3-1

Time Value of Time Value of MoneyMoney

Time Value of Time Value of MoneyMoney

3-2

The Time Value of MoneyThe Time Value of MoneyThe Time Value of MoneyThe Time Value of Money

The Interest Rate

Simple Interest

Compound Interest

Amortizing a Loan

The Interest Rate

Simple Interest

Compound Interest

Amortizing a Loan

3-3

Obviously, $10,000 today$10,000 today.

You already recognize that there is TIME VALUE TO MONEYTIME VALUE TO MONEY!!

The Interest RateThe Interest RateThe Interest RateThe Interest Rate

Which would you prefer -- $10,000 $10,000 today today or $10,000 in 5 years$10,000 in 5 years?

3-4

TIMETIME allows you the opportunity to postpone consumption and earn

INTERESTINTEREST.

Why TIME?Why TIME?Why TIME?Why TIME?

Why is TIMETIME such an important element in your decision?

3-5

Types of InterestTypes of InterestTypes of InterestTypes of Interest

Compound InterestCompound Interest

Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).

Simple InterestSimple Interest

Interest paid (earned) on only the original amount, or principal borrowed (lent).

3-6

Simple Interest FormulaSimple Interest FormulaSimple Interest FormulaSimple Interest Formula

FormulaFormula SI = P0(i)(n)

SI: Simple Interest

P0: Deposit today (t=0)

i: Interest Rate per Period

n: Number of Time Periods

3-7

SI = P0(i)(n)= $1,000(.07)(2)= $140$140

Simple Interest ExampleSimple Interest ExampleSimple Interest ExampleSimple Interest Example

Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

3-8

FVFV = P0 + SI = $1,000 + $140= $1,140$1,140

Future ValueFuture Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.

Simple Interest (FV)Simple Interest (FV)Simple Interest (FV)Simple Interest (FV)

What is the Future Value Future Value (FVFV) of the deposit?

3-9

The Present Value is simply the $1,000 you originally deposited. That is the value today!

Present ValuePresent Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

Simple Interest (PV)Simple Interest (PV)Simple Interest (PV)Simple Interest (PV)

What is the Present Value Present Value (PVPV) of the previous problem?

3-10

0

5000

10000

15000

20000

1st Year 10thYear

20thYear

30thYear

Future Value of a Single $1,000 Deposit

10% SimpleInterest

7% CompoundInterest

10% CompoundInterest

Why Compound Interest?Why Compound Interest?Why Compound Interest?Why Compound Interest?

Fu

ture

Va

lue

(U

.S. D

olla

rs)

3-11

Assume that you deposit $1,000$1,000 at a compound interest rate of 7% for

2 years2 years.

Future ValueFuture ValueSingle Deposit (Graphic)Single Deposit (Graphic)Future ValueFuture ValueSingle Deposit (Graphic)Single Deposit (Graphic)

0 1 22

$1,000$1,000

FVFV22

7%

3-12

FVFV11 = PP00 (1+i)1 = $1,000$1,000 (1.07)= $1,070$1,070

Compound Interest

You earned $70 interest on your $1,000 deposit over the first year.

This is the same amount of interest you would earn under simple interest.

Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)

3-13

FVFV11 = PP00 (1+i)1 = $1,000$1,000 (1.07) = $1,070$1,070

FVFV22 = FV1 (1+i)1 = PP0 0 (1+i)(1+i) = $1,000$1,000(1.07)(1.07)

= PP00 (1+i)2 = $1,000$1,000(1.07)2

= $1,144.90$1,144.90

You earned an EXTRA $4.90$4.90 in Year 2 with compound over simple interest.

Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)

3-14

FVFV11 = P0(1+i)1

FVFV22 = P0(1+i)2

General Future Value Future Value Formula:

FVFVnn = P0 (1+i)n

or FVFVnn = P0 (FVIFFVIFi,n) -- See Table ISee Table I

General Future General Future Value FormulaValue FormulaGeneral Future General Future Value FormulaValue Formula

etc.

3-15

FVIFFVIFi,n is found on Table I at the end

of the book

Valuation Using Table IValuation Using Table IValuation Using Table IValuation Using Table I

Period 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.145 1.1663 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469

3-16

FVFV22 = $1,000 (FVIFFVIF7%,2)= $1,000 (1.145)

= $1,145$1,145 [Due to Rounding]

Using Future Value TablesUsing Future Value TablesUsing Future Value TablesUsing Future Value Tables

Period 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.145 1.1663 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469

3-17

Julie Miller wants to know how large her deposit of $10,000$10,000 today will become at a compound annual interest rate of 10% for 5 years5 years.

Story Problem ExampleStory Problem ExampleStory Problem ExampleStory Problem Example

0 1 2 3 4 55

$10,000$10,000

FVFV55

10%

3-18

Calculation based on Table I:FVFV55 = $10,000 (FVIFFVIF10%, 5)

= $10,000 (1.611)= $16,110$16,110 [Due to Rounding]

Story Problem SolutionStory Problem SolutionStory Problem SolutionStory Problem Solution

Calculation based on general formula:FVFVnn = P0 (1+i)n

FVFV55 = $10,000 (1+ 0.10)5

= $16,105.10$16,105.10

3-19

Assume that you need $1,000$1,000 in 2 years.2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually.

0 1 22

$1,000$1,000

7%

PV1PVPV00

Present ValuePresent Value Single Deposit (Graphic)Single Deposit (Graphic)Present ValuePresent Value Single Deposit (Graphic)Single Deposit (Graphic)

3-20

PVPV00 = FVFV22 / (1+i)2 = $1,000$1,000 / (1.07)2 =

FVFV22 / (1+i)2 = $873.44$873.44

Present Value Present Value Single Deposit (Formula)Single Deposit (Formula)Present Value Present Value Single Deposit (Formula)Single Deposit (Formula)

0 1 22

$1,000$1,000

7%

PVPV00

3-21

PVPV00 = FVFV11 / (1+i)1

PVPV00 = FVFV22 / (1+i)2

General Present Value Present Value Formula:

PVPV00 = FVFVnn / (1+i)n

or PVPV00 = FVFVnn (PVIFPVIFi,n) -- See Table IISee Table II

General Present General Present Value FormulaValue FormulaGeneral Present General Present Value FormulaValue Formula

etc.

3-22

PVIFPVIFi,n is found on Table II at the end

of the book

Valuation Using Table IIValuation Using Table IIValuation Using Table IIValuation Using Table II

Period 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .763 .735 5 .747 .713 .681

3-23

PVPV22 = $1,000$1,000 (PVIF7%,2)= $1,000$1,000 (.873)

= $873$873 [Due to Rounding]

Using Present Value TablesUsing Present Value TablesUsing Present Value TablesUsing Present Value Tables

Period 6% 7% 8%1 .943 .935 .9262 .890 .873 .8573 .840 .816 .7944 .792 .763 .7355 .747 .713 .681

3-24

Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000$10,000 in 5 years5 years at a discount rate of 10%.

Story Problem ExampleStory Problem ExampleStory Problem ExampleStory Problem Example

0 1 2 3 4 55

$10,000$10,000PVPV00

10%

3-25

Calculation based on general formula: PVPV00 = FVFVnn / (1+i)n

PVPV00 = $10,000$10,000 / (1+ 0.10)5

= $6,209.21$6,209.21

Calculation based on Table I:PVPV00 = $10,000$10,000 (PVIFPVIF10%, 5)

= $10,000$10,000 (.621)= $6,210.00$6,210.00 [Due to Rounding]

Story Problem SolutionStory Problem SolutionStory Problem SolutionStory Problem Solution

3-26

Project Evaluation: Project Evaluation: Alternative MethodsAlternative MethodsProject Evaluation: Project Evaluation: Alternative MethodsAlternative Methods

Payback Period (PBP)

Internal Rate of Return (IRR)

Net Present Value (NPV)

Profitability Index (PI)

Payback Period (PBP)

Internal Rate of Return (IRR)

Net Present Value (NPV)

Profitability Index (PI)

3-27

Proposed Project DataProposed Project DataProposed Project DataProposed Project Data

Julie Miller is evaluating a new project for her firm, Basket Wonders (BW). She has determined that the after-tax cash flows for the project will be $10,000; $12,000; $15,000; $10,000; and $7,000, respectively, for each of the Years 1 through 5. The initial cash outlay will be $40,000. The discount rate is 13%.

Julie Miller is evaluating a new project for her firm, Basket Wonders (BW). She has determined that the after-tax cash flows for the project will be $10,000; $12,000; $15,000; $10,000; and $7,000, respectively, for each of the Years 1 through 5. The initial cash outlay will be $40,000. The discount rate is 13%.

3-28

Payback Period (PBP)Payback Period (PBP)Payback Period (PBP)Payback Period (PBP)

PBPPBP is the period of time required for the cumulative

expected cash flows from an investment project to equal

the initial cash outflow.

PBPPBP is the period of time required for the cumulative

expected cash flows from an investment project to equal

the initial cash outflow.

0 1 2 3 4 5

-40 K 10 K 12 K 15 K 10 K 7 K

3-29

(c)10 K 22 K 37 K 47 K 54 K

Payback Solution (#1)Payback Solution (#1)Payback Solution (#1)Payback Solution (#1)

PBPPBP = a + ( b - c ) / d= 3 + (40 - 37) / 10= 3 + (3) / 10= 3.3 Years3.3 Years

PBPPBP = a + ( b - c ) / d= 3 + (40 - 37) / 10= 3 + (3) / 10= 3.3 Years3.3 Years

0 1 2 3 4 5

-40 K 10 K 12 K 15 K 10 K 7 K

CumulativeInflows

(a)

(-b) (d)

3-30

Payback Solution (#2)Payback Solution (#2)Payback Solution (#2)Payback Solution (#2)

PBPPBP = 3 + ( 3K ) / 10K= 3.3 Years3.3 Years

Note: Take absolute value of last negative cumulative cash flow

value.

PBPPBP = 3 + ( 3K ) / 10K= 3.3 Years3.3 Years

Note: Take absolute value of last negative cumulative cash flow

value.

CumulativeCash Flows

-40 K 10 K 12 K 15 K 10 K 7 K

0 1 2 3 4 5

-40 K -30 K -18 K -3 K 7 K 14 K

3-31

PBP Acceptance CriterionPBP Acceptance CriterionPBP Acceptance CriterionPBP Acceptance Criterion

Yes! The firm will receive back the initial cash outlay in less than 3.5 years. [3.3 Years < 3.5 Year Max.]

Yes! The firm will receive back the initial cash outlay in less than 3.5 years. [3.3 Years < 3.5 Year Max.]

The management of Basket Wonders has set a maximum PBP of 3.5 years for projects of this type.

Should this project be accepted?

3-32

Internal Rate of Return (IRR)Internal Rate of Return (IRR)Internal Rate of Return (IRR)Internal Rate of Return (IRR)

IRR is the discount rate that equates the present value of the future net cash

flows from an investment project with the project’s initial cash outflow.

CF1 CF2 CFn (1+IRR)1 (1+IRR)2 (1+IRR)n

+ . . . ++ICO =

3-33

$15,000 $10,000 $7,000

IRR SolutionIRR Solution IRR SolutionIRR Solution

$10,000 $12,000

(1+IRR)1 (1+IRR)2

Find the interest rate (IRR) that causes the discounted cash flows to equal $40,000.

+ +

++$40,000 =

(1+IRR)3 (1+IRR)4 (1+IRR)5

3-34

IRR Solution (Try 10%)IRR Solution (Try 10%)IRR Solution (Try 10%)IRR Solution (Try 10%)

$40,000$40,000 = $10,000(PVIF10%,1) + $12,000(PVIF10%,2) + $15,000(PVIF10%,3) + $10,000(PVIF10%,4) + $ 7,000(PVIF10%,5)

$40,000$40,000 = $10,000(.909) + $12,000(.826) + $15,000(.751) + $10,000(.683) + $ 7,000(.621)

$40,000$40,000 = $9,090 + $9,912 + $11,265 + $6,830 + $4,347

= $41,444$41,444 [[Rate is too low!!Rate is too low!!]]

3-35

IRR Solution (Try 15%)IRR Solution (Try 15%)IRR Solution (Try 15%)IRR Solution (Try 15%)

$40,000$40,000 = $10,000(PVIF15%,1) + $12,000(PVIF15%,2) + $15,000(PVIF15%,3) + $10,000(PVIF15%,4) + $ 7,000(PVIF15%,5)

$40,000$40,000 = $10,000(.870) + $12,000(.756) + $15,000(.658) + $10,000(.572) + $ 7,000(.497)

$40,000$40,000 = $8,700 + $9,072 + $9,870 + $5,720 + $3,479

= $36,841$36,841 [[Rate is too high!!Rate is too high!!]]

3-36

IRR Acceptance CriterionIRR Acceptance CriterionIRR Acceptance CriterionIRR Acceptance Criterion

No! The firm will receive 11.57% for each dollar invested in this project at a cost of 13%. [ IRR < Hurdle Rate ]

No! The firm will receive 11.57% for each dollar invested in this project at a cost of 13%. [ IRR < Hurdle Rate ]

The management of Basket Wonders has determined that the hurdle rate

is 13% for projects of this type.

Should this project be accepted?

3-37

Net Present Value (NPV)Net Present Value (NPV)Net Present Value (NPV)Net Present Value (NPV)

NPV is the present value of an investment project’s net cash

flows minus the project’s initial cash outflow.

CF1 CF2 CFn (1+k)1 (1+k)2 (1+k)n

+ . . . ++ - ICOICONPV =

3-38

Basket Wonders has determined that the appropriate discount rate (k) for this

project is 13%.

$10,000 $7,000

NPV SolutionNPV Solution NPV SolutionNPV Solution

$10,000 $12,000 $15,000 (1.13)1 (1.13)2 (1.13)3

+ +

+ - $40,000$40,000(1.13)4 (1.13)5

NPVNPV = +

3-39

NPV SolutionNPV SolutionNPV SolutionNPV Solution

NPVNPV = $10,000(PVIF13%,1) + $12,000(PVIF13%,2) + $15,000(PVIF13%,3) + $10,000(PVIF13%,4) + $ 7,000(PVIF13%,5) - $40,000$40,000

NPVNPV = $10,000(.885) + $12,000(.783) + $15,000(.693) + $10,000(.613) + $ 7,000(.543) - $40,000$40,000

NPVNPV = $8,850 + $9,396 + $10,395 + $6,130 + $3,801 - $40,000$40,000

= - $1,428$1,428

3-40

NPV Acceptance CriterionNPV Acceptance CriterionNPV Acceptance CriterionNPV Acceptance Criterion

No! The NPV is negative. This means that the project is reducing shareholder

wealth. [Reject Reject as NPVNPV < 00 ]

No! The NPV is negative. This means that the project is reducing shareholder

wealth. [Reject Reject as NPVNPV < 00 ]

The management of Basket Wonders has determined that the required

rate is 13% for projects of this type.

Should this project be accepted?

3-41

Profitability Index (PI)Profitability Index (PI)Profitability Index (PI)Profitability Index (PI)

PI is the ratio of the present value of a project’s future net cash flows to the project’s initial cash outflow.

CF1 CF2 CFn (1+k)1 (1+k)2 (1+k)n

+ . . . ++ ICOICOPI =

PI = 1 + [ NPVNPV / ICOICO ]

<< OR >>

3-42

PI Acceptance CriterionPI Acceptance Criterion PI Acceptance CriterionPI Acceptance Criterion

No! The PIPI is less than 1.00. This means that the project is not profitable.

[Reject Reject as PIPI < 1.001.00 ]

No! The PIPI is less than 1.00. This means that the project is not profitable.

[Reject Reject as PIPI < 1.001.00 ]

PIPI = $38,572 / $40,000

= .9643 (Method #1, 13-33)

Should this project be accepted?

3-43

Evaluation SummaryEvaluation Summary

Method Project Comparison Decision

PBP 3.3 3.5 Accept

IRR 11.47% 13% Reject

NPV -$1,424 $0 Reject

PI .96 1.00 Reject

Basket Wonders Independent Project