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Chapter 5
The Time Value of Money
Foundations of FinanceArthur J. Keown John D. MartinJ. William Petty David F. Scott, Jr.
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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.
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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.
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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.
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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.
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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
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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.
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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
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Future Value
- The amount a sum will grow in a certain number of years when compounded at a specific rate.
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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
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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
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Future Value
• Future Value can be increased by:• Increasing number of years of
compounding• Increasing the interest or
discount rate
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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
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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
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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
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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
i= the interest rate
PV = the present value of the future sum of money
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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
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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
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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
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Annuity
• Series of equal dollar payments for a specified number of years.
• Ordinary annuity payments occur at the end of each period
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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.
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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
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Illustration of a 5yr $500 Annuity Compounded at 6%
5
5006% 1 2 3 40
500500 500 500
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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
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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
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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.
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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
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Annuities Due
• Ordinary annuities in which all payments have been shifted forward by one time period.
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Amortized Loans
• Loans paid off in equal installments over time– Typically Home Mortgages– Auto Loans
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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?
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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.
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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
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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
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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)
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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%.