is
loaned to someone else or is invested in a
financial product that promises earnings
after a certain period of time.
• Example
– Interest earned on money saved in a bank.
Simple Interest
• Occurs when there is no interest earned on
top of interest that was earned in the
previous periods.
A = Principal (1+rt)
Compound Interest
• Generating earnings (interest) from
previously generated earnings.
5 % or 0.05
Php 25.00
Php 500.00
Php 26.25
Php 525.00
Php 27.56
Php 551.25
Php 578.81
Future Value
• FVN = PV (1 + I)N
Where FV = future value
N = no. of periods
I = interest rate
PV = Amount of Outflow at time 0
Future Value
• FVN = PV (1 + I)N
• FV3 = Php 500 (1+0.05)3
= Php 500 (1.157625)
= Php 578.81
Present Value
• The current worth of a future sum of
money or stream of cash flow given a
specified rate of return.
• PV = FVn/(1+I)N
• FV = Php 578.81/(1.05)3
FV = Php 578.81/1.157625
FV = Php 499.99 ~ Php 500.00
When is it most appropriate to
compute for the Present Value?
• Suppose that a broker is trying to convince you
to invest in a bond that will pay Php 578.81 in
three years. Currently, banks are offering 5%
interest rate on 3-year time deposits. You now
have two options: buy the bond or buy the time
deposit. The 5% offered by the bank is your
opportunity cost.
• Opportunity Cost is the rate of return an
investor can earn on an alternative investment.
Solution
• Two Options
Bond offered by the Broker
that will pay you Php 578.81
in 3 years
Invest in a time deposit that
earns 5% in three years
In this case, need to know at what price should we pay for the bond?
First, determine the PV of Php 578.81.
If the broker offers the bond at a price lower than the PV of the bond
which is Php 500.00 then the investor may purchase the bond as this
will yield him a gain.
However, if the broker offers the bond at a price higher than the PV of
the bond then the investor may choose to invest his money in a time
deposit instead.
Solution
• Two Options
Bond offered by the Broker that will pay you Php 578.81 in 3 years
Selling the bond lower than the PV will yield a gain.
Example: Selling the bond at Php 450.00. Determine the FV of the bond.
FV = PV (1 + I)N
FV = 450 (1+0.05)3
FV = 520.93 ~ Php 521
Compare the FV of Php 500.00 versus the FV
of Php 450.00. It is clear that the investor will
yield a gain as the Php 450.00 bond will only
give him Php 521 but the broker promised
him a return of Php 578.81 in 3 years.
Solution
• Two Options
Bond offered by the Broker that will pay you Php 578.81 in 3 years
Selling the bond higher than the PV will yield a loss.
Example: Selling the bond at Php 550.00. Determine the FV of the bond.
FV = PV (1 + I)N
FV = 550 (1+0.05)3
FV = Php 636.70
Compare the FV of Php 500.00 versus the FV of
Php 550.00. Now, the investor will yield a loss
as the Php 550.00 bond will supposedly give
him more than what the broker promised him
of a return of Php 578.81 in 3 years.
Annuities
• An annuity is a series of equal payments
at fixed intervals for a specified number of
periods.
• Two Types of Annuity
– Ordinary Annuity (payment made at the end
of the period
– Annuity Due (payment made at the beginning
of the period)
Annuities
• Ordinary Annuity
5%
Php 300
Php 300
Php 300
Annuities
• Annuity Due
5%
Php 300
Php 300
Php 300
FV of Ordinary Annuity
•
•
•
•
FVAN = PMT [(1+I)N-1/I]
FVA3 = 300 [(1+0.05)3-1/0.05]
FVA3 = 300 (3.1525)
FVA3= Php 945.75
FV of Annuity Due
• FVAdue = FVAordinary (1+I)
• FVAdue = 945.75 (1.05)
• FVAdue = Php 993.04
Annuities
• Ordinary Annuity
– FVA3= Php 945.75
• Annuity Due
– FVAdue = Php 993.04
Annuities
• Ordinary Annuity
5%
1
Php 300
2
Php 300
Php 300
Annuities
• Annuity Due
5%
1
Php 300
2
Php 300
3
Php 300
Annuities
• Ordinary Annuity
– FVA3= Php 945.75
• Annuity Due
– FVAdue = Php 993.04
Present Value of Ordinary Annuity
• PVAN = PMT {1-[1/(1+I)N]/I}
• PVA3 = 300 {1-[1/(1+0.05)3/0.05}
• PVA3 = Php 817.20
Perpetuities
• A perpetuity is a stream of equal
payments at fixed intervals expected to
continue forever.
• PV of a perpetuity = Annual Payment/Rate per annum
Example = Php 450.00/0.03 or 3%
PV of a perpetuity = Php 15,000.00
• The PV of the perpetuity increases when the rate per
annum decreases.
Perpetuities
• The PV of the perpetuity increases when the rate per
annum decreases.
• PV of a perpetuity = Annual Payment/Rate per annum
Example = Php 450.00/0.02 or 2%
PV of a perpetuity = Php 22,500.00
Uneven Cash Flows
•
Annuity Plus Additional Final Payment
•
0
1
2
Php 0
Php 100
Php 100
•
3
Php 100
4
Php 100
5
Php 100 + Php 1000
Irregular Cash Flows
0
1
Php 0
Php 100
2
Php 300
3
Php 300
4
Php 300
5
Php 500
Uneven Cash Flows
• PV = CF1/(1+I)1 + CF2/(1+I)2…
• FV = CF1x(1+r)N-1 + CF1x(1+r)N-1 …
Compute for the Present Value of the uneven
cash flow stream below:
• 5 Periods, Interest is 12%
• Cash flows are as follow:
– Period 1 Php 100.00
– Period 2 Php 300.00
– Period 3 Php 300.00
– Period 4 Php 300.00
– Period 5 Php 500.00
Formula: PV = CF1/(1+I)1 + CF2/(1+I)2…
Present the PV of each period and what is the total PV
of the uneven cash flows after 5 periods?
Compute for the FV of the uneven cash flow
stream below:
• 5 Periods, Interest is 12%
• Cash flows are as follow:
– Period 1 Php 100.00
– Period 2 Php 300.00
– Period 3 Php 300.00
– Period 4 Php 300.00
– Period 5 Php 500.00
Formula: FV = CF1x(1+r)N-1 + CF1x(1+r)N-1 …
Present the FV of each period and what is the total FV
of the uneven cash flows after 5 periods?
Semiannual and Other Compounding Periods
FVN= PV (1+I)N
Given:
PV = Php 1,000.00
Interest = 6%
No. of Periods = 5 Years
a.
b.
•
Divide the interest rate by 2 (semiannual), making it 3%.
Multiply the number of periods by 2, making it 10.
FV10= 1000 (1+0.03)10
FV10= Php 1,343.92
If compounded annually, the amount will be Php
1,338.23 only.
Effects of Compounding on
Interest Rates
• The value of the interest rate stated in the contract will differ
if the interest rate is compounded several times annually.
–
Nominal, Quoted, or Stated Interest Rate (INOM) – the rate stated
in a contract.
–
Effective Annual Rate or Equivalent Annual Rate (EFF%) – is the
annual rate of interest actually being earned.
If a loan or an investment is compounded more than once a year, the
Effective Annual Rate will be higher than the Nominal Rate.
–
The annual percentage rate (APR) is the amount of interest on
your total loan amount that you'll pay annually. A lower APR
could translate to lower monthly payments.
Effective Annual Rate (EFF%)
• EFF% = [1 + (INOM/M)]M-1.0
• M is the number of compounding periods
per year.
• Therefore, a 10% nominal interest rate, if
compounded semi-annually, would be:
EFF% = (1+0.05)2 – 1.0
=1.1025 – 1.0
= 0.1025
EFF% =10.25%
• The annual rate of interest actually being
earned is not 10% but 10.25%.
Amortized Loans
• Loan amortization – the payment of a loan
in installments over a specified period of
time.
• An amortized loan is a loan that is to be
re-paid in equal payments over specified
period of time.
Illustration
• Daisy is considering applying for a home
improvement loan in the amount of Php
100,000.00 with an interest rate of 6%. The
term of the loan is 5 years. Equal
payments at the end of every year should
be made until the loan (including interest)
is paid off.
Amortized Loans
• Amortization Schedule – a table showing
payments to be made, the due dates, and
the breakdown of each payment – the
portion that goes to the principal and how
much goes to interest.
Illustration
• Daisy is considering applying for a home
improvement loan in the amount of Php
100,000.00 with an interest rate of 6%. The
term of the loan is 5 years. Equal payments at
the end of every year should be made until
the loan (including interest) is paid off.
Beginning Amount = Php 100,000.00
Interest Rate = 6%
Term/Period = 5 Years
Time Line of Payments
0
P100,000.00
1
PMT
2
PMT
3
PMT
4
PMT
5
PMT
The sum of the PVs of all PMTs should equal to Php
100,000.00
P100,000.00 = PMT/(1.06)1 + PMT/(1.06)2…
How to solve for the PMT per
period?
• PMT = P [r(1+r)n / (1+r)n-1]
PMT = annual amount per period
P = Principal or loan amount
r = Interest rate per period
n = total no. of payment or periods
PMT = 100,000 [0.06(1+0.06)5 / (1+0.06)5-1]
PMT = Php 23,739.64
Amortization Schedule
Year Beginning
Amount
Payment
Interest Repayment of
Principal
Ending
Balance
1
Php 100,000.00
23,739.64 6,000.00
17,739.64
82,260.36
2
82,260.36
23,739.64 4,935.62
18,804.02
63,456.34
3
63,456.34
23,739.64 3,807.38
19,932.26
43,524.08
4
43,524.08
23,739.64 2,611.44
21,128.20
22,395.89
5
22,395.89
23,739.64 1,343.75
22,395.89
0.00
•Amount of Interest is solve by determining the product between the
Beginning Amount and Interest Rate.
Php 100,000 x 6% = Php 6,000.00
82,260.36 x 6% = Php 4,935.62
•Repayment of Principal = PMT – Interest
•Ending Balance of the Period = Beginning Amount – Repayment of Principal
Next meeting Module 8 Sources and Uses of Funds
END