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Future value Present value Rates of return Amortization

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CHAPTER 2Time Value of Money

- Future value
- Present value
- Rates of return
- Amortization

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.

0

1

2 Year

i%

100

0

1

2

3

i%

100

100

100

0

1

2

3

i%

-50

100

75

50

0

1

2

3

10%

100

FV = ?

Finding FVs (moving to the right

on a time line) is called compounding.

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.

After 3 years:

FV3= PV(1 + i)3

= $100(1.10)3

= $133.10.

In general,

FVn= PV(1 + i)n.

Three Ways to Find FVs

- Solve the equation with a regular calculator.
- Use a spreadsheet.

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

0

1

2

3

10%

100

PV = ?

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

3

1

PV

=

$100

1.10

=

$100

0.7513

=

$75.13.

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.

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

0

1

2

3

10%

100

100

100

110

121

FV= 331

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:

0

1

2

3

10%

100

100

100

90.91

82.64

75.13

248.69 = PV

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:

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)

0

1

2

3

10%

100

100

100

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

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)

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.

4

0

1

2

3

10%

100

300

300

-50

90.91

247.93

225.39

-34.15

530.08 = PV

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

2- using NPV in excel .

Spreadsheet Solution

ABCDE

101234

2100300300-50

3530.09

Excel Formula in cell A3:

=NPV(10%,B2:E2)

- 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.

- 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.