INTRODUCTION TO CORPORATE FINANCE Laurence Booth W. Sean Cleary Prepared by Ken Hartviksen and Robert Ironside

INTRODUCTION TOINTRODUCTION TO CORPORATE FINANCECORPORATE FINANCELaurence Booth Laurence Booth •• W. Sean Cleary W. Sean Cleary

Prepared byPrepared by

Ken Hartviksen and Robert IronsideKen Hartviksen and Robert Ironside

CHAPTER 5CHAPTER 5 Time Value of MoneyTime Value of Money

CHAPTER 5 – Time Value of Money 5 - 3

Lecture AgendaLecture Agenda

• Learning ObjectivesLearning Objectives• Important TermsImportant Terms• CompoundingCompounding• DiscountingDiscounting• Annuities and LoansAnnuities and Loans• PerpetuitiesPerpetuities• Effective Rates of ReturnEffective Rates of Return• Summary and ConclusionsSummary and Conclusions

– Concept Review QuestionsConcept Review Questions– Practice ProblemsPractice Problems


Learning ObjectivesLearning Objectives

• Understand the importance of the time value of Understand the importance of the time value of moneymoney

• Understand the difference between simple Understand the difference between simple interest and compound interest interest and compound interest

• Know how to solve for present value, future Know how to solve for present value, future value, time or ratevalue, time or rate

• Understand annuities and perpetuitiesUnderstand annuities and perpetuities• Know how to construct an amortization tableKnow how to construct an amortization table


Important Chapter TermsImportant Chapter Terms

• AmortizeAmortize• AnnuityAnnuity• Annuity dueAnnuity due• Basis pointBasis point• Cash flowsCash flows• Compound interestCompound interest• Compound interest factor Compound interest factor

(CVIF)(CVIF)• Discount rateDiscount rate• DiscountingDiscounting• Effective rate Effective rate

• LesseeLessee• Medium of exchangeMedium of exchange• MortgageMortgage• Ordinary annuitiesOrdinary annuities• PerpetuitiesPerpetuities• Present value interest Present value interest

factor (PVIF)factor (PVIF)• ReinvestedReinvested• Required rate of returnRequired rate of return• Simple interestSimple interest• Time value of moneyTime value of money


The Time Value of Money ConceptThe Time Value of Money Concept

• Cannot directly compare $1 today with $1 to Cannot directly compare $1 today with $1 to be received at some future datebe received at some future date– Money received today can be invested to earn a rate Money received today can be invested to earn a rate

of returnof return– Thus $1 today is worth more than $1 to be received at Thus $1 today is worth more than $1 to be received at

some future datesome future date

• The interest rate or discount rate is the The interest rate or discount rate is the variable that equates a present value today variable that equates a present value today with a future value at some later date with a future value at some later date


Opportunity CostOpportunity Cost

Opportunity cost = Alternative useOpportunity cost = Alternative use

– The opportunity cost of money is the interest rate that The opportunity cost of money is the interest rate that would be earned by investing itwould be earned by investing it

– It is the underlying reason for the time value of moneyIt is the underlying reason for the time value of money– Money today can be invested to be some greater Money today can be invested to be some greater

amount in the futureamount in the future– Conversely, if you are promised a cash flow in the Conversely, if you are promised a cash flow in the

future, it’s present value today is less than what is future, it’s present value today is less than what is promised!promised!


Choosing from Investment AlternativesChoosing from Investment Alternatives Required Rate of Return or Discount RateRequired Rate of Return or Discount Rate

• You have three choices:You have three choices:1.1. $20,000 received today$20,000 received today

2.2. $31,000 received in 5 years$31,000 received in 5 years

3.3. $3,000 per year indefinitely$3,000 per year indefinitely

• To make a decision, you need to know what To make a decision, you need to know what interest rate to useinterest rate to use

• This interest rate is known as your This interest rate is known as your required required rate of returnrate of return or or discount ratediscount rate..


Simple InterestSimple Interest

• Simple interest is interest paid or received on Simple interest is interest paid or received on only the initial investment (or principal)only the initial investment (or principal)

• At the end of the investment period, the At the end of the investment period, the principal plus interest is receivedprincipal plus interest is received

0 1 2 3 … n

I1 I2 I3 In+P

0 1 2 3 … n

I1 I2 I3 In+P


Simple InterestSimple Interest ExampleExample

PROBLEM:PROBLEM:Invest $1,000 today for a Invest $1,000 today for a five-year term and receive five-year term and receive 8 percent annual simple 8 percent annual simple interest.interest.

Year Beginning Amount Ending Amount1 $1,000 $1,0802 1,080 1,1603 1,160 1,2404 1,240 1,3205 1,320 $1,400

400,1$

400$000,1$

)80$5(000,1$

)08.000,1$5(000,1$5

Value

k)P(nPe n)Value (timSOLUTION:SOLUTION:Annual interest = $1,000 Annual interest = $1,000 × × .08 = $80 per year..08 = $80 per year.


Simple InterestSimple InterestGeneral FormulaGeneral Formula

k)P(nPe n)Value (tim [ 5-1]

Where:P = principal investedn = number of yearsk = interest rate


Compound InterestCompound Interest Compounding (Computing Future Values)Compounding (Computing Future Values)

• Simple interest problems are rare; in Simple interest problems are rare; in finance we are most interested in finance we are most interested in compound interestcompound interest

• Compound interest is interest that is Compound interest is interest that is earned on the principal amount invested earned on the principal amount invested and on any accrued interestand on any accrued interest


Compound InterestCompound Interest ExampleExample

PROBLEM:PROBLEM:Invest $1,000 today for a five-year term and receive 8 Invest $1,000 today for a five-year term and receive 8 percent annual compound interest. How much will the percent annual compound interest. How much will the accumulated value be at time 5?accumulated value be at time 5?

SOLUTION:SOLUTION:

YearBeginning

AmountEnding Amount

1 $1,000.00 $1,080.00

2 1,080.00 1,166.40

3 1,166.40 1,259.71

4 1,259.71 1,360.49

5 1,360.49 1,469.33 33.469,1$)08.1(

49.360,1$)08.1()08.1)(08.1)(08.1)(08.1(

71.259,1$)08.1()08.1)(08.1)(08.1(

40.166,1$)08.1()08.1)(08.1(

080,1$)08.1(

1

55

44

33

22

11

PFV

PPFV

PPFV

PPFV

PFV

k)(PValueFuture n


Compound InterestCompound Interest Example of Interest Earned on InterestExample of Interest Earned on Interest

PROBLEM:PROBLEM:Invest $1,000 today for a five-year term and receive 8 percent annual Invest $1,000 today for a five-year term and receive 8 percent annual compound interest.compound interest.

The Interest earned on Interest Effect:The Interest earned on Interest Effect:Interest (year 1) = $1,000 Interest (year 1) = $1,000 × .08 = $80× .08 = $80

Interest (year 2 ) =($1,000 + $80)×.08 = $86.40Interest (year 2 ) =($1,000 + $80)×.08 = $86.40

Interest (year 3) = ($1,000+$80+$86.40) × .08 = $93.31Interest (year 3) = ($1,000+$80+$86.40) × .08 = $93.31

Year Beginning Amount Ending AmountInterest earned

in the year1 $1,000.00 $1,080.00 $80.002 1,080.00 1,166.40 $86.403 1,166.40 1,259.71 $93.314 1,259.71 1,360.49 $100.785 1,360.49 1,469.33 $108.84


Compound InterestCompound InterestGeneral FormulaGeneral Formula

10n

n k)(PVFV [ 5-2]

factor interest compound theasknown is 1 nk)(

Where:FV= future valueP = principal investedn = number of yearsk = interest rate


Compound InterestCompound InterestSolution Using a Financial Calculator (TI BA II Plus)Solution Using a Financial Calculator (TI BA II Plus)

PMT PV I/Y N

Input the following variables:

0 → ; -1,000 → ; 10 → ; and 5 →

CPT FVPress (Compute) and then

PMT refers to regular paymentsFV is the future valueI/Y is the period interest rateN is the number of periods

PV is entered with a negative sign to reflect investors must pay money now to get money in the future.

Answer = $1,610.51


Compound InterestCompound Interest Simple versus Compound InterestSimple versus Compound Interest

• Compounding of interest magnifies the returns Compounding of interest magnifies the returns on an investmenton an investment

• RReturns are magnifiedeturns are magnified

• The longer they are compoundedThe longer they are compounded• The higher the rate they are compoundedThe higher the rate they are compounded

(See Figure 5-1 to compare simple and compound interest effects over (See Figure 5-1 to compare simple and compound interest effects over time)time)


Compound InterestCompound Interest Simple versus Compound InterestSimple versus Compound Interest

FIGURE 5-1

DO

LL

AR

S

Simple Compound

8,000

7,000

6,000

5,000

4,000

3,000

2,000

1,000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

YEARS


Compound InterestCompound Interest Compounded Returns over Time for Various Asset ClassesCompounded Returns over Time for Various Asset Classes

Annual Arithmetic

Average (%)

Annual Geometric Mean (%)

Yeark-End Value, 2005 ($)

Government of Canada treasury bills 5.20 5.11 $29,711Government of Canada bonds 6.62 6.24 61,404Canadian stocks 11.79 10.60 946,009U.S. stocks 13.15 11.76 1,923,692

Source: Data are from the Canadian Institute of Actuaries

Table 5-1 Ending Wealth of $1,000 Invested From 1938 to 2005 in Various Asset Classes


Compound InterestCompound Interest Discounting (Computing Present Values)Discounting (Computing Present Values)

1

1

)1(0 nnnn

k)(FV

k

FVPV

[ 5-3]


Computing Present ValueComputing Present Value

• The The present Valuepresent Value is an amount today that equates to is an amount today that equates to some larger amount in the futuresome larger amount in the future

• ExampleExample: We know we want $1,000,000 when we : We know we want $1,000,000 when we retire 40 years from today. If we can earn a 10% retire 40 years from today. If we can earn a 10% return on our money, how much should we invest return on our money, how much should we invest today? today?

0

40

1,000,000

1.10

= $22,094.93

nn

FVPV =

1+k

=

Calculator Approach:1,000,000 FV0PMT40 N10 I/YCPT PV 22,094.93


Compound InterestCompound Interest Determining Rates of Return or Holding PeriodsDetermining Rates of Return or Holding Periods

nn kPVFV )1(


Calculating the Rate of ReturnCalculating the Rate of Return

• If we know the present value, the future value and the If we know the present value, the future value and the number of time periods, we can calculate the rate of return number of time periods, we can calculate the rate of return we have earnedwe have earned

• For example, suppose we invested $5,000 six years ago For example, suppose we invested $5,000 six years ago Today, it is worth $10,000. What is the annually Today, it is worth $10,000. What is the annually compounded rate of returned?compounded rate of returned?

0

1

0

1

6

1

1

10,0001

5,000

12.25%

n

n

nn

FV = PV k

FVk =

PV

Calculator Approach:10,000 FV0 PMT5,000 +/- PV6 NCPT I/Y 12.25%


• Thus far, we have dealt only with single Thus far, we have dealt only with single payments, either today or in the futurepayments, either today or in the future

• An annuity is a stream of payments that An annuity is a stream of payments that continues for a finite period of timecontinues for a finite period of time

• If the payment occurs at the end of the If the payment occurs at the end of the period, it is an period, it is an ordinary annuityordinary annuity

• If the payment occurs at the start of the time If the payment occurs at the start of the time period, it is an period, it is an annuity dueannuity due

Annuities and PerpetuitiesAnnuities and Perpetuities AnnuitiesAnnuities


Difference Between Annuity TypesDifference Between Annuity Types

0 1 2 3

$100 $100 $100

$100 $100 $100

0 1 2 3

$100$100

Ordinary Annuity

Annuity Due


Annuities and PerpetuitiesAnnuities and Perpetuities Ordinary AnnuitiesOrdinary Annuities

11

k

k)(PMTFV

n

n

[ 5-4]

)1(

11

0

kk

PMTPVn

[ 5-5]


Annuities and PerpetuitiesAnnuities and Perpetuities Annuities DueAnnuities Due

)111

k (k

k)(PMTFV

n

n

[ 5-6]

k)(1 )1(

11

0

kk

PMTPVn

[ 5-7]


Future Value of an Ordinary AnnuityFuture Value of an Ordinary Annuity

• Assume that we want to save $2,000 at the Assume that we want to save $2,000 at the endend of of each year for the next 10 years. If we can earn 10% each year for the next 10 years. If we can earn 10% on our investments, how much will we have saved?on our investments, how much will we have saved?

10

1 1

1.10 12,000

0.10

$31,874.85

n

OrdinaryAnnuity

kFV = PMT

k

Calculator Approach:2,000 PMT0 PV10 N10 I/YCPT FV 31,874.85


Future Value of an Annuity DueFuture Value of an Annuity Due

• Assume that we want to save $2,000 at the Assume that we want to save $2,000 at the beginningbeginning of each year for the next 10 years. If we can earn 10% of each year for the next 10 years. If we can earn 10% on our investments, how much will we have saved?on our investments, how much will we have saved?

10

1 11

1.10 12,000 1.10

0.10

$35,062.33

n

AnnuityDue

kFV = PMT k

k

Calculator Approach:2nd BGN 2nd Set

2,000 PMT0 PV10 N10 I/YCPT FV 35,062.33


Relationship Between An Annuity Due and An Relationship Between An Annuity Due and An Ordinary AnnuityOrdinary Annuity

• The future value of the ordinary annuity is $ The future value of the ordinary annuity is $ 31,874.8531,874.85• The FV of the annuity due is $ The FV of the annuity due is $ 35,062.3335,062.33• The interest rate is 10%The interest rate is 10%• Now calculate how much larger the annuity due is Now calculate how much larger the annuity due is

compared to the ordinary annuitycompared to the ordinary annuity

1 0

0

%

35,062.33 31,874.85

31,874.85

10%

P P

P


Present Value of an Ordinary AnnuityPresent Value of an Ordinary Annuity

• You have just won a lottery. The Lottery Corporation gives You have just won a lottery. The Lottery Corporation gives you two options. You can take $1,000,000 at the you two options. You can take $1,000,000 at the end end of of each year for 25 years or a lump sum of $10,000,000 each year for 25 years or a lump sum of $10,000,000 today. If the appropriate discount rate is 10%, what today. If the appropriate discount rate is 10%, what should you do?should you do?

25

1 1

1 1.101,000,000

0.10

$9,077,040.02

n

OrdinaryAnnuity

kPV = PMT

k

Calculator Approach:1,000,000 PMT

0 FV25 N10 I/YCPT PV 9,077,040.02


Present Value of an Annuity DuePresent Value of an Annuity Due

Calculator Approach:2nd BGN 2nd Set

1,000,000 PMT0 FV25 N10 I/YCPT PV 9,984,744.02

25

1 11

1 1.101,000,000 1.10

0.10

$9,984,744.02

n

AnnuityDue

kPV = PMT k

k

• Lets continue with the example from the previous page, Lets continue with the example from the previous page, but now the Lottery Corporation gives you the option of but now the Lottery Corporation gives you the option of taking $1,000,000 at the taking $1,000,000 at the beginningbeginning of each year for 25 of each year for 25 years or a lump sum of $10,000,000 today. If the years or a lump sum of $10,000,000 today. If the appropriate discount rate is 10%, what should you do?appropriate discount rate is 10%, what should you do?


Annuities and PerpetuitiesAnnuities and Perpetuities PerpetuitiesPerpetuities

• A perpetuity is a stream of cash flows that A perpetuity is a stream of cash flows that goes on forevergoes on forever

• Examples of perpetuities in financial markets Examples of perpetuities in financial markets includes:includes:– Common stockCommon stock– Preferred stockPreferred stock– Consol bonds (bonds with no maturity date)Consol bonds (bonds with no maturity date)

0 1 2 3

$100 $100 $100


Annuities and PerpetuitiesAnnuities and Perpetuities PV of a PerpetuityPV of a Perpetuity

Where:

PV0 = Present value of the perpetuity

PMT = the periodic cash

K = the discount rate

[ 5-8] 0 k

PMTPV


Perpetuity: An ExamplePerpetuity: An Example

• While acting as executor for a distant relative, While acting as executor for a distant relative, you discover a $1,000 Consol Bond issued by you discover a $1,000 Consol Bond issued by Great Britain in 1814, issued to help fund the Great Britain in 1814, issued to help fund the Napoleonic War. If the bond pays annual Napoleonic War. If the bond pays annual interest of 3.0% and other long U.K. interest of 3.0% and other long U.K. Government bonds are currently paying 5%, Government bonds are currently paying 5%, what would each $1,000 Consol Bond sell for what would each $1,000 Consol Bond sell for in the market?in the market?


Perpetuity: SolutionPerpetuity: Solution

0

$1,000 0.03

0.05$30

0.05$600

PMTPV

k


Nominal Versus Effective Interest RatesNominal Versus Effective Interest Rates

• So far, we have assumed annual So far, we have assumed annual compoundingcompounding

• When rates are compounded annually, the When rates are compounded annually, the quoted ratequoted rate and the and the effective rateeffective rate are equal are equal

• As the number of compounding periods per As the number of compounding periods per year increases, the effective rate will become year increases, the effective rate will become larger than the quoted ratelarger than the quoted rate


Nominal versus Effective RatesNominal versus Effective Rates Determining Effective Annual RatesDetermining Effective Annual Rates

• Effective rate Effective rate for a period is the rate at which for a period is the rate at which a dollar invested grows over that perioda dollar invested grows over that period

1)1( m

m

QRk

[ 5-9]

Determining effective annual rate for a given compound intervalDetermining effective annual rate for a given compound interval


Nominal versus Effective RatesNominal versus Effective Rates Determining Effective Annual RatesDetermining Effective Annual Rates

1 QRek[ 5-10]

Determining effective annual rate when compounding is conducted Determining effective annual rate when compounding is conducted on a continuous basison a continuous basis


Nominal versus Effective RatesNominal versus Effective Rates Effective Rates for “Any” PeriodEffective Rates for “Any” Period

11 -)m

QR(k f

m

[ 5-11]

Determining effective rate for any period, given any quoted ratesDetermining effective rate for any period, given any quoted rates


Calculating the Effective RateCalculating the Effective Rate

1 1m

Effective

QRk

m

Where:

kEffective = Effective annual interest rate

QR = the quoted interest rate

M = the number of compounding periods per year


Example: Effective Rate CalculationExample: Effective Rate Calculation

• A bank is offering loans at 6%, compounded monthly. A bank is offering loans at 6%, compounded monthly. What is the effective annual interest rate on its loans?What is the effective annual interest rate on its loans?

12

1 1

.061 1

12

6.17%

m

Effective

QRk

m


Continuous CompoundingContinuous Compounding

• When compounding occurs continuously, we When compounding occurs continuously, we calculate the effective annual rate using calculate the effective annual rate using ee, , the base of the natural logarithms the base of the natural logarithms (approximately 2.7183)(approximately 2.7183)

1QREffectivek e


10% Compounded At Various Frequencies10% Compounded At Various Frequencies

Compounding Compounding FrequencyFrequency

Effective Annual Effective Annual Interest RateInterest Rate

22 10.25%10.25%

44 10.3813%10.3813%

1212 10.4713%10.4713%

5252 10.5065%10.5065%

365365 10.5156%10.5156%

ContinuousContinuous 10.5171%10.5171%


Calculating the Rate of ReturnCalculating the Rate of Return

• If we know the present value, the future value and the If we know the present value, the future value and the number of time periods, we can calculate the rate of return number of time periods, we can calculate the rate of return we have earnedwe have earned

• For example, suppose we invested $5,000 six years ago For example, suppose we invested $5,000 six years ago Today, it is worth $10,000. What is the annually Today, it is worth $10,000. What is the annually compounded rate of returned?compounded rate of returned?

0

1

0

1

6

1

1

10,0001

5,000

12.25%

n

n

nn

FV = PV k

FVk =

PV

Calculator Approach:10,000 FV0 PMT5,000 +/- PV6 NCPT I/Y 12.25%


Calculating the Number of PeriodsCalculating the Number of Periods

• If we know the present value, the future value and If we know the present value, the future value and the rate of return, we can calculate the number of the rate of return, we can calculate the number of time periods the money needs to be invested for.time periods the money needs to be invested for.

• For example, suppose we invested $25,000 at 8%. For example, suppose we invested $25,000 at 8%. Today, it is worth $40,000. How long has the Today, it is worth $40,000. How long has the money been invested?money been invested?

0

0

1

ln

ln 1

40,000ln

25,000

ln 1.08

6.11

n

n

n

FV = PV k

FVPV

n =k

years

Calculator Approach:40,000 FV0PMT25,000 +/- PV8.0 I/YCPT N 6.11 years


Calculating the Quoted RateCalculating the Quoted Rate

• If we know the effective annual interest rate, (kIf we know the effective annual interest rate, (kEffEff) ) and we know the number of compounding periods, and we know the number of compounding periods, (m) we can solve for the Quoted Rate, as follows:(m) we can solve for the Quoted Rate, as follows:

1

1 1mEffQR k m


When Payment & Compounding Periods DifferWhen Payment & Compounding Periods Differ

• When the number of payments per year is When the number of payments per year is different from the number of compounding different from the number of compounding periods per year, you must calculate the periods per year, you must calculate the interest rate per payment period, using the interest rate per payment period, using the following formula following formula

1 1

m

f

PerPeriod

QRk

m

Where:f = the payment frequency per year


Loan AmortizationLoan Amortization

• A blended payment loan is repaid in equal A blended payment loan is repaid in equal periodic paymentsperiodic payments

• However, the amount of principal and interest However, the amount of principal and interest varies each periodvaries each period

• Assume that we want to calculate an Assume that we want to calculate an amortization table showing the amount of amortization table showing the amount of principal and interest paid each period for a principal and interest paid each period for a $5,000 loan at 10% repaid in three equal $5,000 loan at 10% repaid in three equal annual instalments. annual instalments.


Loan Amortization: SolutionLoan Amortization: Solution

• First calculate the annual paymentsFirst calculate the annual payments

3

1 1

1 1

5,000

1 1.10

0.10

$2,010.57

n

Annuity

Annuity

n

kPV PMT

k

PVPMT

k

k

Calculator Approach:5,000 PV

0 FV3 N10 I/YCPT PMT $2,010.57


Amortization TableAmortization Table

PeriodPeriod Principal: Principal: Start of Start of PeriodPeriod

PaymenPaymentt

InterestInterest PrincipalPrincipal Principal:Principal:

End of End of PeriodPeriod

11 5,000.005,000.00 2010.52010.577

500.00500.00 1,510.571,510.57 3,489.433,489.43

22 3,489.433,489.43 2010.52010.577

348.94348.94 1,661.631,661.63 1,827.801,827.80

33 1,827.801,827.80 2010.52010.577

182.78182.78 1,827.781,827.78 00


Calculating the Balance O/SCalculating the Balance O/S

• At any point in time, the balance outstanding At any point in time, the balance outstanding on the loan (the principal not yet repaid) is on the loan (the principal not yet repaid) is the PV of the loan payments not yet made.the PV of the loan payments not yet made.

• For example, using the previous example, we For example, using the previous example, we can calculate the balance outstanding at the can calculate the balance outstanding at the end of the first year, as shown on the next end of the first year, as shown on the next slideslide


Calculating the Balance O/S after the 1Calculating the Balance O/S after the 1stst Year Year

1

2

1 1

1 1.102,010.57

.10

$3,489.42

n

t

kPV PMT

k


Loan or Mortgage ArrangementsLoan or Mortgage Arrangements MortgagesMortgages

• MortgagesMortgages – a loan involving equal – a loan involving equal ‘blended’ payments (interest and ‘blended’ payments (interest and principal) over a specified payment period principal) over a specified payment period

• Important to distinguish between “term” Important to distinguish between “term” and “amortization period”and “amortization period”– TermTerm – the period for which investors can ‘lock in’ – the period for which investors can ‘lock in’

at a fixed rateat a fixed rate

– Amortization periodAmortization period – the period over which the – the period over which the loan is to be repaidloan is to be repaid


Canadian Residential MortgagesCanadian Residential Mortgages

• By law, banks in Canada can only compound By law, banks in Canada can only compound the interest twice per year on a conventional the interest twice per year on a conventional mortgage, but payments are typically made mortgage, but payments are typically made at least monthlyat least monthly

• To solve for the payment, you must first To solve for the payment, you must first calculate the correct periodic interest ratecalculate the correct periodic interest rate


Canadian Residential MortgagesCanadian Residential Mortgages

• For example, suppose we want to calculate the For example, suppose we want to calculate the monthly payment on a $100,000 mortgage amortized monthly payment on a $100,000 mortgage amortized over 25 years with a 6% annual interest rate.over 25 years with a 6% annual interest rate.

• First, calculate the monthly interest rate:First, calculate the monthly interest rate:

2

12

1 1

.061 1

2

.004938622 0.4938622%

m

f

PerPeriod

QRk

m

or


Calculating the Monthly PaymentCalculating the Monthly Payment

• Now, calculate the monthly payment on the mortgageNow, calculate the monthly payment on the mortgage

0

0

300

1 1

1 1

100,000

1 1.004938622

.004938622

$639.81

n

t

tn

kPV PMT

k

PVPMT

k

k

Calculator Approach:100,000 PV

0FV300 N.4938622 I/YCPT PMT $639.81


Summary and ConclusionsSummary and Conclusions

In this chapter you have learned:In this chapter you have learned:– To compare cash flows that occur at different points in timeTo compare cash flows that occur at different points in time– To determine economically equivalent future values from values To determine economically equivalent future values from values

that occur in previous periods through compounding.that occur in previous periods through compounding.– To determine economically equivalent present values from cash To determine economically equivalent present values from cash

flows that occur in the future through discountingflows that occur in the future through discounting– To find present value and future values of annuities, andTo find present value and future values of annuities, and– To determine effective annual rates of return from quoted To determine effective annual rates of return from quoted

interest rates.interest rates.


Practice Problem 1Practice Problem 1Loan Payments Loan Payments

Your sister has been forced to borrow money Your sister has been forced to borrow money to pay her tuition this year. If she makes to pay her tuition this year. If she makes annual payments on the loan at year end for annual payments on the loan at year end for the next three years, and the loan is for the next three years, and the loan is for $2,500 at a simple interest rate of 6 percent, $2,500 at a simple interest rate of 6 percent, how much will she pay each year?how much will she pay each year?


Practice Problem 1Practice Problem 1Loan Payments Loan Payments

Your sister has been forced to borrow money Your sister has been forced to borrow money to pay her tuition this year. If she makes to pay her tuition this year. If she makes annual payments on the loan at year end for annual payments on the loan at year end for the next three years, and the loan is for the next three years, and the loan is for $2,500 at a simple interest rate of 6 percent, $2,500 at a simple interest rate of 6 percent, how much will she pay each year?how much will she pay each year?

33.983$

00.150$33.833$

)06.500,2($3

500,2$

PMT

[ 5-5]


CopyrightCopyright

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