1. Time Value of Money 1: Analyzing Single Cash Flows
Chapter 04
Time Value of Money 1:
Analyzing Single Cash Flows
McGraw-Hill/Irwin
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Introduction Time Value of Money (TVM)
The values of money change depend on time.
Powerful financial decision-making tool
Used by financial and nonfinancial business managers
Key to making sound personal financial decisions
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3. > $Today Introduction (cont.) TVM Basic Concept:
$1 today is worth more than $1 next year
TVM Decision Based on:
Size of cash flows
Time between cash flows
Rate of return
$Today
$ Next Year
>
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4. Organizing Cash Flows
Cash flows (cash inflow & outflow) timing key to successful business operations
Cash flow analysis
Time line shows magnitude of cash flows at different points in time
Monthly
Quarterly
Semi-annually
Annually
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5. Organizing Cash Flows Cash flow analysis *Inflow = Cash received
a positive number
*Outflow = Cash going out
a negative number
Inflow
Positive #
Outflow
Negative #
Organization
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6. Time Line Example Outflow Inflow
A helpful tool for analysis of cash flows is the time line, which shows the magnitude of cash flows at different points in time
Cash we receive is called an inflow and is represented by a positive number
Cash that leaves us is called an outflow and is represented by a negative number
Outflow
Inflow
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7. Future Value Value of an investment after one or more periods
For example: the $105 payment your bank credits to your account one year from the original $100 investment at 5% annual interest
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8. Single-period Future Value
Concept: Interest is earned on principal
Today’s cash flow + Interest = Value in 1 year
Formula:
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9. Single-period Future Value Example
Assumptions:
Invest $100 today
Earn 5% interest annually (one period)
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10. Compounding & Future Value
Concept: Compounding
Interest is earned on both principal and interest
Today’s cash flow + Interest on Principal and Interest on Interest = Value in 2 years
Formula:
To compute the two-year compounded future
value, simply use the one-year equation (4-1) twice. Suppose you leave the money invested for two years. What is the Future Value?
The total interest of $10.25 represents $10 earned on the original $100 investment plus $0.25 earned on the $5 first year’s interest
This represents compounding, i.e. earning interest on interest
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11. Compounding & Future Value Example
Assumptions:
Invest $100 today
Earn 5% interest for more than one period
To compute the two-year compounded future
value, simply use the one-year equation (4-1) twice. Suppose you leave the money invested for two years. What is the Future Value?
The total interest of $10.25 represents $10 earned on the original $100 investment plus $0.25 earned on the $5 first year’s interest
This represents compounding, i.e. earning interest on interest
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12. The Power of Compounding
Compound interest is powerful wealth-building tool  exponential growth
Let’s illustrate the power of compounding. Figure 4.2 shows the original $100 deposited, the cumulative interest earned on that deposit, the cumulative interest-on-interest earned. By the 27th year, the money from the interest-on-interest exceeds the interest earned on the original deposit. By the 40th year, interest-on-interest contributes more than double the interest on the deposit.
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13. Present Value Opposite of Future Value Future Value = Compounding
Present Value = Discounting
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14. Present Value Concept: Discounting
Value today of sum expected to be received in future
Next period’s valuation ÷ One period of discounting
Formula:
Example: If I want to end up with $100 in an account at the end of one year at 5% per year, I would need to deposit how much?
This process is called discounting.
Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV)
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15. Present Value Example Assumptions: Banks pays $105 in 1 year
Interest rate = 5% interest
Example: If I want to end up with $100 in an account at the end of one year at 5% per year, how much would I need to deposit?
This process is called discounting. Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV).
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16. 4-16

17. Present Value Over Multiple Periods
Concept: Discounting
Reverse of compounding over multiple periods
Formula:
Example: If I want to end up with $100 in an account at the end of one year at 5% per year, I would need to deposit how much?
This process is called discounting.
Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV)
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18. Present Value Over Multiple Periods Example
Assumptions:
$100 payment five years in the future
Interest rate = 5% interest
Example: If I want to end up with $100 in an account at the end of one year at 5% per year, I would need to deposit how much?
This process is called discounting.
Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV)
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19. Present Value with Multiple Rates
Concept: Discounting
Value today of sum expected to be received in future -- variable rates of interest over time
Formula:
Example: If I want to end up with $100 in an account at the end of one year at 5% per year, I would need to deposit how much?
This process is called discounting.
Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV)
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20. Present Value with Multiple Rates Example
Assumptions:
Banks pays $2,500 at end of 3rd year
Interest rate year 1 = 7%
Interest rate year 2 = 8%
Interest rate year 3 = 8.5%
Example: If I want to end up with $100 in an account at the end of one year at 5% per year, I would need to deposit how much?
This process is called discounting.
Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV)
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21. Present Value & Future Value
Concepts: Discounting & Compounding
Move cash flows around in time
Use PV Calculation to discount the Cash Flow
Use FV Calculation to compound the Cash Flow
Businesses often need to move cash flows around in time. That is no problem – we can use FV and PV to do that.
Example: We expect to receive a cash flow of $200 at the end of 3 years. What is the value of the cash flow if we move it to year 2 using a discount rate of 6%? Since we are moving the cash flow one year earlier, we use the present value calculation.
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22. PV & FV Example
Assumptions PV:
Expected cash flow of $200 in 3 years
Decision: change receipt of CF to 2 years (one year earlier)
Discount rate = 6%
PV Calculation to Discount the Cash Flow for 1 year:
Businesses often need to move cash flows around in time. That is no problem – we can use FV and PV to do that.
Example: We expect to receive a cash flow of $200 at the end of 3 years. What is the value of the cash flow if we move it to year 2 using a discount rate of 6%? Since we are moving the cash flow one year earlier, we use the present value calculation.
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23. PV & FV Example Assumptions FV:
Expected cash flow of $200 in 3 years
Decision: change receipt of CF to 5 years later
Compound rate = 6%
FV Calculation to Compound the Cash Flow for 5 years:
What about moving the $200 cash flow to year 5? Since this requires moving
the cash flow later in time by two years, we use the future value equation.
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24. Rule of 72 Concept: Compound Interest
How much time for an amount to double?
Formula: 72 / i = Time for amount to double
The Rule of 72 is based on compounding and shows that the amount of time needed for an amount to double can be approximated by dividing 72 by the interest rate. For example, at 6% it should take 12 years for any amount to double. 12 years = 72/6
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25. Rule of 72 Example Assumptions: Rule of 72 calculation:
Interest rate = 6% interest
Rule of 72 calculation:
72 = Amount of time for amount to double 6
72 / 6 = 12 years
The Rule of 72 is based on compounding and shows that the amount of time needed for an amount to double can be approximated by dividing 72 by the interest rate. For example, at 6% it should take 12 years for any amount to double. 12 years = 72/6
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26. Interest Rate to Double an Investment
Remember that this rule provides only a mathematical approximation. It’s more accurate with lower interest rates. After all, with a 72 percent interest rate, the rule predicts that it will take one year to double the money. However, we know that it actually takes a 100 percent rate to double money in one year.
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27. Computing Interest Rates
Concept: Solving for Interest Rate
Complex Calculation – Use financial calculator
Formula:
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28. Computing Interest Rates Example
Assumptions:
Bought asset for $350
Sold asset for $475
Timeframe: 3 years
Interest Rate Computation – Use financial calculator
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29. Solving for Time Concept: Solving for Time Assumptions/Known Data:
Starting Cash Flow
Interest Rate
Future Cash Flow
Complex calculation – use financial calculator
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30. Solving for Time Example
Question: When interest rates are 9%, how long will it take $5,000 to double?
Assumptions:
Interest = 9%
PV = -5,000
PMT = 0
FV =10,000
Solution: 8.04 years
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