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Cash flow analysis is an extension of the basic TVM concepts applied to compound interest problems when payments occur in regular periods and do not have the same value. Any financial investment can be represented as an initial investment of money and a series of later cash flows that occur in regular periods of time. Each flow of money can be positive (received) or negative (paid out) and considered as a cash flow. Common cash flow problems usually involve the calculation of the Internal Rate of Return (IRR) or the Net Present Value (NPV). The NPV expresses the amount of money resulting from the summation of the initial investment (CF0) and the present value of each anticipated cash flow (CFj) calculated to the time of the initial investment. The IRR is the discounted rate applied to all future cash flows that cause NPV = 0.
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hp calculators
HP 12C Internal Rate of Return
Cash flow and IRR calculations
Cash flow diagrams
The HP12C cash flow approach
Practice with solving cash flow problems related to IRR
How to modify cash flow entries

hp calculators
HP 12C Internal Rate of Return
Cash Flow and IRR calculations
Cash flow analysis is an extension of the basic TVM concepts applied to compound interest problems when payments
occur in regular periods and do not have the same value. Any financial investment can be represented as an initial
investment of money and a series of later cash flows that occur in regular periods of time. Each flow of money can be
positive (received) or negative (paid out) and considered as a cash flow. Common cash flow problems usually involve the
calculation of the Internal Rate of Return (IRR) or the Net Present Value (NPV).
The NPV expresses the amount of money resulting from the summation of the initial investment (CF0) and the present
value of each anticipated cash flow (CFj) calculated to the time of the initial investment. The IRR is the discounted rate
applied to all future cash flows that cause NPV = 0.
The expression that calculates the Internal Rate of Return is:
k
1− (1+ IRR)−nj
0 = CF +
CF

0
j×
 ×(1+ IRR) nj
Figure 1

IRR
1


j=

Cash flow diagrams
The cash flow diagram in Figure 1 illustrates one of the many possible situations that can be handled by the HP12C.
Composition Period
gK Last Cash Flow
gK Intermediate Cash Flow
gJ Initial Cash Flow
ga Number of consecutive occurrences of CFj
Figure 2
The HP12C cash flow approach
In the HP12C each cash flow amount is stored in its corresponding register in memory. For each cash flow amount there
is a related register to store the number of consecutive occurrences of this amount. This approach is shown below:
Registers
Cash flow
Nj
R0
CF0
N0
R1
CF1
N1
...
...
...
R6
CF6
N6
R7
CF7
N7
...
...
...
R.8
CF18
N18
R.9
CF19
N19
Figure 3
FV
CF20
N20
hp calculators
- 2 -
HP 12C Internal Rate of Return - Version 1.0

hp calculators
HP 12C Internal Rate of Return
The HP12C memory organization al ows up to 20 different cash flow amounts plus the initial investment to be stored and
handled according to the diagram in Figure 2. If any cash flow amount repeats consecutively, then it can be stored as a
grouped cash flow CFj and its corresponding Nj holds the number of occurrences, up to 99. TVM register n is used as an
index to control CF operations.
The keys to enter cash flow data are:
gJ - stores the number in the display in R0 and sets n to zero
gK - adds 1 unit to current n contents (j) and then stores the number in the display in Rj
ga - stores the number in the display(∗) in Nj; n contents (j) are not changed
(∗) The number in the display must be a positive integer from 1 to 99, otherwise ga returns
to the display and
no operation is performed.
If the last available register has already been used, gK adds 1 unit to current n contents and stores the number in
the display in TVM register FV. Any attempt to add a cash flow amount with gK after FV has already been used
or when n contents refer to a register that is not available causes
to be shown in the display and no
operation is performed.
Practice solving IRR problems
Example 1: The cash flow diagram below represents a possible investment and you were chosen to determine if it is
feasible. The success of this investment dictates your future in the company, so the analysis must be
precise and error free. What is the correct keystroke sequence to fill the HP12C registers with all data?
$ 178,500.00
$ 20,000.00
CF6
CF3
$ 7,000.00
$ 12,000.00
CF
CF4
1
3 consecutive occurrences
N3
$-10,000.00
$-8,000.00
$-130,000.00
CF
CF
CF
2
5
0
Figure 4
Solution:
Clearing all registers is not necessary to start cash flow analysis because only the registers updated with
cash flow data are used.
130000 Þ gJ
7000 gK
10000 Þ gK
Figure 5
hp calculators
- 3 -
HP 12C Internal Rate of Return - Version 1.0

hp calculators
HP 12C Internal Rate of Return
The next cash flow amount occurs three times in a sequence, so it can be entered as a grouped cash flow.
20000 gK
3 ga
Figure 6
The remaining data is entered with the following keystroke sequence:
12000 gK
8000 Þ gK
178500 gK
Figure 7
Answer:
The keystrokes presented above indicate the correct entries.
Example 2: The cash flow diagram had all of its information used to compose the cash flow data in the HP12C
memory. Show how to check that they were entered correctly.
Solution:
Now that all data is entered, checking for its correctness is possible in two ways. The most common way is
the sequential check and the keystroke sequence for this checking is as follows:
:w
Figure 8
This is the number of the last register used to store the cash flow data. It will be needed later.
:gK
Figure 9
This is the amount of CF6. The sequential checking works backwards, and each time :gK is
pressed, n is decreased by one unit. Now check CF5, CF4 and when checking CF3 verify N3 as wel .
:gK
:gK
:ga
Figure 10
hp calculators
- 4 -
HP 12C Internal Rate of Return - Version 1.0

hp calculators
HP 12C Internal Rate of Return
This is the N3 value. Whenever Nj needs to be checked, it must be recalled first. Now check the CF3 value:
:gK
Figure 11
Continue checking CF2, CF1 and stop when CF0 is shown in the display.
:gK
:gK
:gK
Figure 12
Recal n contents to the display:
:w
Figure 13
Answer:
The entries are correct.
Example 3: The investment is considered attractive if it shows at least 8% of internal rate of return. Calculate the IRR.
Solution:
To perform either IRR or NPV calculations, n must have its contents restored to the correct value:
6 w fL
(flashing)
Figure 14
Answer:
Yes, the investment is attractive based on its 9.37% internal rate of return.
How to modify cash flow entries
If it happens that a cash flow entry was wrongly entered, modifying its amount is not difficult and there is no need to
enter all data again. In fact there are two ways for doing this.
hp calculators
- 5 -
HP 12C Internal Rate of Return - Version 1.0

hp calculators
HP 12C Internal Rate of Return
Example 4: Update the amount of CF2 for $-9,500.00 and compute the new IRR after this change.
Solution 1: Type in the correct amount and store it in R2:
9500 Þ ?2 fL
Figure 15
Solution 2: Set n register to (j-1), type in the correct amount, press gK, then restore n prior to compute IRR:
1n 9500 Þ gK 6n fL
Figure 16
Answer:
The investment is still attractive based on revised IRR of 9.42%.
To modify a wrongly entered Nj, it is necessary to change the value stored in the register n.
Example 5: Now change both N3 and N4 to 2 and calculate the IRR again. The cash flow diagram now looks like this:
$ 178,500.00
$ 20,000.00
CF6
CF
$ 12,000.00
3
$ 7,000.00
CF4
CF1
2 consecutive occurrences
2 consecutive occurrences
N3
N4
$-9,500.00
$-8,000.00
CF2
CF
$-130,000.00
5
CF0
Figure 17
Solution:
For each correction, set n to match j, type in the correct Nj and press ga. After all corrections, set n to
its original value and press fL.
3n 2 ga 4n 2 ga 6 n fL
Figure 18
Answer:
The newly computed IRR is 8.77%.
hp calculators
- 6 -
HP 12C Internal Rate of Return - Version 1.0

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