Chapter 4 The Time Value of Money

Chapter 4The Time Value of Money

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Chapter Outline

4.1 The Timeline

4.2 The Three Rules of Time Travel

4.3 The Power of Compounding

4.4 Valuing a Stream of Cash Flows

4.5 The Net Present Value of a Stream of Cash Flows

4.6 Perpetuities, Annuities, and other Special Cases

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Chapter Outline (cont’d)

4.7 Solving Problems with a Spreadsheet Program

4.8 Solving for Variables Other Than Present Value or Future Value

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Learning Objectives

1. Draw a timeline illustrating a given set of cash flows.

2. List and describe the three rules of time travel.

3. Calculate the future value of:

a. A single sum.

b. An uneven stream of cash flows, starting either now or sometime in the future.

c. An annuity, starting either now or sometime in the future.

d. Several cash flows occurring at regular intervals that grow at a constant rate each period.

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Learning Objectives (cont'd)

4. Calculate the present value of:

a. A single sum.

b. An uneven stream of cash flows, starting either now or sometime in the future.

c. An infinite stream of identical cash flows.

d. An annuity, starting either now or sometime in the future.

e. An infinite stream of cash flows that grow at a constant rate each period.

f. Several cash flows occurring at regular intervals that grow at a constant rate each period.

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Learning Objectives (cont'd)

5. Given four out of the following five inputs for an annuity, compute the fifth: (a) present value, (b) future value, (c) number of periods, (d) periodic interest rate, (e) periodic payment.

6. Given three out of the following four inputs for a single sum, compute the fourth: (a) present value, (b) future value, (c) number of periods, (d) periodic interest rate.

7. Given cash flows and present or future value, compute the internal rate of return for a series of cash flows.

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4.1 The Timeline

A timeline is a linear representation of the timing of potential cash flows.

Drawing a timeline of the cash flows will help you visualize the financial problem.

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4.1 The Timeline (cont’d)

Assume that you loan $10,000 to a friend. You will be repaid in two payments, one at the end of each year over the next two years.

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Differentiate between two types of cash flows

Inflows are positive cash flows.

Outflows are negative cash flows, which are indicated with a – (minus) sign.

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Assume that you are lending $10,000 today and that the loan will be repaid in two annual $6,000 payments.

The first cash flow at date 0 (today) is represented as a negative sum because it is an outflow.

Timelines can represent cash flows that take place at the end of any time period.

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Example 4.1

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Example 4.1 (cont’d)

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4.2 Three Rules of Time Travel Financial decisions often require combining

cash flows or comparing values. Three rules govern these processes.

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The 1st Rule of Time Travel

A dollar today and a dollar in one year are not equivalent.

It is only possible to compare or combine values at the same point in time. Which would you prefer: A gift of $1,000 today or

$1,210 at a later date? To answer this, you will have to compare the

alternatives to decide which is worth more. One factor to consider: How long is “later?”

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The 2nd Rule of Time Travel

To move a cash flow forward in time, you must compound it.

Suppose you have a choice between receiving $1,000 today or $1,210 in two years. You believe you can earn 10% on the $1,000 today, but want to know what the $1,000 will be worth in two years. The time line looks like this:

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(1 ) (1 ) (1 ) (1 )

times

n

nFV C r r r C r

n

The 2nd Rule of Time Travel (cont’d)

Future Value of a Cash Flow

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Using a Financial Calculator: The Basics TI BA II Plus

Future Value

Present Value

I/Y Interest Rate per Year

Interest is entered as a percent, not a decimal For 10%, enter 10, NOT .10

FV

I/Y

PV

18

N

FV2ND

Using a Financial Calculator: The Basics (cont'd) TI BA II Plus

Number of Periods

2nd → CLR TVM

Clears out all TVM registers

Should do between all problems

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I/Y2ND

I/Y2ND # ENTER

.2ND # ENTER

Using a Financial Calculator: Setting the keys TI BA II Plus

2ND → P/Y Check P/Y

2ND → P/Y → # → ENTER Sets Periods per Year to #

2ND → FORMAT → # → ENTER Sets display to # decimal places

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Using a Financial Calculator

TI BA II Plus Cash flows moving in opposite directions must

have opposite signs.

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N

I/Y

PV

FV

2

10

1,000

-1,210CPT

Financial Calculator Solution

Inputs: N = 2 I = 10 PV = 1,000

Output: FV = −1,210

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The 2nd Rule of Time Travel—Alternative Example To move a cash flow forward in time, you

must compound it.

Suppose you have a choice between receiving $5,000 today or $10,000 in five years. You believe you can earn 10% on the $5,000 today, but want to know what the $5,000 will be worth in five years. The time line looks like this:

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0 3 4 521

$5,000 $5, 500 $6,050 $6,655 $7,321 $8,053x 1.10 x 1.10 x 1.10 x 1.10 x 1.10

The 2nd Rule of Time Travel—Alternative Example (cont’d)

In five years, the $5,000 will grow to:$5,000 × (1.10)5 = $8,053

The future value of $5,000 at 10% for five years is $8,053.

You would be better off forgoing the gift of $5,000 today and taking the $10,000 in five years.

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Financial Calculator Solution

Inputs: N = 5 I = 10 PV = 5,000

Output: FV = –8,052.55

N

I/Y

PV

FV

5

10

5,000

-8,052.55CPT

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The 3rd Rule of Time Travel

To move a cash flow backward in time, we must discount it.

Present Value of a Cash Flow

(1 ) (1 )

nn

CPV C r

r

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Example 4.2

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Example 4.2 (cont’d)

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Example 4.2 Financial Calculator Solution Inputs:

N = 10 I = 6 FV = 15,000

Output: PV = –8,375.92

N

I/Y

FV

PV

10

6

15,000

-8,375.92CPT

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The 3rd Rule of Time Travel—Alternative Example Suppose you are offered an investment that

pays $10,000 in five years. If you expect to earn a 10% return, what is the value of this investment?

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The 3rd Rule of Time Travel—Alternative Example (cont’d) The $10,000 is worth:

$10,000 ÷ (1.10)5 = $6,209

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Alternative Example: Financial Calculator Solution Inputs:

N = 5 I = 10 FV = 10,000

Output: PV = –6,209.21

N

I/Y

FV

PV

5

10

10,000

-6,209.21CPT

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Combining Values Using the Rules of Time Travel Recall the 1st rule: It is only possible to

compare or combine values at the same point in time. So far we’ve only looked at comparing.

Suppose we plan to save $1000 today, and $1000 at the end of each of the next two years. If we can earn a fixed 10% interest rate on our savings, how much will we have three years from today?

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Combining Values Using the Rules of Time Travel (cont'd) The time line would look like this:

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Combining Values Using the Rules of Time Travel (cont'd)

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Example 4.3

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Example 4.3 (cont'd)

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Example 4.3 Financial Calculator Solution

CF 1,000

↓ 1,000

2

10NPV

2,735.54

ENTER

ENTER

↓ ENTER

ENTER

↓ CPT

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0 3 4 521

$10,000$5,000

Combining Values Using the Rules of Time Travel—Alternative Example Assume that an investment will pay you

$5,000 now and $10,000 in five years. The time line would like this:

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0 3 4 521

$6,209 $10,000$5,000

$11,209 ÷ 1.105

Combining Values Using the Rules of Time Travel—Alternative Example (cont'd) You can calculate the present value of the combined

cash flows by adding their values today.

The present value of both cash flows is $11,209.

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0 3 4 521

$5,000 $8,053x 1.105

$10,000

$18,053

Combining Values Using the Rules of Time Travel—Alternative Example (cont'd) You can calculate the future value of the

combined cash flows by adding their values in Year 5.

The future value of both cash flows is $18,053.

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0 3 4 521

$11,209 $18,053÷ 1.105

0 3 4 521

$11,209 $18,053x 1.105

Present Value

Future Value

Combining Values Using the Rules of Time Travel—Alternative Example (cont'd)

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4.3 The Power of Compounding: An Application Compounding

Interest on Interest As the number of time periods increases, the future

value increases, at an increasing rate since there is more interest on interest.

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Figure 4.1 The Power of Compounding

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4.4 Valuing a Stream of Cash Flows Based on the first rule of time travel we can

derive a general formula for valuing a stream of cash flows: if we want to find the present value of a stream of cash flows, we simply add up the present values of each.

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4.4 Valuing a Stream of Cash Flows (cont’d)

Present Value of a Cash Flow Stream

0 0

( ) (1 )

N Nn

n nn n

CPV PV C

r

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Example 4.4

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50


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(1 ) nnFV PV r

Future Value of Cash Flow Stream

Future Value of a Cash Flow Stream with a Present Value of PV

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Future Value of Cash Flow Stream—Alternative Example What is the future value in three years of the

following cash flows if the compounding rate is 5%?

0 321

$2,000 $2,000 $2,000

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Future Value of Cash Flow Stream—Alternative Example (cont'd)

Or

0 321

$2,000

$2,000x 1.05 x 1.05

$2,315x 1.05

$2,205

$2,000x 1.05 x 1.05

$2,100$6,620x 1.05

0 321

$2,000

x 1.05$4,100$2,100

$4,305

$2,000 $2,000

x 1.05

$6,305

x 1.05$6,620

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4.5 Net Present Value of a Stream

of Cash Flows Calculating the NPV of future cash flows allows us to evaluate an investment decision.

Net Present Value compares the present value of cash inflows (benefits) to the present value of cash outflows (costs).

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Example 4.5

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0 321

$1,000$3,000 $2,000

4.5 Net Present Value of a Stream

of Cash Flows—Alternative Example Would you be willing to pay $5,000 for the

following stream of cash flows if the discount rate is 7%?

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Compute the Present Value of the Benefits and the Present Value of the Cost… The present value of the benefits is:

3000 / (1.05) + 2000 / (1.05)2 + 1000 / (1.05)3 = 5366.91

The present value of the cost is $5,000, because it occurs now.

The NPV = PV(benefits) – PV(cost)

= 5366.91 – 5000 = 366.91

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Alternative Example Financial Calculator Solution

On a present value basis, the benefits exceed the costs by $366.91.

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4.6 Perpetuities, Annuities, and Other Special Cases When a constant cash flow will occur at

regular intervals forever it is called a perpetuity.

The value of a perpetuity is simply the cash flow divided by the interest rate.

Present Value of a Perpetuity

( in perpetuity) C

PV Cr

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Example 4.6

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Annuities

When a constant cash flow will occur at regular intervals for N periods it is called an annuity.

Present Value of an Annuity 1 1 (annuity of for periods with interest rate ) 1

(1 )

NPV C N r C

r r

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Example 4.7

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Future Value of an Annuity

(annuity) V (1 )

1 1 (1 )

(1 )

1 (1 ) 1

N

NN

N

FV P r

Cr

r r

C rr

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Example 4.7 Financial Calculator Solution Since the payments begin today, this is an

Annuity Due.

First, put the calculator on “Begin” mode:

2ND PMT 2ND ENTER 2ND CPT

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Example 4.7 Financial Calculator Solution (cont'd) Then:

$15 million > $12.16 million, so take the lump sum.

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Example 4.8

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Example 4.8 Financial Calculator Solution Since the payments begin in one year, this is

an Ordinary Annuity. Be sure to put the calculator back on “End” mode:

2ND PMT 2ND ENTER 2ND CPT

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Example 4.8 Financial Calculator Solution (cont'd) Then

N

I/Y

PMT

FV

30

10

10,000

-1,644,940CPT

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Growing Perpetuities

Assume you expect the amount of your perpetual payment to increase at a constant rate, g.

Present Value of a Growing Perpetuity

(growing perpetuity)

C

PVr g

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Example 4.9

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Growing Annuities

The present value of a growing annuity with the initial cash flow c, growth rate g, and interest rate r is defined as:

Present Value of a Growing Annuity

1 1 1

( ) (1 )

Ng

PV Cr g r

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Example 4.10

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4.7 Solving Problems with a Spreadsheet Program Spreadsheets simplify the calculations of

TVM problems NPER RATE PV PMT FV

1 1 V 1 0

(1 ) (1 )

NPER NPER

FVNPV P PMT

RATE RATE RATE

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Example 4.11

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Example 4.12

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4.8 Solving for Variables Other Than Present Values or Future Values Sometimes we know the present value or

future value, but do not know one of the variables we have previously been given as an input. For example, when you take out a loan you may know the amount you would like to borrow, but may not know the loan payments that will be required to repay it.

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Example 4.13

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4.8 Solving for Variables Other Than Present Values or Future Values (cont’d) In some situations, you know the present

value and cash flows of an investment opportunity but you do not know the internal rate of return (IRR), the interest rate that sets the net present value of the cash flows equal to zero.

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Example 4.14

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Example 4.15

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4.8 Solving for Variables Other Than Present Values or Future Values (cont’d) In addition to solving for cash flows or the

interest rate, we can solve for the amount of time it will take a sum of money to grow to a known value.

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Example 4.16

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Chapter 4 The Time Value of Money

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Transcript of Chapter 4 The Time Value of Money