Chapter 4

Principles of
Finance
The Time Value of
Money
(Part 2)
 4- 2

Future Value of Multiple Payment Stream

You are an auto dealer you have two choice to sell a car take
$14,000 cash now, or receive three payments: $2,000 now
and $3,000 at the end of the first year $4,000 at the end of
second year and $5,000 at the end of third year. If interest rate
of money you can earn is 5%, which deal do you prefer?
 4- 3

Future Value of Multiple Payment Streams

With unequal periodic cash flows, treat

each of the cash flows as a lump sum and
calculate its future value over the relevant
number of periods.
Sum up the individual future values to get
the future value of the multiple payment
streams.
 4- 4

Topics Covered

Annuity Due

Perpetuity ‫أبدية‬

Future and Present Value of Annuity Due

Future and Present Value of Perpetuity

Applications
 4- 5

Annuity

Question Answer

What is annuity?? A series of equal cash flows at regular

periods.
What is ordinary annuity?? Cash flows at end of periods.

What is annuity due?? Cash flows at start of periods.

 4- 6

Future Value of Annuity Stream

Annuity ???
A series of equal cash flows at regular
interval across time is an annuity.

Brain Test
Equal Car Installments Yes
Electricity bill No
House rent monthly Yes
Monthly Gas bill No

Ordinary Annuity End of the period CF

Payments or receipt at the end of each period
 4- 7

Future Value of Annuity Stream

Toyota motors has following car sale plan.

Receive $5000 at the end of every year for five years, average investment rate in the
market is 10%. Calculate the future value of the plan?
 4- 8

Future Value of Annuity Stream

On sale of a laptop Dell have following receipt options
i. Receive cash payment of $3400
ii. Receive $1000 year-end payments for next 5 years,
Market investment rate is 5% per annum, evaluate which option is better?

FV $3400 (1 .05) 5

$4339
 4- 9

Future Value of Annuity Application

The formula for calculating the future value of an ordinary annuity stream is as
follows:

FV = CF X (1+r)n - 1
r
FVIFA
 4- 10

Future Value of Annuity Application cont.

Jill has been faithfully depositing $2,000 at the end of each year since the
past 10 years into an account that pays 8% per year. How much money
will she have accumulated in the account?

Use Formula

FV=$2000*[((1.08)^10 - 1)/.08]
= $28,973.13
Use Table
FV=$2000*14.4866
= $28,973.13
 4- 11

Ordinary Annuity Vs. Annuity Due

A cash flow stream such as rent, lease, and
insurance payments, which involves equal
periodic cash flows that begin right away or at
the beginning of each time interval is known as
an annuity due.
 4- 12

Future Value of Annuity Due

Add one more period to previous formula

FV Annuity Due > FV Ordinary Annuity

 4- 13

Future Value of Annuity Due cont.

Example – Retirement Plan
You plan to deposit your saving $3,000 at the start of
every year for 20 years and then retire. Given a 8%
rate of interest, what will be the FV of your retirement
account?
Formula

Table FV= $3000(45.7620) (1.08)

=$148,268.76
 4- 14

PV of Multiple Cash Flows

Example
Your auto dealer gives you the choice to pay $15,500 cash
now, or make three payments: $8,000 now and $4,000 at
the end of the following two years. If your cost of money
is 8%, which do you prefer?
Immediatepay ment 8,000.00
4 , 000
PV1 (1 .08)1
3,703.70
4 , 000
PV2 (1 .08) 2
3,429.36
Total PV $15,133.06
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 4- 15

PV of Multiple Cash Flows

PVs can be added together to evaluate

multiple cash flows.

C1 C2
PV (1 r ) 1
(1 r ) 2 ....
 4- 16

Present Value of Annuity Stream

You are purchasing a car. You are scheduled to make 5
annual installments of $5,000 per year. Given a rate of
interest of 10%, what is the price you are paying for the car
(i.e. what is the PV)?
 4- 17

Present Value of Annuity Stream cont.

1
1 n
1 r
PV PMT
r

1
1 PVIFA
1 r
n

Present Value Interest Factor of an Annuity:

The present value of $1 a year for each of t years
 4- 18

Present Value of Annuity Application cont.

1
1 n
1 r
PV PMT
r
 4- 19

Perpetuities & Annuities

A Perpetuity is an equal periodic cash flow

stream that will never end.

PV of Perpetuity Formula

PMT
PV r

PMT = Cash payment

r = interest rate
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 4- 20

Perpetuities & Annuities

Example - Perpetuity
In order to create an endowment fund, which pays
$100,000 per year, forever, how much money
must be set aside today in the rate of interest is
10%?

100 , 000
PV .10 $1,000,000

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 4- 21

Perpetuities & Annuities

Example - continued
If the first perpetuity payment will not be received
until three years from today, how much money
needs to be set aside today?

1, 000 , 000
PV (1 .10 ) 3
$751,315

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 4- 22

Example: Annuity
You are purchasing a car. You are scheduled to
make 3 annual installments of $4,000 per year.
Given a rate of interest of 10%, what is the price you
are paying for the car (i.e. what is the PV)?
1
1 n
1 r
PV PMT
r

PV 4 ,000 1
.10
1
.10 ( 1 .10 ) 3

PV $9,947.41
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 4- 23

Three Loan Payment Methods

Loan payments can be structured in one of 3
ways:
1) Discount loan
• Principal and interest is paid in lump sum at end
2) Interest-only loan
• Periodic interest-only payments, principal due at
end.
3) Amortized loan
• Equal periodic payments of principal and interest

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 4- 24

Three Loan Payment Methods (cont.)

Example: Discount versus Interest-only versus Amortized
loans

Abdullah wants to borrow $40,000 for a period of 5 years.

The lenders offers him a choice of three payment structures:
1) Pay all of the interest (10% per year) and principal in one lump sum at
the end of 5 years;
2) Pay interest at the rate of 10% per year for 4 years and then a final
payment of interest and principal at the end of the 5th year;
3) Pay 5 equal payments at the end of each year inclusive of interest and
part of the principal.

Under which of the three options will Abdullah pay the least interest and
why? Calculate the total amount of the payments and the amount of interest
paid under each alternative.

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 4- 25

Three Loan Payment Methods (cont.)

Abdullah wants to borrow $40,000 for a period of 5 years.
Method 1: Discount Loan.
Since all the interest and the principal is paid at the end of 5
years we can use the FV of a lump sum equation to calculate
the payment required, i.e.
FV = PV x (1 + r)n
FV5 = $40,000 x (1+0.10)5
= $40,000 x 1.61051
= $64, 420.40
Interest paid = Total payment - Loan amount
Interest paid = $64,420.40 - $40,000 = $24,420.40
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 4- 26

Three Loan Payment Methods (cont.)

Abdullah wants to borrow $40,000 for a period of 5 years.
Method 2: Interest-Only Loan.
Annual Interest Payment (Years 1-4)
= $40,000 x 0.10 = $4,000
Year 5 payment
= Annual interest payment + Principal payment
= $4,000 + $40,000 = $44,000
Total payment = $16,000 + $44,000 = $60,000
Interest paid = $20,000

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 4- 27

Three Loan Payment Methods (cont.)

Abdullah wants to borrow $40,000 for a
period of 5 yr.
Method 3: Amortized Loan.
n = 5; I = 10%; PV=$40,000
1
1 n
1 r
PV PMT
r
PMT = $10,551.86
Total payments = 5*$10,551.8 = $52,759.31
Interest paid = Total Payments - Loan Amount
= $52,759.31-$40,000
Interest paid = $12,759.31
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 4- 28

Three Loan Payment Methods (cont.)

Loan Type Total Payment Interest Paid

Discount Loan $64,420.40 $24,420.40

Interest-only Loan $60,000.00 $20,000.00
Amortized Loan $52,759.31 $12,759.31

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 4- 29

Amortization Schedules
Example: Loan amortization schedule.
Prepare a loan amortization schedule for the
amortized loan option given in example above.
What is the loan payoff amount at the end of 2
years?

PV = $40,000; n=5; i=10%; FV=0;

PMT = $10,551.89

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 4- 30

Amortization Schedules (cont.)

Year Beg. Bal Payment Interest Prin. Red End. Bal

1 40,000.00 10,551.89 4,000.00 6,551.89 33,448.11

2 33,448.11 10,551.89 3,344.81 7,207.08 26,241.03

3 26,241.03 10,551.89 2,624.10 7,927.79 18,313.24

4 18,313.24 10,551.89 1,831.32 8,720.57 9,592.67

5 9,592.67 10,551.89 959.27 9,592.67 0

The loan payoff amount at the end of 2 years is

$26,241.03

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 4- 31

Topics Covered

Problems
&

Cases

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 4- 32

Problem
Q-Sam Hinds, a local dentist, is going to remodel the dental
reception area and two new workstations. He has contacted
IKEA, and the new equipment and cabinetry will cost $18,000.
IKEA will finance the equipment purchase at 7.5% over a six-
year period of time. What will Hinds have to pay in annual
payments for this equipment?
1
1 n
1 r
PV PMT
r

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 4- 33

Problem
Q-The Stack has just written and recorded the single
greatest rock song ever made. The boys in the band believe
that the royalties from this song will pay the band a
handsome $200,000 every year forever. The record studio is
also convinced that the song will be a smash hit and that the
royalty estimate is accurate. The record studio wants to pay
the band upfront and not make any more payments for the
song. What should the record company offer the band if
they use
5% discount rate,
7.5% discount rate, or
10% discount rate?

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 4- 34

Problem
County Ranch Insurance Company wants to offer a
guaranteed annuity in units of $500, payable at the end of
each year for twenty-five years. The company has a strong
investment record and can consistently earn 7% on its
investments after taxes. If the company wants to make 1%
on this contract, what price should it set on it? Assume
that it is an ordinary annuity and that the price is the same
as present value.
(Use 6% as the discount rate)
 4- 35

Case 1
Q-What is the difference between a
series of payments and an annuity?
What are the two specific
characteristics of a series of payments
that make them an annuity?
Answer-

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 4- 36

Case 2
Q What effect on the future value of an
-

annuity does increasing the interest rate

have? Does a change from 4% to 6% have
the same dollar impact as a change from
6% to 8%?
Answer

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 4- 37

Case 3
Q What
- effect on the present value of an annuity does
increasing the interest rate have? Does a decrease from 7%
to 5% have the same dollar impact as a decrease from 5% to
3%?

Answer

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 4- 38

Case 4
Q-Is the present value always less than the
future value???
Answer
 4- 39

Topics Covered

Problems
&

Cases
 4- 40

MCQ Problem
If a perpetuity is worth $1,000 and rate
is 15.5%, what is the cash flow?

a. $155
b. $157
c. $150
d. $160
 4- 41

MCQ Problem
The value of a payment, if the payment were made at
some point in the future is called the

a. Time value of money

b. Principal
c. Present value
d. Future value
 4- 42

MCQ Problem
An investor deposits £600 in a bank and plans to leave
it there for four years. The value of the account after
four years if it earns 10 percent interest compounded
annually will be £

a. £ 856.78
b. £ 878.46
c. £ 915.34
d. £ 934.23
e. £ 978.99
 4- 43

MCQ Problem
Which equation represents the general relationship
between future values and present values

a. FV = PV(1+r)^n
b. PV= FV(1+r) ^n
d. PV=FV(1xr) ^n
e. FV=PV(1xr) ^n
 4- 44

MCQ Problem
A (an) ............ is a finite series of equal cash flows
made at regular intervals.

a. IRA
b. Annuity
c. Perpetuity
d. Annual regularity
 4- 45

MCQ Problem
An annuity with the first cash flow occurring
immediately is called a(n)

a. First annuity
b. Cash annuity
c. Simple annuity
d. Annuity due
 4- 46

MCQ Problem
If one speaks of an annuity without any qualification,
a(n) ……….. is being discussed

a. First annuity
b. Cash annuity
c. Simple annuity
d. Annuity due
 4- 47

MCQ Problem
Calculate the present value of an annuity of $100 for
two periods with a 12% rate of interest

a. $ 88.59
b. $100.45
c. $123.90
d. $ 169.05
 4- 48

MCQ Problem
Calculate the present value factor for a five-period
annuity with an interest rate of 12% per period

a. 1.2008
b. 2.6554
c. 3.6048
d. 4.7665
 4- 49

MCQ Problem
The present value of a perpetuity is given by

a. PV= r/C
b. PV= CF/r
c. PV= C x r
d. PV = C/ (1+r)