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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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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


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%? $50 at the end of Years 1 through 3.

0

1

2

3

10%

100

FV = ?

Finding FVs (moving to the right

on a time line) is called compounding.


After 1 year: $50 at the end of Years 1 through 3.

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.


After 3 years: $50 at the end of Years 1 through 3.

FV3 = PV(1 + i)3

= $100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.


Three Ways to Find FVs $50 at the end of Years 1 through 3.

  • 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%? $50 at the end of Years 1 through 3.

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

0

1

2

3

10%

100

PV = ?


Solve FV $50 at the end of Years 1 through 3.n = PV(1 + i )n for PV:

3

1

PV

=

$100

1.10

=

$100

0.7513

=

$75.13.


Finding the Time to Double $50 at the end of Years 1 through 3.

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.


What’s the difference between an $50 at the end of Years 1 through 3.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


What s the fv of a 3 year ordinary annuity of 100 at 10
What’s the FV of a 3-year ordinary annuity of $100 at 10%? $50 at the end of Years 1 through 3.

0

1

2

3

10%

100

100

100

110

121

FV = 331


Ordinary annuity
ordinary $50 at the end of Years 1 through 3.annuity


FV Annuity Formula $50 at the end of Years 1 through 3.

  • 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? $50 at the end of Years 1 through 3.

0

1

2

3

10%

100

100

100

90.91

82.64

75.13

248.69 = PV


PV Annuity Formula $50 at the end of Years 1 through 3.

  • 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 $50 at the end of Years 1 through 3.

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)


Find the fv and pv if the annuity were an annuity due
Find the FV and PV if the $50 at the end of Years 1 through 3.annuity were an annuity due.

0

1

2

3

10%

100

100

100


PV and FV of Annuity Due $50 at the end of Years 1 through 3.

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


Annuity due
annuity $50 at the end of Years 1 through 3.due


Excel function for annuities due
Excel Function for Annuities Due $50 at the end of Years 1 through 3.

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 $50 at the end of Years 1 through 3.

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 cash $50 at the end of Years 1 through 3.flow 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 $50 at the end of Years 1 through 3.

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

2- using NPV in excel .


Spreadsheet Solution $50 at the end of Years 1 through 3.

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 $50 at the end of Years 1 through 3.

  • 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 $50 at the end of Years 1 through 3.

  • 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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