ch3 time value of money

8/6/2019 Ch3 Time Value of Money

1/60

Chapter 3Chapter 3

Time Value ofTime Value of

MoneyMoney

Time Value ofTime Value ofMoneyMoney


2/60

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

x The Interest Rate

x

Simple Interestx Compound Interest

x Amortizing a Loan

x Compounding More ThanOnce per Year

x The Interest Rate

x

Simple Interestx Compound Interest

x Amortizing a Loan

x Compounding More ThanOnce per Year


3/60

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

You already recognize that there isTIME VALUE TO MONEYTIME VALUE TO MONEY!!

The Interest RateThe Interest RateThe Interest RateThe Interest Rate

Which would you prefer -- $10,000$10,000

todaytoday or$10,000 in 5 years$10,000 in 5 years?


4/60

TIMETIME allows you the opportunityto

postpone consumption and earnINTERESTINTEREST.

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

Why is TIMETIME such an important

element in your decision?


5/60

Types of InterestTypes of InterestTypes of InterestTypes of Interest

x Compound InterestCompound Interest

Interest paid (earned) on any previousinterest earned, as well as on theprincipal borrowed (lent).

x Simple InterestSimple Interest

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


6/60

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


7/60

xSI = P0

(i)(n)

= $1,000(.07)(2)= $140$140

Simple Interest ExampleSimple Interest ExampleSimple Interest ExampleSimple Interest Example

x Assume that you deposit $1,000 in anaccount earning 7% simple interest for

2 years. What is the accumulatedinterestat the end of the 2nd year?


8/60

FVFV = P0 + SI

= $1,000+ $140=$1,140$1,140

x Future ValueFuture Valueis the value at some futuretime of a present amount of money, or aseries of payments, evaluated at a given

interest rate.

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

x What is the Future ValueFuture Value (FVFV) of thedeposit?


9/60

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

x Present ValuePresent Valueis the current value of afuture amount of money, or a series ofpayments, evaluated at a given interest

rate.

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

x What is the Present ValuePresent Value (PVPV) of theprevious problem?


10/60

0

0

5000

10000

15000

20000

1st Year 10th

Year

20th

Year

30th

Year

Future Value of a Single $1,000 Depos

10% SimpleInterest

7% CompoundInterest

10% CompoundInterest

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

Future

Va

lue(U.S.D

olla

rs)


11/60

1

Assume that you deposit $1,000$1,000 ata compound interest rate of7% for

2 years2 years.

Future ValueFuture Value

Single Deposit (Graphic)Single Deposit (Graphic)


Single Deposit (Graphic)Single Deposit (Graphic)

0 1 22

$1,000$1,000

FVFV22

7%


12/60

2

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.


Single Deposit (Formula)Single Deposit (Formula)




13/60

3

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

FVFV22 = FV1 (1+i)1

= PP00 (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$4.90in Year 2 withcompound over simple interest.






14/60

4

FVFV11 = P0(1+i)1

FVFV22 = P0(1+i)2

General Future ValueFuture Value Formula:

FVFVnn = P0 (1+i)n

or FVFVnn

= P0

(FVIFFVIFi,n

) -- See Table ISee Table I

General FutureGeneral Future

Value FormulaValue Formula

General FutureGeneral Future


etc.


15/60

5

FVIFFVIFi,nis 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 1.080

2 1.124 1.145 1.1663 1.191 1.225 1.260

4 1.262 1.311 1.360

5 1.338 1.403 1.469


16/60

6

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 1.080

2 1.124 1.145 1.1663 1.191 1.225 1.260

4 1.262 1.311 1.360

5 1.338 1.403 1.469


17/60

7

Julie Miller wants to know how large her deposit

of$10,000$10,000 today will become at a compound

annual interest rate of10% for5 years5 years.

Story Problem ExampleStory Problem ExampleStory Problem ExampleStory Problem Example

0 1 2 3 4 55

$10,000$10,000

FVFV55

10%


18/60

8

x 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

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

n

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

= $16,105.10$16,105.10


19/60

9

We will use the Rule-of-72Rule-of-72..

Double Your Money!!!Double Your Money!!!Double Your Money!!!Double Your Money!!!

Quick! How long does it take to double$5,000 at a compound rate of 12% per

year (approx.)?


20/60

0

Approx. Years to Double = 7272/ i%

7272 / 12% = 6 Years6 Years

[Actual Time is 6.12 Years]

The Rule-of-72The Rule-of-72The Rule-of-72The Rule-of-72

Quick! How long does it take to double$5,000 at a compound rate of 12% per

year (approx.)?


21/60

1

Assume that you need $1,000$1,000in 2 years.2 years. Letsexamine the process to determine how much youneed to deposit today at a discount rate of7%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)


22/60

2

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

=FVFV

22/ (1+i)2 =

$873.44$873.44

Present ValuePresent Value


Present ValuePresent Value


0 1 22

$1,000$1,000

7%

PVPV00


23/60

3

PVPV00= FVFV11 / (1+i)1

PVPV00 = FVFV22 / (1+i)2

General Present ValuePresent Value Formula:

PVPV00 = FVFVnn / (1+i)n

or PVPV00

= FVFVnn

(PVIFPVIFi,n

) -- See Table IISee Table II

General PresentGeneral Present


General PresentGeneral Present


etc.


24/60

4

PVIFPVIFi,nis 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


25/60

5

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 .926

2 .890 .873 .8573 .840 .816 .794

4 .792 .763 .735

5 .747 .713 .681


26/60

6

Julie Miller wants to know how large of adeposit to make so that the money will growto $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$10,000

PVPV00

10%


27/60

7

x 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

x 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


28/60

8

Types of AnnuitiesTypes of AnnuitiesTypes of AnnuitiesTypes of Annuities

x Ordinary AnnuityOrdinary Annuity: Payments or receiptsoccur at the end of each period.

x Annuity DueAnnuity Due: Payments or receiptsoccur at the beginning of each period.

xAn AnnuityAn Annuityrepresents a series of equalpayments (or receipts) occurring over a

specified number of equidistant periods.


29/60

9

Examples of AnnuitiesExamples of Annuities

x Student Loan Payments

x Car Loan Paymentsx Insurance Premiums

x Mortgage Paymentsx Retirement Savings


30/60

0

Parts of an AnnuityParts of an AnnuityParts of an AnnuityParts of an Annuity

0 1 2 3

$100 $100 $100

(Ordinary Annuity)

EndEnd of

Period 1

EndEnd of

Period 2

Today EqualEqual Cash Flows

Each 1 Period Apart

EndEnd of

Period 3


31/60

1

Parts of an AnnuityParts of an AnnuityParts of an AnnuityParts of an Annuity

0 1 2 3

$100 $100 $100

(Annuity Due)

BeginningBeginning of

Period 1


Period 2

Today EqualEqual Cash Flows

Each 1 Period Apart


Period 3


32/60

2

FVAFVAnn = R(1+i)n-1 + R(1+i)n-2 +

... + R(1+i)1+ R(1+i)0

Overview of anOverview of an

Ordinary Annuity -- FVAOrdinary Annuity -- FVA



R R R

0 1 2 nn n+1

FVAFVAnn

R = PeriodicCash Flow

Cash flows occur at the end of the period

i% . . .


33/60

3

FVAFVA33 = $1,000(1.07)2 +$1,000(1.07)1 + $1,000(1.07)0

= $1,145+$1,070+$1,000 =

$3,215$3,215

Example of anExample of an




$1,000 $1,000 $1,000

0 1 2 33 4

$3,215 = FVA$3,215 = FVA33

7%

$1,070

$1,145



34/60

4

Hint on Annuity ValuationHint on Annuity Valuation

The future value of an ordinaryannuity can be viewed as

occurring at the endendof the lastcash flow period, whereas thefuture value of an annuity due

can be viewed as occurring atthe beginningbeginningof the last cash

flow period.


35/60

5

FVAFVAnn = R (FVIFAi%,n)

FVAFVA33 = $1,000 (FVIFA7% ,3)

= $1,000 (3.215) = $3,215$3,215

Valuation Using Table IIIValuation Using Table IIIValuation Using Table IIIValuation Using Table III

Period 6% 7% 8%1 1.000 1.000 1.000

2 2.060 2.070 2.080

3 3.184 3.215 3.2464 4.375 4.440 4.506

5 5.637 5.751 5.867


36/60

6

FVADFVADnn = R(1+i)n + R(1+i)n-1 +... + R(1+i)2+ R(1+i)1

= FVAFVAnn (1+i)

Overview View of anOverview View of an

Annuity Due -- FVADAnnuity Due -- FVAD

Overview View of anOverview View of an


R R R R R

0 1 2 3 n-1n-1 n

FVADFVADnn

i% . . .

Cash flows occur at the beginning of the period


37/60

7

FVADFVAD33 = $1,000(1.07)3 +$1,000(1.07)2 + $1,000(1.07)1

= $1,225+$1,145+$1,070 =

$3,440$3,440





$1,000 $1,000 $1,000 $1,070

0 1 2 33 4

$3,440 = FVAD$3,440 = FVAD33

7%

$1,225

$1,145



38/60

8

FVADFVADnn = R (FVIFAi%,n)(1+i)

FVADFVAD33 = $1,000 (FVIFA7% ,3)(1.07)

= $1,000 (3.215)(1.07) = $3,440$3,440

Valuation Using Table IIIValuation Using Table IIIValuation Using Table IIIValuation Using Table III

Period 6% 7% 8%1 1.000 1.000 1.000

2 2.060 2.070 2.080

3 3.184 3.215 3.2464 4.375 4.440 4.506

5 5.637 5.751 5.867


39/60

9

PVAPVAnn = R/(1+i)1 + R/(1+i)2

+ ... + R/(1+i)n


Ordinary Annuity -- PVAOrdinary Annuity -- PVA



R R R

0 1 2 nn n+1

PVAPVAnn

R = PeriodicCash Flow

i% . . .



40/60

0

PVAPVA33 = $1,000/(1.07)1

+$1,000/(1.07)2 + $1,000/(1.07)3

= $934.58 + $873.44 + $816.30=$2,624.32$2,624.32





$1,000 $1,000 $1,000

0 1 2 33 4

$2,624.32 = PVA$2,624.32 = PVA33

7%

$934.58$873.44$816.30



41/60

1

Hint on Annuity ValuationHint on Annuity Valuation

The present value of an ordinaryannuity can be viewed as

occurring at the beginningbeginningof thefirst cash flow period, whereasthe future value of an annuity

due can be viewed as occurringat the endendof the first cash flow

period.


42/60

2

PVAPVAnn = R (PVIFAi%,n)

PVAPVA33 = $1,000 (PVIFA7% ,3)

= $1,000 (2.624) = $2,624$2,624

Valuation Using Table IVValuation Using Table IVValuation Using Table IVValuation Using Table IV

Period 6% 7% 8%1 0.943 0.935 0.926

2 1.833 1.808 1.783

3 2.673 2.624 2.5774 3.465 3.387 3.312

5 4.212 4.100 3.993


43/60

3

PVADPVADnn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 =

PVAPVAnn (1+i)


Annuity Due -- PVADAnnuity Due -- PVAD



R R R R

0 1 2 n-1n-1 n

PVADPVADnn

R: PeriodicCash Flow

i% . . .



44/60

4

PVADPVADnn = $1,000/(1.07)0+ $1,000/(1.07)1+

$1,000/(1.07)2 = $2,808.02$2,808.02





$1,000.00 $1,000 $1,000

0 1 2 33 4

$2,808.02$2,808.02 = PVADPVADnn

7%

$ 934.58

$ 873.44



45/60

5

PVADPVADnn = R (PVIFAi%,n)(1+i)

PVADPVAD33 = $1,000 (PVIFA7% ,3)(1.07)

= $1,000 (2.624)(1.07) = $2,808$2,808

Valuation Using Table IVValuation Using Table IVValuation Using Table IVValuation Using Table IV

Period 6% 7% 8%1 0.943 0.935 0.926

2 1.833 1.808 1.783

3 2.673 2.624 2.5774 3.465 3.387 3.312

5 4.212 4.100 3.993


46/60

6

1. Read problem thoroughly

2. Create a time line

3. Put cash flows and arrows on time line4. Determine if it is a PV or FV problem

5. Determine if solution involves a single

CF, annuity stream(s), or mixed flow6. Solve the problem

7. Check with financial calculator (optional)

Steps to Solve Time ValueSteps to Solve Time Value

of Money Problemsof Money Problems

Steps to Solve Time ValueSteps to Solve Time Value

of Money Problemsof Money Problems


47/60

7

Julie Miller will receive the set ofcashflows below. What is the Present ValuePresent Valueat a discount rate of10%10%.

Mixed Flows ExampleMixed Flows ExampleMixed Flows ExampleMixed Flows Example

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $100

PVPV00

10%10%


48/60

8

1. Solve a piece-at-a-timepiece-at-a-time bydiscounting each

piecepiece back to t=0.

2. Solve a group-at-a-timegroup-at-a-time by firstbreaking problem into groups

of annuity streams and any singlecash flow groups. Then discounteach groupgroup back to t=0.

How to Solve?How to Solve?How to Solve?How to Solve?


49/60

9

Piece-At-A-TimePiece-At-A-TimePiece-At-A-TimePiece-At-A-Time

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $100

10%

$545.45$545.45$495.87$495.87

$300.53$300.53$273.21$273.21

$ 62.09$ 62.09

$1677.15$1677.15 == PVPV00of the Mixed Flowof the Mixed Flow


50/60

0

Group-At-A-Time (#1)Group-At-A-Time (#1)Group-At-A-Time (#1)Group-At-A-Time (#1)

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $100

10%

$1,041.60$1,041.60$ 573.57$ 573.57

$ 62.10$ 62.10

$1,677.27$1,677.27== PVPV00of Mixed Flowof Mixed Flow [Using Tables][Using Tables]

$600(PVIFA10% ,2) = $600(1.736) = $1,041.60

$400(PVIFA10% ,2)(PVIF10% ,2) = $400(1.736)(0.826) = $573.57

$100 (PVIF10% ,5

) = $100 (0.621) = $62.10


51/60

1

Group-At-A-Time (#2)Group-At-A-Time (#2)Group-At-A-Time (#2)Group-At-A-Time (#2)

0 1 2 3 4

$400 $400 $400 $400$400 $400 $400 $400

PVPV00equals

$1677.30.$1677.30.

0 1 2

$200 $200$200 $200

0 1 2 3 4 5

$100$100

$1,268.00$1,268.00

$347.20$347.20

$62.10$62.10

PlusPlus

PlusPlus

F fF fF fF f


52/60

2

General Formula:

FVn

= PVPV00

(1 + [i/m])mn

n: Number of Yearsm: Compounding Periods per Year

i: Annual Interest RateFVn,m: FV at the end of Year n

PVPV00: PV of the Cash Flow today

Frequency ofFrequency of

CompoundingCompounding

Frequency ofFrequency of

CompoundingCompounding


53/60

3

Julie Miller has $1,000$1,000 to invest for2Years at an annual interest rate of

12%.

Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2)

= 1,254.401,254.40Semi FV2 = 1,0001,000(1+ [.12/2])

(2)(2)

= 1,262.481,262.48

Impact of FrequencyImpact of FrequencyImpact of FrequencyImpact of Frequency


54/60

4

Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2)

= 1,266.771,266.77

Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2)

= 1,269.731,269.73

Daily FV2 = 1,0001,000(1+[.12/365])(365)(2)

= 1,271.201,271.20

Impact of FrequencyImpact of FrequencyImpact of FrequencyImpact of Frequency

Eff ti A lEff ti A lEff ti A lEff ti A l


55/60

5

Effective Annual Interest Rate

The actual rate of interest earned

(paid) after adjusting the nominalratefor factors such as the numberofcompounding periods per year.

(1 + [ i/ m ] )m- 1

Effective AnnualEffective Annual

Interest RateInterest Rate

Effective AnnualEffective Annual

Interest RateInterest Rate

BW Eff tiBW Eff tiBW Eff tiBW Eff ti


56/60

6

Basket Wonders (BW) has a $1,000CD at the bank. The interest rate

is 6%compounded quarterly for 1year. What is the Effective Annual

Interest Rate (EAREAR)?

EAREAR = ( 1 + 6%/ 4 )4 - 1= 1.0614 - 1 = .0614 or6.14%!6.14%!

BWs EffectiveBWs Effective

Annual Interest RateAnnual Interest Rate

BWs EffectiveBWs Effective

Annual Interest RateAnnual Interest Rate


57/60

7

1. Calculate the payment per period.

2. Determine the interest in Period t.

(Loan Balance at t-1) x (i% / m)3. Compute principal paymentprincipal payment in Period t.

(Payment-Interestfrom Step 2)

4. Determine ending balance in Period t.(Balance -principal paymentprincipal paymentfrom Step 3)

5. Start again at Step 2 and repeat.

Steps to Amortizing a LoanSteps to Amortizing a LoanSteps to Amortizing a LoanSteps to Amortizing a Loan


58/60

8

Julie Miller is borrowing $10,000$10,000 at acompound annual interest rate of12%.

Amortize the loan ifannual payments are

made for5 years.Step 1:Payment

PVPV00 = R (PVIFA i%,n)

$10,000$10,000 = R (PVIFA 12% ,5)

$10,000$10,000 = R (3.605)

RR = $10,000$10,000 / 3.605 = $2,774$2,774

Amortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan Example


59/60

9

Amortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan Example

End ofYear

Payment Interest Principal EndingBalance

0 --- --- --- $10,000

1 $2,774 $1,200 $1,574 8,4262 2,774 1,011 1,763 6,663

3 2,774 800 1,974 4,689

4 2,774 563 2,211 2,478

5 2,775 297 2,478 0

$13,871 $3,871 $10,000

[Last Payment Slightly Higher Due to Rounding]


60/60

Usefulness of AmortizationUsefulness of Amortization

2.2. Calculate Debt OutstandingCalculate Debt Outstanding --The quantity of outstandingdebt may be used in financingthe day-to-day activities of thefirm.

1.1. Determine Interest ExpenseDetermine Interest Expense --Interest expenses may reduce

taxable income of the firm.

ch3 time value of money

Documents