© 2012 McGrawHill Ryerson Ltd.Chapter 5 -1 Example: You deposit $1,200 in your bank account today;...

© 2012 McGrawHill Ryerson Ltd. Chapter 5 -1

Example:

You deposit $1,200 in your bank account today; $1,400 one

year later; and $1,000 two years from today. If your bank

offers you an 8% interest rate on your account, how much

money will you have in the account three years from

today?

LO3


Future value example

0 1 2

$1,000

8%

$1,200

8%

$1,400

38%

?

??

???

LO3


0 1 2

$1,000

8%

$1,200

8%

$1,400

3

8%

$1,000x(1+0.08)1=$1,080.00

$1,400x(1+0.08)2=$1,632.96$1,200x(1+0.08)3=$1,511.65

FV = $4,224.61

LO3


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?

Option 1: $15,500 todayOption 2: $8,000 today; $4,000 at the end of one year; and

$4,000 at the end of two years.

We can only compare these two options if both of them are converted for the same time period. Today will be an ideal one to do that.

LO3

© 2012 McGrawHill Ryerson Ltd. Chapter 5-5

0 1

$4,000

8%

28%

$4,000/(1+0.08)1=$3,703.70

$4,000

$4,000/(1+0.08)2=$3,429.36

PV = $15,133.06

$8,000

LO3

Present value of option # 2:


Compare the present value of the two options at present:

◦ Option 1: $15,500◦ Option 2: $15,133

Thus option 2 is a better choice for the buyer. That means, option 1 is a better payment for the seller.

LO3


Annuities: Cash flows of equal amount every period for a limited number of periods◦ Example: Loan payments for automobile, periodic

earnings from lottery wins, etc.

Perpetuities: Cash flows of equal amount every period for an unlimited number of periods◦ Example: Property tax payments, preferred stocks, etc.

LO3


The PV of a perpetuity is calculated by dividing the level cash flow by the interest rate.

PV of a perpetuity =

Cr

0 1 3 2 ……. ∞

CCCCC

Note: This formula gives you the present value of a perpetuity starting one period from now

PV

LO3


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

PV of perpetuity =

$100,000

0.10

0 1 … 2 ……. ∞

$100,000

PV $100,000

$100,000

= $1,000,000

LO3


Example contd.:If the first perpetuity payment will not be received until four years from today, how much money needs to be set aside today?

PV at end of Year 3 =

$100,0000.10

1 2 4 3 5 ∞10% 10% 10%

$100,000

PV $100,000 $100,000

= $1,000,000

….…

PV today =$1,000,00

0(1.10)3= $751,315

0

LO3


Present Value: The PV of a t period annuity with cash flow of C and discount rate r is given by:

PV of t-period annuity =

0 1 3 2 ……. t

CCCCCPV

trrrC

)1(

11

Note: This formula gives you the present value of an annuity starting one period from now. This is called a regular annuity.

LO3


Present Value Interest Factor of Annuity:

In the present value formula for an annuity

the term in the parentheses is called the Present Value Interest Factor of an annuity (PVIFA).

There is a table that can be used to find present values of $1 annuities for different rates and time periods, as shown in the next slide

trrrC

)1(

11

LO3


Example: You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year, with the first payment one year from now. Given a rate of interest of 10%, what is the price you are paying for the car?

0 1 2

$4,000

10%

$4,000

10%

$4,000

310%

PV of 3-period annuity 41.947,9$)10.01(10.0

1

10.0

1000,4$

3

Or, from the PVIFA table = $4,000 x 2.487 = $9,947.41

LO3


Future Value: The FV of a t period annuity with cash flow of C and discount rate r is given by:

0 1 3 2 ……. t

FV

CCCC

FV of t-period annuity =

C x [(1 + r)t – 1]r

Note: This formula gives you the present value of an annuity starting one period from now. This is called a regular annuity.

LO3


Future Value Interest Factor of Annuity:

◦ In the present value formula for an annuity, the term

[(1 + r)t – 1]r

is called the Future Value Interest Factor of an annuity (FVIFA).

◦ There is a table that can be used to find future values of $1 annuities for different rates and time periods, as shown in the next slide.

LO3


Annuities Due:

A level stream of payments starting immediately, is known as an annuity due. The difference between an annuity due and a regular annuity is shown in the following example.

LO3


A growing perpetuity with a constant growth rate of ‘g’ has a PV that can be shown as:

0 1 3 2 ……. ∞

C(1+g)3

C(1+g)2

C(1+g)CPV

PV of a growing perpetuity =

Cr - g

LO3


A growing annuity with a growth rate of ‘g’ has a PV that can be shown as:

0 1 3 2 ……. t

C(1+g)3

C(1+g)2

C(1+g)CPV

T

r

g

gr

CannuitygrowingofPV

1

111

LO3

© 2012 McGrawHill Ryerson Ltd.Chapter 5 -1 Example: You deposit $1,200 in your bank account today;...

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