# Chapter 5 The Time Value of Money

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Chapter 5 The Time Value of Money
Foundations of Finance Arthur J. Keown John D. Martin J. William Petty David F. Scott, Jr. Chapter 5 The Time Value of Money

Foundations of Finance
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. Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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. Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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. Foundations of Finance Pearson Prentice Hall

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. Foundations of Finance Pearson Prentice Hall

Foundations of Finance
Simple Interest Interest is earned on principal \$100 invested at 6% per year 1st year interest is \$6.00 2nd year interest is \$6.00 3rd year interest is \$6.00 Total interest earned: \$18.00 Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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. Foundations of Finance Pearson Prentice Hall

Foundations of Finance
Compound Interest Interest is earned on previously earned interest \$100 invested at 6% with annual compounding 1st year interest is \$6.00 Principal is \$106.00 2nd year interest is \$6.36 Principal is \$112.36 3rd year interest is \$6.74 Principal is \$119.11 Total interest earned: \$19.11 Foundations of Finance Pearson Prentice Hall

Foundations of Finance
Future Value - The amount a sum will grow in a certain number of years when compounded at a specific rate. Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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 Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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 Foundations of Finance Pearson Prentice Hall

Foundations of Finance
Future Value Future Value can be increased by: Increasing number of years of compounding Increasing the interest or discount rate Foundations of Finance Pearson Prentice Hall

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 Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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 Foundations of Finance Pearson Prentice Hall

Future Value Using Calculators
Using any four inputs you will find the 5th. Set to P/YR = 1 and END mode. INPUTS OUTPUT N 10 I/YR 6 -100 PV PMT FV 179.10 Foundations of Finance Pearson Prentice Hall

Foundations of Finance Pearson Prentice Hall

Foundations of Finance
Present Value The current value of a future payment PV = 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 i= the interest rate PV = the present value of the future sum of money Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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 Foundations of Finance Pearson Prentice Hall

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 i= the interest rate n= number of compounding periods Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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 Foundations of Finance Pearson Prentice Hall

Present Value Using Calculators
Using any four inputs you will find the 5th. Set to P/YR = 1 and END mode. INPUTS OUTPUT N 10 PV -55.84 I/YR 6 PMT FV 100.00 Foundations of Finance Pearson Prentice Hall

Foundations of Finance
Annuity Series of equal dollar payments for a specified number of years. Ordinary annuity payments occur at the end of each period Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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. Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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 = \$ \$ \$ \$ \$500 = \$2,818.50 Foundations of Finance Pearson Prentice Hall

Illustration of a 5yr \$500 Annuity Compounded at 6%
1 2 3 4 5 6% 500 500 500 500 500 Foundations of Finance Pearson Prentice Hall

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 years FVIFi,n= future-value interest factor or sum of annuity of \$1 for n years PMT= the annuity payment deposited or received at the end of each year i= the annual interest (or discount) rate n = the number of years for which the annuity will last Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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 Foundations of Finance Pearson Prentice Hall

Future Value of an Annuity Using Calculators
Using any four inputs you will find the 5th. Set to P/YR = 1 and END mode. INPUTS OUTPUT N 5 FV -2,818.55 PV I/YR 6 PMT 500 Foundations of Finance Pearson Prentice Hall

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. Foundations of Finance Pearson Prentice Hall

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 Foundations of Finance Pearson Prentice Hall

Foundations of Finance
Annuities Due Ordinary annuities in which all payments have been shifted forward by one time period. Foundations of Finance Pearson Prentice Hall

Foundations of Finance
Amortized Loans Loans paid off in equal installments over time Typically Home Mortgages Auto Loans Foundations of Finance Pearson Prentice Hall

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? Foundations of Finance Pearson Prentice Hall

Future Value Using Calculators
Using any four inputs you will find the 5th. Set to P/YR = 1 and END mode. INPUTS OUTPUT N 4 PMT -2,101.59 PV 6,000 I/YR 15 FV Foundations of Finance Pearson Prentice Hall

Foundations of Finance
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. Foundations of Finance Pearson Prentice Hall

Amortization Schedule
Yr. Annuity Interest Principal Balance 1 \$2,101.58 \$900.00 \$1,201.58 \$4,798.42 2 719.76 1,381.82 3,416.60 3 512.49 1,589.09 1,827.51 4 274.07 Foundations of Finance Pearson Prentice Hall

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 tables N = 5 years, with semiannual compounding or 10 Use 10 for number of periods, 4% each Foundations of Finance Pearson Prentice Hall

Foundations of Finance
Perpetuity An annuity that continues forever is called perpetuity The present value of a perpetuity is PV = PP/i PV = present value of the perpetuity PP = constant dollar amount provided by the of perpetuity i = annuity interest (or discount rate) Foundations of Finance Pearson Prentice Hall

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%. Foundations of Finance Pearson Prentice Hall