foundation of finance

Chapter 5

The Time Value of Money

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

Chapter 5 The Time Value of Money

Pearson Prentice HallFoundations of Finance5 - 2

Learning Objectives

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

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

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



Learning Objectives

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

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

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



Learning Objectives

• Determine the present value of an uneven stream of payments

• Determine the present value of a perpetuity.

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



Principles Used in this Chapter

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



Simple Interest

Interest is earned on principal

$100 invested at 6% per year1st year interest is $6.002nd year interest is $6.003rd year interest is $6.00Total interest earned: $18.00



Compound Interest

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



Compound Interest

Interest is earned on previously earned interest

$100 invested at 6% with annual compounding

1st year interest is $6.00 Principal is $106.002nd year interest is $6.36 Principal is $112.36 3rd year interest is $6.74 Principal is $119.11Total interest earned: $19.11



Future Value

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



Future Value

FV1 = PV (1 + i)Where FV1 = the future of the investment at

the end of one year

i= the annual interest (or discount) rate

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



Future Value

What will an investment be worth in 2 years?

$100 invested at 6%FV2= PV(1+i)2 = $100 (1+.06)2

$100 (1.06)2 = $112.36



Future Value

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

compounding• Increasing the interest or

discount rate



Future Value Using Tables

FVn = PV (FVIFi,n)Where FVn = the future of the investment at

the end of n year

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

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

i= the interest rate

n= number of compounding periods



Future Value

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

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

FVn= PV (FVIF8%,7yr)= $500 (1.714) = $857



Future 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 Value

The current value of a future paymentPV = FVn {1/(1+i)n}

Where FVn = the future of the investment at the end of n years

n= number of years until payment is received


PV = the present value of the future sum of money



Present Value

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

PV = $500 {1/(1+.06)10}= $500 (1/1.791)= $500 (.558)= $279



Present Value Using Tables

PVn = FV (PVIFi,n)Where PVn = the present value of a future sum of

money

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

PVIF = Present Value interest factor of $1


n= number of compounding periods



Present Value

What is the present value of $100 to be received in 10 years if the discount rate is 6%? PVn = FV (PVIF6%,10yrs.)

= $100 (.558)= $55.80



Annuity

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

• Ordinary annuity payments occur at the end of each period



Compound Annuity

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



Compound Annuity

FV5 = $500 (1+.06)4 + $500 (1+.06)3

+$500(1+.06)2 + $500 (1+.06) + $500

= $500 (1.262) + $500 (1.191) + $500 (1.124) + $500 (1.090) + $500

= $631.00 + $595.50 + $562.00 +$530.00 + $500

= $2,818.50



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

5

5006% 1 2 3 40

500500 500 500



Future Value of an Annuity

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

the end of the nth yearsFVIFi,n= future-value interest factor or sum of

annuity of $1 for n yearsPMT= the annuity payment deposited or

received at the end of each yeari= the annual interest (or discount) raten = the number of years for which the

annuity will last



Compounding Annuity

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

FV = PMT {(FVIFi,n-1)/ i } Simplified this equation is:FV5 = PMT(FVIFAi,n)

= $500(5.637)= $2,818.50



Present Value of an Annuity

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

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



Present Value of an Annuity

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

PV = PMT(PVIFAi,n)= $500(4.212)= $2,106



Annuities Due

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



Amortized Loans

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



Payments and Annuities

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



Amortization of a Loan

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

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



Amortization Schedule

Yr. Annuity Interest Principal Balance

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

2 $2,101.58 719.76 1,381.82 3,416.60

3 $2,101.58 512.49 1,589.09 1,827.51

4 $2,101.58 274.07 1,827.51



Compounding Interest with Non-annual periods

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

Example:8% a year, with semiannual compounding for

5 years.8% / 2 = 4% column on the tablesN = 5 years, with semiannual compounding

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



Perpetuity

• An annuity that continues forever is called perpetuity

• The present value of a perpetuity is PV = PP/iPV = present value of the perpetuityPP = constant dollar amount

provided by the of perpetuityi = annuity interest (or discount

rate)



The Multinational Firm

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

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

• Inflation rate outside US difficult to predict• Inflation rate in Argentina in 1989 was

4,924%, in 1990 dropped to 1,344%, and in 1991 it was only 84%.

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