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CHAPTER 2 Time Value of Money. Future value Present value Rates of return Amortization. Introduction. In fact, of all the concepts used in finance, none is more important than the time value of money, which is also called discounted cash flow (DCF) analysis.

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Presentation Transcript
slide1

CHAPTER 2Time Value of Money

  • Future value
  • Present value
  • Rates of return
  • Amortization
introduction
Introduction

In fact, of all the concepts used in finance, none is more important than the time value of money, which is also called discounted cash flow (DCF) analysis.

PV : present value, or beginning amount, in your account

i : interest rate

INT : dollars of interest you earn

FV : future value

n : number of periods involved in the analysis

slide3
Time lines show timing of cash flows.

0

1

2

3

i%

CF0

CF1

CF2

CF3

Tick marksat ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

time line for uneven cfs 50 at t 0 and 100 75 and 50 at the end of years 1 through 3
Time line for uneven CFs: -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3.

0

1

2

3

i%

-50

100

75

50

what s the fv of an initial 100 after 3 years if i 10
What’s the FV of an initial $100 after 3 years if i = 10%?

0

1

2

3

10%

100

FV = ?

Finding FVs (moving to the right

on a time line) is called compounding.

slide8

After 1 year:

FV1 = PV + INT1 = PV + PV (i)

= PV(1 + i)

= $100(1.10)

= $110.00.

After 2 years:

FV2 = PV(1 + i)2

= $100(1.10)2

= $121.00.

slide9

After 3 years:

FV3 = PV(1 + i)3

= $100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.

slide10

Three Ways to Find FVs

  • Solve the equation with a regular calculator.
  • Use a spreadsheet.
what s the pv of 100 due in 3 years if i 10
What’s the PV of $100 due in 3 years if i = 10%?

Finding PVs is discounting, and it’s the reverse of compounding.

0

1

2

3

10%

100

PV = ?

slide12

Solve FVn = PV(1 + i )n for PV:

3

1

PV

=

$100

1.10

=

$100

0.7513

=

$75.13.

slide13

Finding the Time to Double

0

1

2

?

20%

2

-1

FV = PV(1 + i)n

$2 = $1(1 + 0.20)n

(1.2)n = $2/$1 = 2

nLN(1.2) = LN(2)

n = LN(2)/LN(1.2)

n = 0.693/0.182 = 3.8.

slide14

What’s the difference between an ordinaryannuity and an annuitydue?

Ordinary Annuity

0

1

2

3

i%

PMT

PMT

PMT

Annuity Due

0

1

2

3

i%

PMT

PMT

PMT

PV

FV

slide17

FV Annuity Formula

  • The future value of an annuity with n periods and an interest rate of i can be found with the following formula:
what s the pv of this ordinary annuity
What’s the PV of this ordinary annuity?

0

1

2

3

10%

100

100

100

90.91

82.64

75.13

248.69 = PV

slide19

PV Annuity Formula

  • The present value of an annuity with n periods and an interest rate of i can be found with the following formula:
special function for annuities
Special Function for Annuities

For ordinary annuities, this formula in cell A3 gives 248.96:

=PV(10%,3,-100)

A similar function gives the future value of 331.00:

=FV(10%,3,-100)

slide22

PV and FV of Annuity Due

vs. Ordinary Annuity

  • PV of annuity due:
    • = (PV of ordinary annuity) (1+i)
    • = (248.69) (1+ 0.10) = 273.56
  • FV of annuity due:
    • = (FV of ordinary annuity) (1+i)
    • = (331.00) (1+ 0.10) = 364.1
excel function for annuities due
Excel Function for Annuities Due

Change the formula to:

=PV(10%,3,-100,0,1)

The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due:

=FV(10%,3,-100,0,1)

uneven cash flow streams
Uneven Cash Flow Streams

We will use Payment (PMT) for annuity situations where the cash flows are equal amounts, and we will use the term

Cash flow (CF) to denote uneven cash flows.

what is the pv of this uneven cash flow stream
What is the PV of this uneven cashflow stream?

4

0

1

2

3

10%

100

300

300

-50

90.91

247.93

225.39

-34.15

530.08 = PV

how to find pv of this uneven cash
How to find PV of this uneven cash

1- We could find the PV of each individual cash flow using the numerical.

2- using NPV in excel .

slide28

Spreadsheet Solution

A B C D E

1 0 1 2 3 4

2 100 300 300 -50

3 530.09

Excel Formula in cell A3:

=NPV(10%,B2:E2)

home work
HOME WORK
  • Find the future value of the following annuities. The first payment in these annuities is made at the end of Year 1;
  • a. $400 per year for 10 years at 10 percent.
  • b. $200 per year for 5 years at 5 percent.
  • c. $400 per year for 5 years at 0 percent.
  • d. Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.
home work1
HOME WORK
  • Find the present value of the following ordinary annuities:
  • a. $400 per year for 10 years at 10 percent.
  • b. $200 per year for 5 years at 5 percent.
  • c. $400 per year for 5 years at 0 percent.
  • d. Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.
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