Computing Cash Flows

The calculator solves for these cash-flow values:

• Net present value (NPV) is the total present value of all cash flows,

including inflows (cash received) and outflows (cash paid out). A

positive NPV value indicates a profitable investment.

• Net future value (NFV) is the total future value of all cash flows. A

positive NFV value also indicates a profitable investment.

• Payback (PB) is the time required to recover the initial cost of an

investment, disregarding the present value of the cash inflows (time

value of money).

• Discounted payback (DPB) is the time required to recover the initial

cost of an investment using the present value of the cash inflows

(time value of money).

• Internal rate of return (IRR) is the interest rate at which the net

present value of the cash flows is equal to 0.

• Modified internal rate of return (MOD) considers the reinvestment

of cash when solving for IRR.

Computing NPV, NFV, PB, and DPB

1. Press ( to display the current discount rate (I).

2. Key in a value and press !.

3. Press # to display the current net present value (NPV).

4. To compute the net present value for the series of cash flows

entered, press %.

5. To compute the net future value (NFV), press #. The NFV value

appears.

6. To compute payback (PB), press #. The PB value appears.

7. To compute the payback discounted over time (DBP), press #. The

DBP value appears.

Computing IRR and MOD

1. Press ). The IRR variable and current value are displayed (based on

the current cash-flow values).

2. To compute the internal rate of return, press %. The calculator

displays the IRR value.

3. To select the reinvestment rate (RI), press #.

4. Key in the reinvestment rate value and press !.

5. To compute the modified internal rate of return, press #. The

calculator displays the MOD value.

When solving for IRR, the calculator performs a series of complex,

iterative calculations that can take seconds or even minutes to complete.

The number of possible IRR solutions depends on the number of sign

changes in your cash-flow sequence.

• When a sequence of cash flows has no sign changes, no IRR solution

exists. The calculator displays Error 5.

• When a sequence of cash flows has only one sign change, only one

IRR solution exists, which the calculator displays.

• When a sequence of cash flows has two or more sign changes:

– At least one solution exists.

– As many solutions can exist as there are sign changes.

When more than one solution exists, the calculator displays the one

closest to zero. Because the displayed solution has no financial

meaning, you should use caution in making investment decisions

based on an IRR computed for a cash-flow stream with more than

one sign change.

The time line reflects a sequence of cash flows with three sign

changes, indicating that one, two, or three IRR solutions can exist.

• When solving complex cash-flow problems, the calculator might not

find PB, DPB, IRR, and MOD, even if a solution exists. In this case, the

calculator displays Error 7 (iteration limit exceeded).

Example: Solving for Unequal Cash Flows

These examples show you how to enter and edit unequal cash-flow data

to calculate:

• Net present value (NPV)

• Net future value (NFV)

• Payback (PB)

• Discounted payback (DPB)

• Internal rate of return (IRR)

• Modified internal rate of return (MOD)

A company pays $7,000 for a new machine, plans a 20% annual return on

the investment, and expects these annual cash flows over the next six

years:

As the time line shows, the cash flows are a combination of equal and

unequal values. As an outflow, the initial cash flow (CFo) appears as a

negative value.

Year Cash Flow Number Cash Flow Estimate

Purchase CFo -$7,000

1 C01 3,000

2–5 C02 5,000 each year

6 C03 4,000

48 Cash Flow Worksheet

Entering Cash-Flow Data

Editing Cash-Flow Data

After entering the cash-flow data, you learn that the $4,000 cash-flow

value should occur in the second year instead of the sixth. To edit, delete

the $4,000 value for year 6 and insert it for year 2.

To Press Display

Select Cash Flow worksheet. ' CFo= 0.00

Enter initial cash flow. 7000 S ! CFo= -7,000.00

Enter cash flow for first year. # 3000 !

#

C01=

F01=

3,000.00

1.00

Enter cash flows for years

two through five.

# 5000 !

# 4 !

C02=

F02=

5,000.00

4.00

Enter cash flow for sixth year. # 4000 !

#

C03=

F03=

4,000.00

1.00

To Press Display

Move to third cash flow. " C03= 4,000.00

Delete third cash flow. & W C03= 0.00

Move to second cash flow. " " C02= 5,000.00

Insert new second cash flow. & X 4000 !

#

C02=

F02=

4,000.00

1.00

Move to next cash flow to

verify data.

#

#

C03=

F03=

5,000.00

4.00

Cash Flow Worksheet 49

Computing NPV, NFV, PB, and DPB

Use an interest rate per period (I) of 20%.

Answers: NPV is $7,266.44. NFV is $21,697.47. PB is 2.00. DPB is 2.60.

Computing IRR and MOD

Answer: IRR is 52.71%. MOD is 35.12%.

Example: Value of a Lease with Uneven Payments

A lease with an uneven payment schedule usually accommodates

seasonal or other anticipated fluctuations in the lessee’s cash position.

A 36-month lease has the following payment schedule and beginning-ofperiod

payments.

To Press Display

Access interest rate variable ( I= 0.00

Enter interest rate per period. 20 ! I= 20.00

Compute net present value. # % NPV= 7,266.44

Compute net future value. # % NFV= 21,697.47

Compute payback. # % PB= 2.00

Compute discounted payback. # % DPB= 2.60

To Press Display

Access IRR. ) IRR= 0.00

Compute internal rate of return. # IRR= 52.71

Select reinvestment rate (RI) # RI= 0.00

Enter reinvestment rate. 20 ! RI= 20.0

Compute modified internal rate of return. # % MOD= 35.12

Number of Months Payment Amount

4 $0

8 $5000

3 $0

If the required earnings rate is 10% per 12-month period with monthly

compounding:

• What is the present value of these lease payments?

• What even payment amount at the beginning of each month would

result in the same present value?

Because the cash flows are uneven, use the Cash Flow worksheet to

determine the net present value of the lease.

The calculator solves for these cash-flow values:

• Net present value (NPV) is the total present value of all cash flows,

including inflows (cash received) and outflows (cash paid out). A

positive NPV value indicates a profitable investment.

• Net future value (NFV) is the total future value of all cash flows. A

positive NFV value also indicates a profitable investment.

• Payback (PB) is the time required to recover the initial cost of an

investment, disregarding the present value of the cash inflows (time

value of money).

• Discounted payback (DPB) is the time required to recover the initial

cost of an investment using the present value of the cash inflows

(time value of money).

• Internal rate of return (IRR) is the interest rate at which the net

present value of the cash flows is equal to 0.

• Modified internal rate of return (MOD) considers the reinvestment

of cash when solving for IRR.

Computing NPV, NFV, PB, and DPB

1. Press ( to display the current discount rate (I).

2. Key in a value and press !.

3. Press # to display the current net present value (NPV).

4. To compute the net present value for the series of cash flows

entered, press %.

5. To compute the net future value (NFV), press #. The NFV value

appears.

6. To compute payback (PB), press #. The PB value appears.

7. To compute the payback discounted over time (DBP), press #. The

DBP value appears.

Computing IRR and MOD

1. Press ). The IRR variable and current value are displayed (based on

the current cash-flow values).

2. To compute the internal rate of return, press %. The calculator

displays the IRR value.

3. To select the reinvestment rate (RI), press #.

4. Key in the reinvestment rate value and press !.

5. To compute the modified internal rate of return, press #. The

calculator displays the MOD value.

When solving for IRR, the calculator performs a series of complex,

iterative calculations that can take seconds or even minutes to complete.

The number of possible IRR solutions depends on the number of sign

changes in your cash-flow sequence.

• When a sequence of cash flows has no sign changes, no IRR solution

exists. The calculator displays Error 5.

• When a sequence of cash flows has only one sign change, only one

IRR solution exists, which the calculator displays.

• When a sequence of cash flows has two or more sign changes:

– At least one solution exists.

– As many solutions can exist as there are sign changes.

When more than one solution exists, the calculator displays the one

closest to zero. Because the displayed solution has no financial

meaning, you should use caution in making investment decisions

based on an IRR computed for a cash-flow stream with more than

one sign change.

The time line reflects a sequence of cash flows with three sign

changes, indicating that one, two, or three IRR solutions can exist.

• When solving complex cash-flow problems, the calculator might not

find PB, DPB, IRR, and MOD, even if a solution exists. In this case, the

calculator displays Error 7 (iteration limit exceeded).

Example: Solving for Unequal Cash Flows

These examples show you how to enter and edit unequal cash-flow data

to calculate:

• Net present value (NPV)

• Net future value (NFV)

• Payback (PB)

• Discounted payback (DPB)

• Internal rate of return (IRR)

• Modified internal rate of return (MOD)

A company pays $7,000 for a new machine, plans a 20% annual return on

the investment, and expects these annual cash flows over the next six

years:

As the time line shows, the cash flows are a combination of equal and

unequal values. As an outflow, the initial cash flow (CFo) appears as a

negative value.

Year Cash Flow Number Cash Flow Estimate

Purchase CFo -$7,000

1 C01 3,000

2–5 C02 5,000 each year

6 C03 4,000

48 Cash Flow Worksheet

Entering Cash-Flow Data

Editing Cash-Flow Data

After entering the cash-flow data, you learn that the $4,000 cash-flow

value should occur in the second year instead of the sixth. To edit, delete

the $4,000 value for year 6 and insert it for year 2.

To Press Display

Select Cash Flow worksheet. ' CFo= 0.00

Enter initial cash flow. 7000 S ! CFo= -7,000.00

Enter cash flow for first year. # 3000 !

#

C01=

F01=

3,000.00

1.00

Enter cash flows for years

two through five.

# 5000 !

# 4 !

C02=

F02=

5,000.00

4.00

Enter cash flow for sixth year. # 4000 !

#

C03=

F03=

4,000.00

1.00

To Press Display

Move to third cash flow. " C03= 4,000.00

Delete third cash flow. & W C03= 0.00

Move to second cash flow. " " C02= 5,000.00

Insert new second cash flow. & X 4000 !

#

C02=

F02=

4,000.00

1.00

Move to next cash flow to

verify data.

#

#

C03=

F03=

5,000.00

4.00

Cash Flow Worksheet 49

Computing NPV, NFV, PB, and DPB

Use an interest rate per period (I) of 20%.

Answers: NPV is $7,266.44. NFV is $21,697.47. PB is 2.00. DPB is 2.60.

Computing IRR and MOD

Answer: IRR is 52.71%. MOD is 35.12%.

Example: Value of a Lease with Uneven Payments

A lease with an uneven payment schedule usually accommodates

seasonal or other anticipated fluctuations in the lessee’s cash position.

A 36-month lease has the following payment schedule and beginning-ofperiod

payments.

To Press Display

Access interest rate variable ( I= 0.00

Enter interest rate per period. 20 ! I= 20.00

Compute net present value. # % NPV= 7,266.44

Compute net future value. # % NFV= 21,697.47

Compute payback. # % PB= 2.00

Compute discounted payback. # % DPB= 2.60

To Press Display

Access IRR. ) IRR= 0.00

Compute internal rate of return. # IRR= 52.71

Select reinvestment rate (RI) # RI= 0.00

Enter reinvestment rate. 20 ! RI= 20.0

Compute modified internal rate of return. # % MOD= 35.12

Number of Months Payment Amount

4 $0

8 $5000

3 $0

If the required earnings rate is 10% per 12-month period with monthly

compounding:

• What is the present value of these lease payments?

• What even payment amount at the beginning of each month would

result in the same present value?

Because the cash flows are uneven, use the Cash Flow worksheet to

determine the net present value of the lease.