chapter 5 the time value of money foundations of finance arthur j. keownjohn d. martin j. william...

Chapter 5

The Time Value of Money

Foundations of FinanceFoundations of FinanceArthur J. KeownArthur J. Keown John D. MartinJohn D. MartinJ. William PettyJ. William Petty David F. Scott, Jr.David F. Scott, Jr.

Pearson Prentice Hall

Foundations of Finance5 - 2

Chapter 5 The Time Value of Money

Learning ObjectivesLearning Objectives

Explain the mechanics of compounding, which is Explain the mechanics of compounding, which is how money grows over a time when it is how money grows over a time when it is invested.invested.

Be able to move money through time using time Be able to move money through time using time value of money tables, financial calculators, and value of money tables, financial calculators, and spreadsheets.spreadsheets.

Discuss the relationship between compounding Discuss the relationship between compounding and bringing money back to present.and bringing money back to present.





Define an ordinary annuity and calculate its Define an ordinary annuity and calculate its compound or future value.compound or future value.

Differentiate between an ordinary annuity and an Differentiate between an ordinary annuity and an annuity due and determine the future and annuity due and determine the future and present value of an annuity due.present value of an annuity due.

Determine the future or present value of a sum Determine the future or present value of a sum when there are nonannual compounding periods.when there are nonannual compounding periods.





• Determine the present value of an uneven Determine the present value of an uneven stream of paymentsstream of payments

• Determine the present value of a perpetuity.Determine the present value of a perpetuity.

• Explain how the international setting Explain how the international setting complicates the time value of money.complicates the time value of money.




Principles Used in this ChapterPrinciples Used in this Chapter

• Principle 2Principle 2: The Time Value of : The Time Value of Money – A Dollar Received Today Is Money – A Dollar Received Today Is Worth More Than a Dollar Received Worth More Than a Dollar Received in The Future. in The Future.




Simple InterestSimple Interest

Interest is earned on principalInterest is earned on principal

$100 invested at 6% per year$100 invested at 6% per year

11stst year year interest is $6.00interest is $6.00

22ndnd year year interest is $6.00interest is $6.00

33rdrd year year interest is $6.00interest is $6.00

Total interest earned: $18.00Total interest earned: $18.00




Compound InterestCompound Interest

• When interest paid on an When interest paid on an investment during the first investment during the first period is added to the principal; period is added to the principal; then, during the second period, then, during the second period, interest is earned on the new interest is earned on the new sum.sum.




Compound InterestCompound Interest

Interest is earned on previously earned Interest is earned on previously earned interestinterest

$100 invested at 6% with annual compounding$100 invested at 6% with annual compounding11stst year interest is $6.00 Principal is $106.00 year interest is $6.00 Principal is $106.0022ndnd year interest is $6.36 Principal is $112.36 year interest is $6.36 Principal is $112.36 33rdrd year interest is $6.74 Principal is $119.11 year interest is $6.74 Principal is $119.11Total interest earned: $19.11Total interest earned: $19.11




Future ValueFuture Value

- The amount a sum will grow in a - The amount a sum will grow in a certain number of years when certain number of years when compounded at a specific rate.compounded at a specific rate.




Future Value Future Value

FVFV1 1 = PV (1 + i)= PV (1 + i)

WhereWhere FV FV1 1 = = the future of the investment at the future of the investment at the end of one year the end of one year

i= i= the annual interest (or discount) the annual interest (or discount) rate rate

PV = PV = the present value, or original the present value, or original amount invested at the beginning amount invested at the beginning

of the first year of the first year





What will an investment be worth in What will an investment be worth in 2 years?2 years?

$100 invested at 6%$100 invested at 6%

FVFV22= PV(1+i)= PV(1+i)2 2 == $100 (1+.06)$100 (1+.06)22

$100 (1.06)$100 (1.06)22 = $112.36 = $112.36





• Future Value can be increased Future Value can be increased by:by:• Increasing number of years of Increasing number of years of

compoundingcompounding• Increasing the interest or discount Increasing the interest or discount

raterate




Future Value Using TablesFuture Value Using Tables

FVFVn n = PV (FVIF= PV (FVIFi,ni,n))

WhereWhere FV FVn n = = the future of the investment at the future of the investment at the end of n year the end of n year

PV = PV = the present value, or original the present value, or original amount invested at the beginning amount invested at the beginning of the first year of the first year

FVIF = FVIF = Future value interest factor or Future value interest factor or the compound sum of $1 the compound sum of $1

i= i= the interest ratethe interest rate

n= n= number of compounding periodsnumber of compounding periods





What is the future value of $500 invested What is the future value of $500 invested at 8% for 7 years? (Assume annual at 8% for 7 years? (Assume annual compounding)compounding)

Using the tables, look at 8% column, 7 Using the tables, look at 8% column, 7 time periods. What is the factor?time periods. What is the factor?

FVFVnn= = PV (FVIFPV (FVIF8%,7yr8%,7yr))

= = $500 (1.714) $500 (1.714)

= $857= $857




Future Value Using CalculatorsFuture Value Using Calculators

Using any four inputs you will find the 5th. Set Using any four inputs you will find the 5th. Set to P/YR = 1 and END mode.to P/YR = 1 and END mode.

INPUTS OUTPUT

N I/YR

PMT

PV

FV

10 6

0

179.10

-100




Future Value Using SpreadsheetsFuture Value Using Spreadsheets

rate (I) = 8%number of periods (n) = 7

payment (PMT) = 0present value (PV) = $500

type (0=at end of period) = 0

Future value = $856.91

Excel formula: FV = (rate, number of periods, payment, present value, type)

Entered in cell d13: = FV(d7,d8,d9,-d10,d11) Notice that present value ($500) took a negative value

If we invest $500 in a bank where it will earn 8 percent compounded annually, how much will it be worth at the end of 7 years?

Spreadsheets and the Time Value of Money




Present ValuePresent Value

The current value of a future paymentThe current value of a future paymentPVPV = FV= FVnn {1/(1+i) {1/(1+i)nn}}

WhereWhere FV FVn n = = the future of the investment at the future of the investment at the end of n years the end of n years

n= n= number of years until payment is number of years until payment is received received


PV = PV = the present value of the future sum the present value of the future sum of money of money





What will be the present value of $500 to be What will be the present value of $500 to be received 10 years from today if the discount rate received 10 years from today if the discount rate is 6%?is 6%?

PV PV = = $500 {1/(1+.06)$500 {1/(1+.06)1010}} = $500 (1/1.791)= $500 (1/1.791) = $500 (.558)= $500 (.558)

= $279= $279




Present Value Using TablesPresent Value Using Tables

PVPVn n = FV (PVIF= FV (PVIFi,ni,n))

WhereWhere PV PVn n = = the present value of a future sum of the present value of a future sum of money money

FV = FV = the future value of an investment at the future value of an investment at the end of an investment period the end of an investment period

PVIF = PVIF = Present Value interest factor of $1Present Value interest factor of $1


n= n= number of compounding periodsnumber of compounding periods





What is the present value of $100 to What is the present value of $100 to be received in 10 years if the be received in 10 years if the discount rate is 6%? discount rate is 6%?

PVPVn n = FV (PVIF= FV (PVIF6%,10yrs.6%,10yrs.))

= $100 (.558)= $100 (.558)

= $55.80= $55.80




Present Value Using CalculatorsPresent Value Using Calculators


INPUTS OUTPUT

N

I/YR

PMT

PV

FV

10

6

0

100.00

-55.84




AnnuityAnnuity

• Series of equal dollar payments for Series of equal dollar payments for a specified number of years.a specified number of years.

• Ordinary annuity payments occur Ordinary annuity payments occur at the end of each periodat the end of each period




Compound AnnuityCompound Annuity

• Depositing or investing an equal Depositing or investing an equal sum of money at the end of each sum of money at the end of each year for a certain number of years year for a certain number of years and allowing it to grow.and allowing it to grow.




Compound AnnuityCompound Annuity

FVFV55 = = $500 (1+.06)$500 (1+.06)4 4 + $500 (1+.06)+ $500 (1+.06)3 3 +$500(1+.06)+$500(1+.06)22 + $500 + $500 (1+.06) + (1+.06) + $500$500

= = $500 (1.262) + $500 (1.191) + $500 (1.262) + $500 (1.191) + $500 (1.124) + $500 (1.090) + $500 (1.124) + $500 (1.090) + $500 $500 = $631.00 + $595.50 + $562.00 += $631.00 + $595.50 + $562.00 + $530.00 + $500$530.00 + $500 = $2,818.50= $2,818.50




Illustration of a 5yr $500 Annuity Illustration of a 5yr $500 Annuity Compounded at 6%Compounded at 6%

5

500

6%1 2 3 40

500500 500 500




Future Value of an AnnuityFuture Value of an Annuity

FVFV = PMT {(FVIF= PMT {(FVIFi,ni,n-1)/ i }-1)/ i }WhereWhere FV FV n n= = the future of an annuity at the future of an annuity at

the end of the nth years the end of the nth years

FVIFFVIFi,ni,n= future-value interest factor or sum of = future-value interest factor or sum of annuity of $1 for n years annuity of $1 for n years

PMT= PMT= the annuity payment deposited or the annuity payment deposited or received at the end of each year received at the end of each year

i= i= the annual interest (or discount) ratethe annual interest (or discount) rate

n = n = the number of years for which the the number of years for which the annuity will last annuity will last




Compounding AnnuityCompounding Annuity

What will $500 deposited in the bank every year for 5 What will $500 deposited in the bank every year for 5 years at 10% be worth?years at 10% be worth?

FVFV = PMT {(FVIF= PMT {(FVIFi,ni,n-1)/ i } -1)/ i }

Simplified this equation is:Simplified this equation is:

FVFV55 = PMT(FVIFA = PMT(FVIFAi,ni,n))

= $500(5.637)= $500(5.637)

= $2,818.50= $2,818.50




Future Value of an Annuity Using Future Value of an Annuity Using CalculatorsCalculators


INPUTS OUTPUT

N

I/YR

PMT

PV

FV5

6

500

0

-2,818.55




Present Value of an AnnuityPresent Value of an Annuity

• Pensions, insurance obligations, Pensions, insurance obligations, and interest received from bonds and interest received from bonds are all annuities. These items all are all annuities. These items all have a present value. have a present value.

• Calculate the present value of an Calculate the present value of an annuity using the present value of annuity using the present value of annuity table.annuity table.




Present Value of an AnnuityPresent Value of an Annuity

Calculate the present value of a $500 annuity Calculate the present value of a $500 annuity received at the end of the year annually for received at the end of the year annually for five years when the discount rate is 6%.five years when the discount rate is 6%.

PV = PMT(PVIFAPV = PMT(PVIFAi,ni,n)) = = $500(4.212)$500(4.212)

= $2,106= $2,106




Annuities DueAnnuities Due

• Ordinary annuities in which all Ordinary annuities in which all payments have been shifted payments have been shifted forward by one time period.forward by one time period.




Amortized LoansAmortized Loans

• Loans paid off in equal installments Loans paid off in equal installments over timeover time– Typically Home MortgagesTypically Home Mortgages– Auto LoansAuto Loans




Payments and AnnuitiesPayments and Annuities

If you want to finance a new If you want to finance a new machinery with a purchase price of machinery with a purchase price of $6,000 at an interest rate of 15% $6,000 at an interest rate of 15% over 4 years, what will your over 4 years, what will your payments be?payments be?




Future Value Using CalculatorsFuture Value Using Calculators


INPUTS OUTPUT

N

I/YR

PMT

PV

FV

4

15

6,000

0

-2,101.59




Amortization of a LoanAmortization of a Loan

• Reducing the balance of a loan via Reducing the balance of a loan via annuity payments is called amortizing.annuity payments is called amortizing.

• A typical amortization schedule looks at A typical amortization schedule looks at payment, interest, principal payment payment, interest, principal payment and balance.and balance.




Amortization ScheduleAmortization Schedule

Yr.Yr. AnnuityAnnuity InterestInterest PrincipalPrincipal BalanceBalance

11 $2,101.58$2,101.58 $900.00$900.00 $1,201.58$1,201.58 $4,798.42$4,798.42

22 $2,101.58$2,101.58 719.76719.76 1,381.821,381.82 3,416.603,416.60

33 $2,101.58$2,101.58 512.49512.49 1,589.091,589.09 1,827.511,827.51

44 $2,101.58$2,101.58 274.07274.07 1,827.511,827.51




Compounding Interest with Non-Compounding Interest with Non-annual periodsannual periods

If using the tables, divide the percentage by the If using the tables, divide the percentage by the number of compounding periods in a year, and number of compounding periods in a year, and multiply the time periods by the number of multiply the time periods by the number of compounding periods in a year.compounding periods in a year.

Example:Example:

8% a year, with semiannual compounding for 5 8% a year, with semiannual compounding for 5 years.years.

8% / 2 = 4% column on the tables8% / 2 = 4% column on the tables

N = 5 years, with semiannual compounding or 10N = 5 years, with semiannual compounding or 10

Use 10 for number of periods, 4% eachUse 10 for number of periods, 4% each




PerpetuityPerpetuity

• An annuity that continues forever is An annuity that continues forever is called perpetuitycalled perpetuity

• The present value of a perpetuity is The present value of a perpetuity is

PV = PP/iPV = PP/i PVPV = present value of the perpetuity = present value of the perpetuity PPPP = constant dollar amount = constant dollar amount provided by the of perpetuity provided by the of perpetuity

ii = annuity interest (or discount = annuity interest (or discount rate) rate)




The Multinational FirmThe Multinational Firm

• Principle 1Principle 1- The Risk Return Tradeoff – We - The Risk Return Tradeoff – We Won’t Take on Additional Risk Unless We Won’t Take on Additional Risk Unless We Expect to Be Compensated with Additional Expect to Be Compensated with Additional ReturnReturn

• The discount rate is reflected in the rate of The discount rate is reflected in the rate of inflation.inflation.

• Inflation rate outside US difficult to predictInflation rate outside US difficult to predict• Inflation rate in Argentina in 1989 was 4,924%, Inflation rate in Argentina in 1989 was 4,924%,

in 1990 dropped to 1,344%, and in 1991 it was in 1990 dropped to 1,344%, and in 1991 it was only 84%.only 84%.

chapter 5 the time value of money foundations of finance arthur j. keownjohn d. martin j. william...

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