1. Cash Flow Diagrams <in> <out> 1 2 3 4 $200 $300 $150 $100
$750
four 1-year periods

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3. (unfortunately, piggy banks don’t pay interest)
Cash Flow Diagrams
Cash flow diagrams provide a simple way to visualize the cash that comes “in” and the cash that goes “out.”
Here, money is deposited into a piggy bank once a year over a four year period.
<out>
<in> 1 2 3 4
$200
$300
$150
$100
$750
(unfortunately, piggy banks don’t pay interest)
four 1-year periods

4. <in> versus <out> is a matter of perspective
Viewpoint of the Piggy Bank
Viewpoint of your Pocket
<out>
<in> 1 2 3 4
$200
$300
$150
$100
$750
<out>
<in> 1 2 3 4
$200
$300
$150
$100
$750

5. Present Amount “P” and Future Amount “F”
P = present amount (amount at t=0) F = future amount (equivalent future amount at t=n of any P at t=0)
n = number of interest periods (usually months or years)
i = interest rate
𝐹= 𝑃 1+𝑖 𝑛
Example: You deposit $10,000 in an account that earns an annual interest rate of 8%. How much will the account be worth in 10 years, assuming annual compounding?
<in>
F = ?
<out>
$10,000
𝐹= 𝑃 1+𝑖 𝑛 =$10,
=$21,589.25

6. Class Problem: You would like to collect a lump sum of $5 million when you retire in 50 years. Assuming you can earn 10% interest (compounded annually), how much do you need to invest now? Before you begin, draw a cash flow diagram.
<in>
F = $5,000,000
| | | | | | | | | | |
<out>
P = ?
𝐹= 𝑃 1+𝑖 𝑛
𝑃=𝐹 1+𝑖 −𝑛
=$5,000, −50
=$𝟒𝟐,𝟓𝟗𝟐.𝟕𝟔

7. Multiple <ins> and <outs>1 2 3
$500
$1,000
$350
$700
P = ?
What is the “present amount” for the cash flow shown here, assuming an annual interest rate of 10% compounded annually?
Stated another way, if you were a lender and wanted to earn 10% on your money, then how much could you lend now <out> considering the payments you will receive <in> over the three year period?
Bring each of the four payments back to the present:
1. $500: 𝑃= 𝐹 1+𝑖 −𝑛 = $ −0 = $500.00
This is the amount you would need at t=0 to generate the cash flow shown, assuming 10% interest.
2. $1,000: 𝑃= 𝐹 1+𝑖 −𝑛 = $1, −1 = $909.09
3. $350: 𝑃= 𝐹 1+𝑖 −𝑛 = $ −2 = $289.26
4. $700: 𝑃= 𝐹 1+𝑖 −𝑛 = $ −3 = $525.92 +
$2,224.27

8. Multiple <ins> and <outs>1 2 3
$500
$1,000
$350
$700
F = ?
Now, reverse the previous problem. Here, think of yourself as depositing the money in an account that earns 10% interest (compounded annually). How much will the cash flow be worth at the end of three years?
HINT: Be careful on the values of “n” that you use.
𝐹= 𝑃 1+𝑖 𝑛
APPLY REPEATEDLY
The $700 draws interest zero years (you withdraw it the same day you deposit it).
𝐹=$ $ $1, $
= $ $ $1, $665.50
𝑭=$2,960.50

9. Class Problem: Your company has the option of purchasing a piece of equipment that will lower labor costs OR continuing to pay your existing labor costs.
OPTION 1 Continue paying existing labor costs, assuming $1,500 is paid out at the end of each year.
OPTION 2 Purchase the new equipment to reduce labor costs:
The equipment costs $5,000.
The equipment is worth $3,000 at the end of 5 years (salvage value of the equipment).
Labor costs will be reduced from $1,500 per year down to $1,000 per year.
Assume that your money has the capacity to earn an annual interest rate of 10% compounded annually.
REQUIRED:
Draw a cash flow diagram for each case.
Evaluate the “present amount” (present cost) for each option.
Choose the best option by comparing the “present costs.” Be careful, since the present cost is really the amount of money that you would need up front to cover your expenses over the five year period.

10. Solution: P
𝑃= − − − − −5 𝑃=
=$𝟓,𝟔𝟖𝟔.𝟏𝟖 P
5000
3000
𝑃= − − − − −5 − −5
𝑃= −
𝑃=$𝟔,𝟗𝟐𝟖.𝟎𝟐
Comparison of Options: Option 1 will cost the company less money and should be chosen based on this analysis. However, there could be other factors to consider, such as the reliability of employees, product quality issues, etc.