Time Value of Money

Chapter 2: The Time Value of Money
Instructor: Fahim Muntaha

Outline
• Time value of Money
• Simple interest rate
• Compound interest rate
• Annuities
• Perpetuities
• Loan amortization
• Uneven cash flow stream

Learning Outcomes
• Will a clear concept of Time value of Money
• Will learn how to calculate Simple interest rate and
Compound interest rate
• Learn about Annuities and Perpetuities
• Learn to calculate and take decision from Loan amortization
schedule

Learning objective
How to draw time lines

Time lines
• Show the timing of cash flows.
• Tick marks occur at the end of periods, so Time
0 is today; Time 1 is the end of the first period
(year, month, etc.) or the beginning of the
second period.
CF0 CF1 CF3CF2
0 1 2 3
i%

Drawing time lines:
i. $100 lump sum due in 2 years;
ii. 3-year $100 ordinary annuity
100 100100
0 1 2 3
i%
ii. 3 year $100 ordinary annuity
100
0 1 2
i%
i. $100 lump sum due in 2 years

Drawing time lines:
Uneven cash flow stream; CF0 = -$50,
CF1 = $100, CF2 = $75, and CF3 = $50
100 5075
0 1 2 3
i%
-50
Uneven cash flow stream

Drawing time lines:
Uneven cash flow stream; CF0 = -$50,
CF1 = $100, CF2 = - $75, and CF3 = $50
100 50-75
0 1 2 3
i%
-50
Uneven positive and negative cash flow stream

Draw a timeline by yourself
Mr. X invested in his business AED 20,000 at the
beginning of the year and expecting that the
business will need to reinvest AED 3000 in the
next year. After that the business will generate
AED 5000 cash inflows at the end of each year.
But in the last year the cash inflow will be
10,000. Business will run for 6 years.
Draw a Time line by yourself.

Drawing time lines:
Uneven cash flow stream; CF0 = -AED 20,000,
CF1 = - 3000, CF2 = 5000, CF3 = 5000, CF4 = 5000,
CF5= 5000, and CF6= 5000,
-3000 10,0005000
0 1 2 6
i%
-20,000
Uneven positive and negative cash flow stream
5000 5000 5000
43
5

Learning objective
Understanding the concept
The Time value of Money

Definition
The Time Value of Money mathematics quantifies the
value of a dollar through time.
• A dollar on hand today is worth more than a dollar to
be received in the future because the dollar on hand
today can be invested to earn interest to yield more
than a dollar in the future.
• This depends upon the rate of return or interest rate
which can be earned on the investment.

Uses of the Time Value of Money
The Time Value of Money has applications in many
areas of Corporate Finance including:
• Capital Budgeting,
• Bond Valuation, and
• Stock Valuation.
• Loan amortization.
• Forecast investment decision.

Areas of Time Value of Money
The Time Value of Money concepts will be grouped into two
areas:
Future Value describes the process of finding what an
investment today will grow to in the future.
Present Value describes the process of determining what a cash
flow to be received in the future is worth in today's dollars.
FV = ?
0 1 2 310%
100
133
0 1 2 310%
PV = ?

Financial Equation to solve FV
FVn = PV ( 1 + i )n
Here,
FV = Future value
PV = Present Value
i= Interest rate per year
n = Number of years

What is the future value (FV) of an initial $100
after 3 years, if I/YR = 10%?
Compounding: Finding the FV of a cash flow or series of cash flows when
compound interest is applied is called compounding.
Compound Interest rate: Interest calculated on the initial principal and also
on the accumulated interest of previous periods of a deposit or loan.
Compound interest can be thought of as “interest on interest,” and will make
a deposit or loan grow at a faster rate than simple interest, which is interest
calculated only on the principal amount.
FV = ?
0 1 2 3
10%
100

Solving for FV:
The arithmetic method
• After 1 year:
– FV1 = PV ( 1 + i ) = $100 (1.10)
= $110.00
• After 2 years:
– FV2 = PV ( 1 + i )2 = $100 (1.10)2
=$121.00
• After 3 years:
– FV3 = PV ( 1 + i )3 = $100 (1.10)3
=$133.10
• After n years (general case):
– FVn = PV ( 1 + i )n

Others methods to solve FV
FV can be solved by using the
• arithmetic,
• financial calculator,
• Compounding and discounting table and
• spreadsheet methods.

Solving FV using Compounding Tables
Financial Equation is
FVn = PV ( 1 + i )n
Here,
FV = Future value
PV = Present Value
n = Number of years
Using compounding Table
FVn = PV X FVIFi,n
Here,
FV = Future value
PV = Present Value
n = Number of years
FVIFi,n = Future value interest
factor for i% n years

Use any of these equations
Using Financial Equation
FVn = PV ( 1 + i )n
FV3 = PV ( 1 + i )3
= $100 (1.10)3
=$133.10
Using Compounding table
FVn = PV X FVIFi,n
FV3 = $100 (1.3310)
=$133.10
What is the future value (FV) of an
initial $100 after 3 years, if I/YR = 10%?

Solve the Problems by yourself
1. You currently have $ 2,000 on hand that you plan to use to
purchase a three years bank certificate of deposit (CD). The
CD pays 4% interest rate annually.
a. How much money will you have when the CD matures?
b. What would be the FV if the CD will mature after 5 years?
c. What if the interest rate increased from 4% to 6% , but
the CD maturity was still three years?
Answers: $2,249.73, $2,433.31, $2,382.03.

What is Present Value
Present Value describes the process of determining what a cash
flow to be received in the future is worth in today's dollars. The
PV shows the value of cash flows in terms of today’s purchasing
power.
Discounting: Finding the PV of a cash flow or series of cash flows
when compound interest is applied is called discounting (the
reverse of compounding).
133
0 1 2 310%
PV = ?

Financial Equation to solve PV
FVn = PV ( 1+ i)n
Here,
FV = Future value
PV = Present Value
i= Interest rate per
year
n = Number of years
PV = FVn/( 1 + i )n
Or
PV = FVn X PVIFi, n
PVIFi, n= Present Value interest
factor for i% n years

What is the present value (PV) of $100 due in 3
years, if I/YR = 10%?
Using Financial Equation
PV = FVn / ( 1 + i )n
= $100 / ( 1 + 0.10 )3
= $75.13
Here,
FV = Future value
PV = Present Value
n = Number of years
Using Discounting table
PV = FVn X PVIFi, n
= $100 X 0.7513
= $75.13
PVIFi, n= Present Value interest
factor for i% n years = 0.7513
(derived from PV table)

Solve the Problems by yourself
1. What amount must pay today in a three year CD paying 4%
interest annually to provide you with $2,249.73 at the end of
CD maturity?
a. What would be the PV if the CD will mature after 5 years?
b. What if the interest rate increased from 4% to 6% , but
the CD maturity was still three years?
Answers: $2,000, $1,849.11, $1,888.92.

Learning objective
Understanding semiannual and other
compounding period

Use this formula
FVn = PV ( 1 + i/m )n X m
Here,
FV = Future value
PV = Present Value
n = Number of years
M = number of the month interest compounded in a year

What is the FV of $100 after 3 years
under 10% semiannual
compounding?
Quarterly compounding?

Solution
$134.49(1.025)$100FV
$134.01(1.05)$100FV
)
2
0.10
1($100FV
)
m
i
1(PVFV
12
3Q
6
3S
32
3S
nm
n






Here,
FV = Future value
PV = Present Value
n = Number of years
M = number of the month
interest compounded in a year

What’s the FV of a 3-year $100 annuity, if the
quoted interest rate is 10%, compounded
semiannually?
• Payments occur annually, but compounding
occurs every 6 months.
• Cannot use normal annuity valuation techniques.
0 1
100
2 3
5%
4 5
100 100
6
1 2 3

Monthly compound interest
1. Calculate the present value on Jan 1, 2011 of $1,500 to be
received on Dec 31, 2011. The market interest rate is 9%.
Compounding is done on monthly basis.
2. A friend of you has won a prize of $10,000 to be paid exactly
after 2 years. On the same day, he was offered $8,000 as a
consideration for his agreement to sell the right to receive the
prize. The market interest rate is 12% and the interest is
compounded on monthly basis. Help him by determining
whether the offer should be accepted or not.

Solution
1. We have,
Future Value FV = $1,500
Compounding Periods n = 12 Interest
Rate i = 9%/12 = 0.75%
Present Value
PV = $1,500 / ( 1 + 0.75% )^12
= $1,500 / 1.0075^12
= $1,500 / 1.093807
= $1,371.36
2. We have,
Future Value FV = $10,000
Compounding Periods n = 2 × 12 = 24
Interest Rate i = 12%/12 = 1%
Present Value
PV = $10,000 / ( 1 + 1% )^2X12)
= $10,000 / 1.01^24
= $10,000 / 1.269735
= $7,875.66
Since the present value of
the prize is less than the
amount offered, it is good to
accept the offer.

Learning objective
Understanding Annuities and Perpetuities
(EVEN CASH FLOW STREAM)

Annuity
An annuity is a series of equal dollar payments that
are made at the end or beginning of equidistant
points in time.
Characteristics
• An annuity is a series of payments
• It has fixed intervals of time.
• It has a limited life time.
• The payments (deposits) may be made weekly, monthly,
quarterly, yearly, or at any other interval of time.
• The valuation of an annuity entails concepts such as time value of
money, interest rate, and future value.

Example of Annuity
Examples of annuities are
• regular deposits to a savings account,
• monthly home mortgage payments,
• monthly insurance payments and
• pension payments etc.
Annuity
PMT PMTPMT
0 1 2 3
i%

Types of Annuity
• There are two types of annuity formulas.
1. Ordinary Annuity
2. Annuity Due:
We will derive the ordinary annuity formula first.

Understanding Ordinary Annuity

Ordinary Annuity
If a series of payment is made at the end od each payment
period then it is an Ordinary Annuity. One formula is based on
the payments being made at the end of the payment period. This
called ordinary annuity.
Ordinary Annuity
PMT PMTPMT
0 1 2 3
i%

10-40
$1000
$1000 (1.04)1n = 1
Sum = FV of annuity
0 1 2 3 4 Interval
number
$1000 $1000 $1000
$1000 (1.04)2
n = 2
$1000 (1.04)3
n = 3
…the sum of the future values of all the payments
Assume that there are four(4) annual $1000 payments
with interest at 4%
Future Value
of an
Ordinary Simple Annuity

• FVn
= FV of annuity at the end of nth period.
• PMT = annuity payment deposited or received at the end
of each period
• i = interest rate per period
• n = number of periods for which annuity will last
FV of an Ordinary Annuity

• Example 6.1 How much money will you
accumulate by the end of year 10 if you
deposit $3,000 each for the next ten years in a
savings account that earns 5% per year?

• FV = $3000 {[ (1+.05)10 - 1] ÷ (.05)}
= $3,000 { [0.63] ÷ (.05) }
= $3,000 {12.58}
= $37,740

Solving for PMT in an Ordinary Annuity
• Instead of figuring out how much money you
will accumulate (i.e. FV), you may like to know
how much you need to save each period (i.e.
PMT) in order to accumulate a certain amount
at the end of n years.
• In this case, we know the values of n, i, and
FVn in equation 6-1c and we need to
determine the value of PMT.

(cont.)
• Example 6.2: Suppose you would like to have
$25,000 saved 6 years from now to pay
towards your down payment on a new house.
If you are going to make equal annual end-of-
year payments to an investment account that
pays 7 percent, how big do these annual
payments need to be?

(cont.)
Here we know,
• FVn
= $25,000;
• n = 6; and
• i=7% and
• we need to determine PMT.

(cont.)
• $25,000 = PMT {[ (1+.07)6 - 1] ÷ (.07)}
= PMT{ [.50] ÷ (.07) }
= PMT {7.153}
$25,000 ÷ 7.153 = PMT = $3,495.03

You save $500 every month for the next
three years. Assume your savings can earn
3% converted monthly. Determine the total
in your account three years from now.
18810.28
Careful about effective Interest
Rate
Solve the FV of Ordinary annuity by yourself

You vow to save $500/month for the
next four months, with your first
deposit one month from today. If
your savings can earn 3% converted
monthly, determine the total in your
account four months from now
2007.51
Solve the FV of Ordinary annuity by yourself

The Present Value of an Ordinary
Annuity
• The present value of an ordinary annuity
measures the value today of a stream of cash
flows occurring in the future.

Annuity (cont.)
• Figure 6-2 shows the lump sum equivalent
($2,106.18) of receiving $500 per year for the
next five years at an interest rate of 6%.
• If you don’t know annuity formula you can
use simple PV formula and Time line.
PV = FVn/( 1 + i )n = FVn ( 1 + i )-n

If you don’t know annuity formula you can use
simple PV formula.

The Formula of Present Value of an
Ordinary Annuity (cont.)
Here
• PMT = annuity payment deposited or received at the end
of each period.
• i = discount rate (or interest rate) on a per period basis.
• n = number of periods for which the annuity will last.

Practice yourself using Ordinary
annuity formula
• Assume that there are four(4) annual $1000
payments with interest at 4% annually
compounded.
According to formula
• PV = PMT [1-{1/(1+i)n}]/ i
=3629.90

Understanding Loan Amortization
schedule using Ordinary annuity
concept

Amortized Loans
• An amortized loan is a loan paid off in equal
payments – consequently, the loan payments
are an annuity.
• Examples: Home mortgage loans, Auto loans

Amortized Loans (cont.)
In an amortized loan,
• the present value can be thought of as the amount borrowed,
• n is the number of periods the loan lasts for,
• i is the interest rate per period,
• future value takes on zero because the loan will be paid of
after n periods, and
• payment is the loan payment that is made.

• Example 6.5 Suppose you plan to get a $9,000
loan from a furniture dealer at 18% annual
interest with annual payments that you will
pay off in over five years. What will your
annual payments be on this loan?

• Using FVOA or PVOA Formula solve the PMT
first
– N = 5
– i = 18.0
– PV = 9000
– FV = 0
– PMT = $2,878.00

The Loan Amortization Schedule
Year Amount Owed
on Principal
at the
Beginning of
the Year (1)
Annuity
Payment
(2)
Interest
Portion
of the
Annuity
(3) = (1) ×
18%
Repaymen
t of the
Principal
Portion of
the
Annuity
(4) =
(2) –(3)
Outstanding
Loan
Balance at
Year end,
After the
Annuity
Payment
(5)
=(1) – (4)
1 $2,878
2 $2,878
3 $2,878
4 $2,878
5 $2,878 $0.00

Year Amount Owed
on Principal
at the
Beginning of
the Year (1)
Annuity
Payment
(2)
Interest
Portion
of the
Annuity
(3) = (1) ×
18%
Repaymen
t of the
Principal
Portion of
the
Annuity
(4) =
(2) –(3)
Outstanding
Loan
Balance at
Year end,
After the
Annuity
Payment
(5)
=(1) – (4)
1 $9,000 $2,878 $1,620.00
2 $2,878
3 $2,878
4 $2,878
5 $2,878 $0.00

Year Amount Owed
on Principal
at the
Beginning of
the Year (1)
Annuity
Payment
(2)
Interest
Portion
of the
Annuity
(3) = (1) ×
18%
Repaymen
t of the
Principal
Portion of
the
Annuity
(4) =
(2) –(3)
Outstanding
Loan
Balance at
Year end,
After the
Annuity
Payment
(5)
=(1) – (4)
1 $9,000 $2,878 $1,620.00 $1,258.00
2 $2,878
3 $2,878
4 $2,878
5 $2,878 $0.00

Year Amount Owed
on Principal
at the
Beginning of
the Year (1)
Annuity
Payment
(2)
Interest
Portion
of the
Annuity
(3) = (1) ×
18%
Repaymen
t of the
Principal
Portion of
the
Annuity
(4) =
(2) –(3)
Outstanding
Loan
Balance at
Year end,
After the
Annuity
Payment
(5)
=(1) – (4)
1 $9,000 $2,878 $1,620.00 $1,258.00 $7,742.00
2 $7,742.00 $2,878
3 $2,878
4 $2,878
5 $2,878 $0.00

Year Amount Owed
on Principal
at the
Beginning of
the Year (1)
Annuity
Payment
(2)
Interest
Portion
of the
Annuity
(3) = (1) ×
18%
Repaymen
t of the
Principal
Portion of
the
Annuity
(4) =
(2) –(3)
Outstanding
Loan
Balance at
Year end,
After the
Annuity
Payment
(5)
=(1) – (4)
1 $9,000 $2,878 $1,620.00 $1,258.00 $7,742.00
2 $7,742 $2,878 $1,393.56 $1,484.44 $6,257.56
3 $6257.56 $2,878 $1,126.36 $1,751.64 $4,505.92
4 $4,505.92 $2,878 $811.07 $2,066.93 $2,438.98
5 $2,438.98 $2,878 $439.02 $2,438.98 $0.00

(cont.)
• We can observe the following from the table:
– Size of each payment remains the same.
– However, Interest payment declines each year as
the amount owed declines and more of the
principal is repaid.

Annuity (cont.)
• Note , it is important that “n” and “i” match. If periods are
expressed in terms of number of monthly payments, the
interest rate must be expressed in terms of the interest rate
per month.

You overhear your friend saying the he is repaying a
loan at $450 every month for the next nine months.
The interest rate he has been charged is 12%
compounded monthly.
i. Calculate the amount of the loan,
ii. Show Monthly Loan Amortization Schedule
iii. and the amount of interest involved.
PV= , Repaid 4,050.00, i = 131.76
Class work/ HW

Amortized Loans with Monthly
Payments
• Many loans such as auto and home loans
require monthly payments. This requires
converting n to number of months and
computing the monthly interest rate.

Payments (cont.)
Example 6.6 You have just found the perfect
home. However, in order to buy it, you will need
to take out a $300,000, 30-year mortgage at an
annual rate of 6 percent. What will your monthly
mortgage payments be?

Payments (cont.)
• Mathematical Formula
• Here annual interest rate = .06, number of
years = 30, m=12, PV = $300,000

Payments (cont.)
$300,000= PMT
$300,000 = PMT [166.79]
$300,000 ÷ 166.79 = PMT
$1798.67 = PMT
1- 1/(1+.06/12)360
.06/12

Annuity Due
• Annuity due is an annuity in which all the cash flows occur at
the beginning of the period. For example, rent payments on
apartments are typically annuity due as rent is paid at the
beginning of the month.
PMT PMT
0 1 2 3
i%
PMT
Annuity Due

Example of Annuity due
If you lease equipment, a vehicle, or
rent property, the typical lease contract
requires payments at the
beginning of each period of
coverage

13 - 76
“Payments…in advance”
“First payment … made today”
“Payments at the beginning of each…..”
“Payments ….. starting now”
Clues to help identify annuities due

What is the difference between an ordinary
annuity and an annuity due?
Ordinary Annuity
PMT PMTPMT
0 1 2 3
i%
PMT PMT
0 1 2 3
i%
PMT
Annuity Due

13 - 78
FVdue = PMT (1+ i)n - 1[ i
]Formula * (1+ i)
PVdue = PMT 1-(1+ i)-n
[ i
]Formula * (1+ i)

Annuities Due: Future Value
• Computation of future value of an annuity due
requires compounding the cash flows for one
additional period, beyond an ordinary annuity.

Annuities Due: Future Value (cont.)
• Recall Example 6.1 where we calculated the
future value of 10-year ordinary annuity of
$3,000 earning 5 per cent to be $37,734.
• What will be the future value if the deposits of
$3,000 were made at the beginning of the
year i.e. the cash flows were annuity due?

Annuities Due: Future Value (cont.)
• FV = $3000 {[ (1+.05)10 - 1] ÷ (.05)} (1.05)
= $3,000 { [0.63] ÷ (.05) } (1.05)
= $3,000 {12.58}(1.05)
= $39,620

Annuities Due: Present Value
• Since with annuity due, each cash flow is
received one year earlier, its present value will
be discounted back for one less period.

Annuities Due: Present Value (cont.)
• Recall checkpoint 6.2 Check yourself problem
where we computed the PV of 10-year
ordinary annuity of $10,000 at a 10 percent
discount rate to be equal to $61,446.
• What will be the present value if $10,000 is
received at the beginning of each year i.e. the
cash flows were annuity due?

Annuities Due: Present Value (cont.)
• PV = $10,000 {[1-(1/(1.10)10] ÷ (.10)} (1.1)
= $10,000 {[ 0.6144] ÷ (.10)}(1.1)
= $10,000 {6.144) (1.1)
= $ 67,590

future value and present value of an annuity
due are larger than that of an ordinary annuity
• The examples illustrate that both the future
value and present value of an annuity due are
larger than that of an ordinary annuity
because, in each case, all payments are
received or paid earlier.

Calculate PV of ANNUITY DUE by
yourself
• 3 year annuity due of $100 at 10%
• 364.10

Learning objective
Understanding Perpetuity

Perpetuities
• A perpetuity is an annuity that continues
forever or has no maturity. For example, a
dividend stream on a share of preferred stock.
There are two basic types of perpetuities:
– Growing perpetuity in which cash flows grow at a
constant rate, g, from period to period.
– Level perpetuity in which the payments are
constant rate from period to period.

Other Perpetuity Examples
• British Consol Bonds
• Canadian Pacific 4%
Perpetual Bonds
• Endowments
– How much can I
withdraw annually
without invading
principal?

Present Value of a Level Perpetuity
• PV = the present value of a level perpetuity
• PMT = the constant dollar amount provided by
the perpetuity
• i = the interest (or discount) rate per period

Present Value of a Growing Perpetuity
• In growing perpetuities, the periodic cash
flows grow at a constant rate each period.
• The present value of a growing perpetuity can
be calculated using a simple mathematical
equation.

Present Value of a Growing Perpetuity
(cont.)
• PV = Present value of a growing perpetuity
• PMTperiod 1 = Payment made at the end of first period
• i = rate of interest used to discount the growing perpetuity’s
cash flows
• g = the rate of growth in the payment of cash flows from
period to period

Growing Perpetuity Example
• Suppose the initial payment is $100 and that this
grows at 3% per year while the discount rate is 5%
• The value of this growing perpetuity is:
000,5$
03.05.
100





gi
PMT
PV

Other Growing Perpetuity Examples
• Stock price = present value
of growing dividend stream

FV & PV of lump sum Payment
• FVn = PV ( 1 + i )n
• FVn = PV ( 1 + i/m )n X m
• PV = FVn/( 1 + i )n
• PV =FVn/( 1 + i/m)nxm

Annuities
Ordinary Annuity Annuity Due

Perpetuity
Level Perpetuity Growing Perpetuity

Learning objective
Understanding uneven cash flow stream

Complex Cash Flow Streams
• The cash flows streams in the business world
may not always involve one type of cash flows.
The cash flows may have a mixed pattern. For
example, different cash flow amounts mixed in
with annuities.
• For example, figure 6-4 summarizes the cash
flows for Marriott.

Year Cash Flow
1 500
2 200
3 -400
4 500
5 500
6 500
7 500
8 500
9 500
10 500
Uneven cash flow problem
• i= 6%
• Find out the
present value of
this project.

Year Cash Flow
1 500
2 200
3 -400
4 500
5 500
6 500
7 500
8 500
9 500
10 500
Split the problem into different parts
Uneven cash
flows
Negative Cash flow
Annuity
OA?
Or
AD?

Year Cash Flow
1 500
2 200
Try to solve step by step
Uneven cash
flows
Find the PV of each cash flow individually

Year Cash Flow
1 500
2 200
Uneven cash
flows
Find the PV of each cash flow individually
PV = FVn/( 1 + i )n
= 500/(1+ 0.06) 1 =?
= 300/(1+ 0.06) 2 =?

Year Cash Flow
3 -400
Negative Cash flow
Find the PV of this negative cash flow

Year Cash Flow
3 -400
Negative Cash flow
Find the PV of this negative cash flow
PV = FVn/( 1 + i )n
= -400/(1+ 0.06) 3 =?

Year Cash Flow
4 500
5 500
6 500
7 500
8 500
9 500
10 500
Annuity
OA?
Or
AD?
Find the PV of this Annuity

Year Cash Flow
4 500
5 500
6 500
7 500
8 500
9 500
10 500
Annuity
Ordinary
Annuity
Find the PV of this Annuity
= 500{ [1- 1/ (1+0.06)7 ] / 0.06 }
= ?

Complex Cash Flow Streams (cont.)

Discount the present value of ordinary annuity
back three years to the present.
PV = FVn/( 1 + i )n = 2791 / (1+0.06) 3
=?

Complex Cash Flow Streams (cont.)
• In this case, we can find the present value of the project by
summing up all the individual cash flows by proceeding in four
steps:
1. Find the present value of individual cash flows in years 1, 2, and 3.
2. Find the present value of ordinary annuity cash flow stream from
years 4 through 10.
3. Discount the present value of ordinary annuity (step 2) back three
years to the present.
4. Add present values from step 1 and step 3.

Julie Miller will receive the set of cash flows
below. What is the Present Value at a
discount rate of 10%.
Mixed Flows Example using tables
0 1 2 3 4 5
$600 $600 $400 $400 $100
PV0
10%

“Group-At-A-Time”
0 1 2 3 4 5
$600 $600 $400 $400 $100
10%
$1,041.60
$ 573.57
$ 62.10
$1,677.27 = PV0 of Mixed Flow [Using Tables]
$600(PVIFA10%,2) = $600(1.736) = $1,041.60
$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57
$100 (PVIF10%,5) = $100 (0.621) = $62.10

Do it by yourself
Year Cash Flows A Cash Flows B
1 100 300
2 400 400
3 400 400
4 400 400
5 300 100
Suppose you have two
investment options A
and B. Interest rate is 8%
compounded annually.
Find the Present value of
these cash flows.
Which one would you
prefer and why?

Summary
• Time value of Money
• Simple interest rate
• Compound interest rate
• Annuities
• Perpetuities
• Loan amortization
QUESTION?

Remarks
• I, r, k can be used to represent interest rate
• T, n can be used to represent time
• PVOA = Present value of an ordinary annuity
• FVOA =Future value of an ordinary annuity
• PVAD = Present value of annuity due
• FVAD = Future value of annuity due

Suggestions
i. What do you understand about FV and compounding?
ii. What do you understand about PV and Discounting?
iii. What is annuity? Describe the features of annuity.
iv. What are different types of annuity?
v. Differentiate between ordinary annuity and annuity due.
vi. What is perpetuity?
vii. What is loan amortization schedule? How this schedule can
help us? (answer by yourself)

Again
• Assign home work
• Fix a date for catch-up class.
• Fix a date for Class test on this chapter

Thank you!
@EICabudhabiEuropean International College-AbuDhabiEIC Abu Dhabi

Time Value of Money

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