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August, 2000 UT Department of Finance
The Time Value of Money
In order to work the problems in this module, the user should have the use of a business calculator such as the Hewlett Packard 17BII.
The author grants individuals a limited license to use this presentation. It is the sole property of the author who holds the corresponding copyrights. The user agrees not to reproduce, duplicate or distribute any copies of this presentation in any form.
The author would like to thank the Innovative Technology Center at The University of Tennessee which supported this project with a grant through the “Teaching with Technology Summer Institute.” She would also like to commend the teachers who helped her design the module.
If you have any comments or suggestions on how to improve this presentation, please e-mail the author at smurphy@utk.edu.– Copyright ©2000 Suzan Murphy
In order to work the problems in this module, the user should have the use of a business calculator such as the Hewlett Packard 17BII.
The author grants individuals a limited license to use this presentation. It is the sole property of the author who holds the corresponding copyrights. The user agrees not to reproduce, duplicate or distribute any copies of this presentation in any form.
The author would like to thank the Innovative Technology Center at The University of Tennessee which supported this project with a grant through the “Teaching with Technology Summer Institute.” She would also like to commend the teachers who helped her design the module.
If you have any comments or suggestions on how to improve this presentation, please e-mail the author at smurphy@utk.edu.– Copyright ©2000 Suzan Murphy
August, 2000 UT Department of Finance
The Time Value of MoneyThe Time Value of Money What is the “Time Value of Money”? Compound Interest Future Value Present Value Frequency of Compounding Annuities Multiple Cash Flows Bond Valuation
What is the “Time Value of Money”? Compound Interest Future Value Present Value Frequency of Compounding Annuities Multiple Cash Flows Bond Valuation
August, 2000 UT Department of Finance
Obviously, $1,000 today.
Money received sooner rather than later allows one to use the funds for investment or consumption purposes. This concept is referred to as the TIME VALUE OF MONEY!!
The Time Value of MoneyThe Time Value of Money
Which would you rather have -- $1,000 today or $1,000 in 5 years?
August, 2000 UT Department of Finance
How can one compare amounts in different time periods?How can one compare amounts in different time periods?
One can adjust values from different time periods using an interest rate.
Remember, one CANNOT compare numbers in different time periods without first adjusting them using an interest rate.
August, 2000 UT Department of Finance
Compound InterestCompound Interest
When interest is paid on not only the principal amount invested, but also on any previous interest earned, this is called compound interest.
FV = Principal + (Principal x Interest) = 2000 + (2000 x .06) = 2000 (1 + i) = PV (1 + i)
Note: PV refers to Present Value or Principal
August, 2000 UT Department of Finance
If you invested $2,000 today in an account that pays 6% interest, with interest compounded annually, how much will be in the account at the end of two years if there are no withdrawals?
Future Value (Graphic)Future Value (Graphic)
0 1 2
$2,000
FV
6%
August, 2000 UT Department of Finance
FV1 = PV (1+i)n = $2,000 (1.06)2
= $2,247.20
Future Value (Formula)Future Value (Formula)
FV = future value, a value at some future point in timePV = present value, a value today which is usually designated as time 0i = rate of interest per compounding period n = number of compounding periods
Calculator Keystrokes: 1.06 (2nd yx) 2 x 2000 =
August, 2000 UT Department of Finance
Future Value (HP 17 B II Calculator)Future Value (HP 17 B II Calculator)
2
6
2000 +/-
N
I%Yr
PV
2,247.20FV
Exit until you get Fin Menu. 2nd, Clear Data.
Choose Fin, then TVM
August, 2000 UT Department of Finance
John wants to know how large his $5,000 deposit will become at an annual compound interest rate of 8% at the end of 5 years.
Future Value ExampleFuture Value Example
0 1 2 3 4 5
$5,000
FV5
8%
August, 2000 UT Department of Finance
Calculator keystrokes: 1.08 2nd yx x 5000 =
Future Value SolutionFuture Value Solution
Calculation based on general formula: FVn = PV (1+i)n
FV5 = $5,000 (1+ 0.08)5
= $7,346.64
August, 2000 UT Department of Finance
Future Value (HP 17 B II Calculator)Future Value (HP 17 B II Calculator)
8
5000 +/-
FV
N
I%Yr
PV
7,346.64
Exit until you get Fin Menu. 2nd, Clear Data.
Choose Fin, then TVM
5
August, 2000 UT Department of Finance
Double Your Money!!!Double Your Money!!!
Quick! How long does it take to double $5,000 at a compound rate of 12% per year
(approx.)?
We will use the “Rule-of-72”.
August, 2000 UT Department of Finance
The “Rule-of-72”The “Rule-of-72”
Quick! How long does it take to double $5,000 at a compound rate of 12% per year
(approx.)?
Approx. Years to Double = 72 / i%
72 / 12% = 6 Years[Actual Time is 6.12 Years]
August, 2000 UT Department of Finance
Present ValuePresent Value
Since FV = PV(1 + i)n.
PV = FV / (1+i)n.
Discounting is the process of translating a future value or a set of future cash flows into a present value.
August, 2000 UT Department of Finance
Assume that you need to have exactly $4,000 saved 10 years from now. How much must you deposit today in an account that pays 6% interest, compounded annually, so that you reach your goal of $4,000?
0 5 10
$4,000
6%
PV0
Present Value (Graphic)Present Value (Graphic)
August, 2000 UT Department of Finance
Present Value (HP 17 B II Calculator)Present Value (HP 17 B II Calculator)
10
6
4000
PV
N
I%Yr
FV
-2,233.57
Exit until you get Fin Menu. 2nd, Clear Data.
Choose Fin, then TVM
August, 2000 UT Department of Finance
Joann needs to know how large of a deposit to make today so that the money will grow to $2,500 in 5 years. Assume today’s deposit will grow at a compound rate of 4% annually.
Present Value ExamplePresent Value Example
0 1 2 3 4 5
$2,500PV0
4%
August, 2000 UT Department of Finance
Calculation based on general formula: PV0 = FVn / (1+i)n
PV0 = $2,500/(1.04)5
= $2,054.81
Calculator keystrokes: 1.04 2nd yx 5 = 2nd 1/x X 2500 =
Present Value SolutionPresent Value Solution
August, 2000 UT Department of Finance
Present Value (HP 17 B II Calculator)Present Value (HP 17 B II Calculator)
5
4
2,500 +/-
N
I%Yr
FV
2,054.81PV
Exit until you get Fin Menu. 2nd, Clear Data.
Choose Fin, then TVM
August, 2000 UT Department of Finance
Finding “n” or “i” when one knows PV and FVFinding “n” or “i” when one knows PV and FV
If one invests $2,000 today and has accumulated $2,676.45 after exactly five years, what rate of annual compound interest was earned?
August, 2000 UT Department of Finance
(HP 17 B II Calculator)(HP 17 B II Calculator)
5
2000 +/-
2,676.45
I%Yr
N
PV
FV
6.00
Exit until you get Fin Menu. 2nd, Clear Data.
Choose Fin, then TVM
August, 2000 UT Department of Finance
General Formula:
FVn = PV0(1 + [i/m])mn
n: Number of Years
m: Compounding Periods per Year
i: Annual Interest Rate
FVn,m: FV at the end of Year n
PV0: PV of the Cash Flow today
Frequency of CompoundingFrequency of Compounding
August, 2000 UT Department of Finance
Frequency of Compounding Example Suppose you deposit $1,000 in an account that
pays 12% interest, compounded quarterly. How much will be in the account after eight years if there are no withdrawals?
PV = $1,000
i = 12%/4 = 3% per quarter
n = 8 x 4 = 32 quarters
August, 2000 UT Department of Finance
Solution based on formula:
FV= PV (1 + i)n
= 1,000(1.03)32
= 2,575.10
Calculator Keystrokes:
1.03 2nd yx 32 X 1000 =
August, 2000 UT Department of Finance
Future Value, Frequency of Compounding (HP 17 B II Calculator)Future Value, Frequency of Compounding (HP 17 B II Calculator)
32
3
1000 +/-
N
I%Yr
PV
2,575.10FV
Exit until you get Fin Menu. 2nd, Clear Data.
Choose Fin, then TVM
August, 2000 UT Department of Finance
AnnuitiesAnnuities
Examples of Annuities Include:Student Loan Payments
Car Loan Payments
Insurance Premiums
Mortgage Payments
Retirement Savings
An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.
August, 2000 UT Department of Finance
FVA3 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0 = $3,215
If one saves $1,000 a year at the end of every year for three years in an account earning 7% interest, compounded annually, how much will one have at the end of the third year?
Example of an Ordinary Annuity -- FVAExample of an Ordinary Annuity -- FVA
$1,000 $1,000 $1,000
0 1 2 3 4
$3,215 = FVA3
End of Year
7%
$1,070
$1,145
August, 2000 UT Department of Finance
Future Value (HP 17 B II Calculator)Future Value (HP 17 B II Calculator)
1,000 +/-
3
7
FV
PMT
N
I%Yr
3,214.90
Exit until you get Fin Menu. 2nd, Clear Data.
Choose Fin, then TVM
August, 2000 UT Department of Finance
PVA3 = $1,000/(1.07)1 + $1,000/(1.07)2 +
$1,000/(1.07)3 = $2,624.32If one agrees to repay a loan by paying $1,000 a year at the end of every year for three years and the discount
rate is 7%, how much could one borrow today?
Example of anOrdinary Annuity -- PVAExample of anOrdinary Annuity -- PVA
$1,000 $1,000 $1,000
0 1 2 3 4
$2,624.32 = PVA3
End of Year
7%
$934.58$873.44 $816.30
August, 2000 UT Department of Finance
Present Value (HP 17 B II Calculator)Present Value (HP 17 B II Calculator)
Exit until you get Fin Menu. 2nd, Clear Data.
Choose Fin, then TVM
PMT1,000
3 N
7 I% Yr
PV -2,624.32
August, 2000 UT Department of Finance
Suppose an investment promises a cash flow of $500 in one year, $600 at the end of two years and $10,700 at the end of the third year. If the discount rate is 5%, what is the value of this investment today?
Multiple Cash Flows ExampleMultiple Cash Flows Example
0 1 2 3
$500 $600 $10,700
PV0
5%
August, 2000 UT Department of Finance
Multiple Cash Flow SolutionMultiple Cash Flow Solution
0 1 2 3
$500 $600 $10,7005%
$476.19$544.22$9,243.06
$10,263.47 = PV0 of the Multiple Cash Flows
August, 2000 UT Department of Finance
Multiple Cash Flow Solution (HP 17 B II Calculator)Multiple Cash Flow Solution (HP 17 B II Calculator)
FIN
Flow(0)=?
Flow(1)=?
Flow(2)=?
CFLO
0
500
600
Exit until you get Fin Menu. 2nd, Clear Data.
Flow(3)=? 10,700
NVP
I%5
Calc
Exit
# Times (2) = 1
Input# Times (1) = 1
Input
Input
Input
Input
Input
August, 2000 UT Department of Finance
Bond Valuation ProblemBond Valuation Problem
Find today’s value of a coupon bond with a maturity value of $1,000 and a coupon rate of 6%. The bond will mature exactly ten years from today, and interest is paid semi-annually. Assume the discount rate used to value the bond is 8.00% because that is your required rate of return on an investment such as this.
Interest = $30 every six months for 20 periods
Interest rate = 8%/2 = 4% every six months
August, 2000 UT Department of Finance
Bond Valuation Solution (HP 17 B II Calculator)Bond Valuation Solution (HP 17 B II Calculator)
Exit until you get Fin Menu. 2nd, Clear Data
FIN TVM
1000
30
4
20
PV
PMT
FV
I% YR
N
-864.09
0 1 2 ……….… 20
30 30 30 1000
August, 2000 UT Department of Finance
Welcome to the Interactive Exercises Choose a problem; select a solution To return to this page (slide 37), use Power Point’s
Navigation Menu Choose “Go” and “By Title”
1122
33
August, 2000 UT Department of Finance
Problem #1
You must decide between $25,000 in cash today or $30,000 in cash to be received two years from now. If you can earn 8% interest on your investments, which is the better deal?
August, 2000 UT Department of Finance
Possible Answers - Problem 1
$25,000 in cash today $30,000 in cash to be received two years fro
m now Either option O.K.
Need a Hint?Need a Hint?
August, 2000 UT Department of Finance
Solution (HP 17 B II Calculator)Problem #1Solution (HP 17 B II Calculator)Problem #1
Exit until you get Fin Menu. 2nd, Clear Data
Choose FIN, then TVM
I%YR
N
FV
-25,720.16PV
30,000
8
2
Compare PV of $30,000, which is $25,720.16 to PV of $25,000. $30,000 to be received 2
years from now is better.
August, 2000 UT Department of Finance
Problem #2
What is the value of $100 per year for four years, with the first cash flow one year from today, if one is earning 5% interest, compounded annually? Find the value of these cash flows four years from today.
August, 2000 UT Department of Finance
Possible Answers - Problem 2
$400 $431.01 $452.56
Need a Hint?
Need a Hint?
August, 2000 UT Department of Finance
Solution (HP 17 B II Calculator)Problem #2Solution (HP 17 B II Calculator)Problem #2
Exit until you get Fin Menu. 2nd, Clear Data
Choose FIN, then TVMPMT
FVA=100(1.05)3 + 100(1.05)2 + 100(1.05)1 + 100(1.05)0
100
I% YR
N
431.01
4
5
FV
0 1 2 3 4
100 100 100 100
August, 2000 UT Department of Finance
Problem #3
What is today’s value of a $1,000 face value bond with a 5% coupon rate (interest is paid semi-annually) which has three years remaining to maturity. The bond is priced to yield 8%.
August, 2000 UT Department of Finance
Possible Solutions - Problem 3
$1,000 $921.37 $1021.37
Need a Hint?Need a Hint?
August, 2000 UT Department of Finance
Solution (HP 17 B II Calculator)Problem #3Solution (HP 17 B II Calculator)Problem #3
Exit until you get Fin Menu. 2nd, Clear Data
FIN TVM
1000
25
4
6
PV
PMT
FV
I% YR
N
921.37
0 1 2 ……….… 12
25 25 25 1000
August, 2000 UT Department of Finance
Congratulations!
You obviously understand this material. Now try the next problem.
The Interactive Exercises are found on slide #37.
August, 2000 UT Department of Finance
Comparing PV to FV
Remember, both quantities must be present value amounts or both quantities must be future value amounts in order to be compared.
August, 2000 UT Department of Finance
How to solve a time value of money problem. The “value four years from today” is a
future value amount. The “expected cash flows of $100 per year
for four years” refers to an annuity of $100. Since it is a future value problem and there
is an annuity, you need to solve for a FUTURE VALUE OF AN ANNUITY.
August, 2000 UT Department of Finance
Valuing a Bond
The interest payments represent an annuity and you must find the present value of the annuity.
The maturity value represents a future value amount and you must find the present value of this single amount.
Since the interest is paid semi-annually, discount at HALF the required rate of return (4%) and TWICE the number of years to maturity (6 periods).
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