Managerial Finance
Jdr
PPT_Chapter_05.pptx
Copyright © 2015 by The McGraw-Hill Companies, Inc. All rights reserved
Chapter 5
The Time Value of Money
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Topics Covered
5.1 Future Values and Compound Interest
5.2 Present Values
5.3 Multiple Cash Flows
5.4 Reducing the Chore of the Calculations (1)
5.5 Level Cash Flows: Perpetuities and Annuities
5.6 Reducing the Chore of the Calculations (2)
5.7 Effective Annual Interest Rates
5.8 Inflation & The Time Value of Money
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Future Values
Future Value
Amount to which an investment will grow after earning interest
Compound Interest
Interest earned on interest
Simple Interest
Interest earned only on the original investment
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Future Values
$
$
$
Example - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100
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Future Values
| Today Future Years | ||||||
| 1 | 2 | 3 | 4 | 5 | 6 | |
| Interest Earned | 6 | 6 | 6 | 6 | 6 | |
| Value | 100 | 106 | 112 | 118 | 124 | 130 |
Example - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100
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Future Values
Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous year’s balance
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Future Values
| Today Future Years | ||||||
| 1 | 2 | 3 | 4 | 5 | ||
| Interest Earned | 6 | 6.36 | 6.74 | 7.15 | 7.57 | |
| Value | 100 | 106 | 112.36 | 119.10 | 126.25 | 133.82 |
Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous year’s balance
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Future Values
Future Value of $100 = FV
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Future Values
= $133.82
Example - FV
What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?
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Future Values with Compounding
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Manhattan Island Sale
Note: The value of Manhattan Island land is well below this figure
Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal?
To answer, determine what $24 is worth in the year 2014, compounded at 8%
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Present Values
Discount Rate
Interest rate used to compute present values of future cash flows
Present Value
Value today of a future cash flow
Discount Factor
Present value of a $1 future payment
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Present Values
Present Value
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Present Values
Discounted Cash Flow (DCF)
Method of calculating present value by discounting future cash flows.
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Future cash flow
Present value
Present Values
$2,572
Example
You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?
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Discount Factor = DF = PV of $1
Discount factors can be used to compute the present value of any cash flow
Present Values
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Present Values with Compounding
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Present Values
Drawing a time line can help us to calculate the present value of the payments to Kangaroo Autos
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The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable.
Time Value of Money (Applications)
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Italy borrowed money for 2 years, but it did not announce an interest rate. It simply offered to sell each IOU for €963.06. What is the interest rate?
Time Value of Money (Applications)
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Value of Free Credit
Implied Interest Rates
Internal Rate of Return
Time necessary to accumulate funds
Time Value of Money (Applications)
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FV of Multiple Cash Flows
Example
You are able to put $1,200 in the bank now, and another $1,400 in 1 year. If you earn an 8% rate of interest, how much will you be able to spend on a computer in 2 years?
________________________________________________________________________________
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PV of Multiple Cash Flows
PVs can be added together to evaluate multiple cash flows
…
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PV of Multiple Cash Flows
Example
Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?
payment 8,000.00
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Present Values
Present Value
Year 0
4000/1.08
4000/1.082
Total
= $3,703.70
= $3,429.36
= $15,133.06
$4,000
$8,000
Year
0 1 2
$ 4,000
$8,000
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Spreadsheets
| Finding the Present Value of Multiple Cash Flows by Using a Spreadsheet | |||
| Time until CF | Cash Flow | Present Value | Formula in Column C |
| 0 | 8000 | $8,000 | =PV ($B$11,A4,0,-B4) |
| 1 | 4000 | $3,703.70 | =PV ($B$11,A5,0,-B5) |
| 2 | 4000 | $3,429.36 | =PV ($B$11,A6,0,-B6) |
| SUM: | $15,133.06 | =SUM(C4:C6) | |
| Discount rate: | .08 |
Example
Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?
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Financial Calculators
n is the number of periods
i is the interest rate, expressed as a percentage (not a decimal)
PV is the present value
PMT is the amount of any recurring payment
FV is the future value
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n
PMT
FV
i
PV
Example
What is the future value of Peter Minuit’s $24 investment if invested at 8% for 388 years.
Financial Calculators
Enter the number listed below the key and then push the financial function key. To get the final answer, then push FV and you will get -$233.17 trillion. Ignore the minus sign and that is the answer.
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388
0
FV
8
24
n
PMT
FV
i
PV
Financial Calculators
To get the final answer, then push PV and you will get -$2,572.02. Ignore the minus sign and that is the answer.
Example:
You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?
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2
0
3000
8
PV
n
PMT
FV
i
PV
Perpetuities & Annuities
Perpetuity
A stream of level cash payments that never ends
Annuity
Level stream of cash flows at regular intervals with a finite maturity
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Perpetuities & Annuities
PV of Perpetuity Formula
C = cash payment
r = interest rate
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Perpetuities & Annuities
Example
In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?
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Perpetuities & Annuities
Example (continued)
If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?
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PV of Annuity Formula
C = cash payment
r = interest rate
t = Number of years cash payment is received
Perpetuities & Annuities
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Perpetuities & Annuities
PV Annuity Factor (PVAF)
The present value of $1 a year for each of t years
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Perpetuities & Annuities
Example
You are purchasing a car. You are scheduled to make 3 annual installments of $8,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e., what is the PV)?
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Perpetuities & Annuities
Example - Annuity
You are purchasing a car. You are scheduled to make 3 annual installments of $8,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?
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Perpetuities & Annuities
Applications
Value of payments
Implied interest rate for an annuity
Calculation of periodic payments
Mortgage payment
Annual income from an investment payout
Future Value of annual payments
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Perpetuities & Annuities
Example: Future value of annual payments
You plan to save $3,000 every year for 4 years. Given an 8% rate of interest, what will be the FV of your account?
$13,518
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Perpetuities & Annuities
Example: Future value of annual payments
You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, how much will you have saved by the time you retire?
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Perpetuities & Annuities
Example
You are purchasing a home and are scheduled to make 30 annual installments of $10,000 per year. Given an interest rate of 5%, what is the price you are paying for the house (i.e. what is the present value)?
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Annuity Due
Level stream of cash flows starting immediately.
How does it differ from an ordinary annuity?
How does the future value differ from an ordinary annuity?
Annuity Due
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Annuity Due: Level stream of cash flows starting immediately.
Example
Suppose you invest $429.59 annually at the beginning of each year at 10% interest. After 50 years, how much would your investment be worth?
Annuities Due: Example
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Annuity Due: Level stream of cash flows starting immediately.
Example
You are purchasing a car. You are scheduled to make 3 annual installments of $8,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?
To get the final answer, push PV and you will get
-$19,894.82. Ignore the minus sign and that is the answer.
Financial Calculators
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3
-8000
0
10
PV
n
PMT
FV
i
PV
Example
You are taking out a mortgage for $100,000. You will pay it back over 30 years paying 1% per month. What is your monthly payment?
To get the final answer, push PMT and you will get
-$1,028.61
Financial Calculators
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360
PMT
0
1
100,000
n
PMT
FV
i
PV
Spreadsheets
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Effective Interest Rates
Effective Annual Interest Rate
Interest rate that is annualized using compound interest
Annual Percentage Rate
Interest rate that is annualized using simple interest
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EAR & APR Calculations
Annual Percentage Rate (APR):
Effective Annual Interest Rate (EAR):
*where MR = monthly interest rate
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Effective annual interest rate: - Interest rate that is annualized using compound interest.
Annual percentage rate: Interest rate that is annualized using simple interest.
Effective Interest Rates
Example
Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?
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Effective Interest Rates
Example
Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?
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Example
Given a 12% APR, what is the Effective Annual Rate, given monthly compounding?
To get the final answer, push FV and you will get 1.1268. Subtract 1, and your answer is 12.68%.
EAR & Financial Calculators
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12
0
FV
1
-1
n
PMT
FV
i
PV
Example
Given a 6.50 % EAR what is the APR, given monthly compounding?
To get the final answer, push i and you will get .5262. Multiply by 12, and your answer is 6.314%.
APR & Financial Calculators
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12
0
1.065
i
-1
n
PMT
FV
i
PV
Inflation
Inflation
Rate at which prices as a whole are increasing
Nominal Interest Rate
Rate at which money invested grows
Real Interest Rate
Rate at which the purchasing power of an investment increases
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Inflation
Annual U.S. Inflation Rates from 1900 - 2013
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Approximation formula:
Inflation
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Inflation
Example
If the interest rate on one year govt. bonds is 6.0% and the inflation rate is 2.0%, what is the real interest rate?
Savings
Bond
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Inflation
Remember:
Current dollar cash flows must be discounted by the nominal interest rate
Real cash flows must be discounted by the real interest rate
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