Sometimes one or more payments are made in advance (leasing is a good example). These agreements call for several of the payments to be made when the transaction is closed. There may be a residual value (salvage value) at the end of the normal term.
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HP 10b Calculator  How to Calculate Multiple Advanced Payments
Calculator symbol key
The procedures in this document use the following text to represent symbol keys:
Key

Description

Text Representation


colored key; shift key

SHIFT

Calculating payment (without residual)
The following procedures calculate the amount of the periodic payment when one or more payments are made in advance. An example for calculating yield, using uneven cash flows (IRR), is also given.

Set to END mode (press SHIFT, then BEG/END if BEGIN annunciator is displayed) and store the number of payments per year.

Key in the total number of payments in the lease and press []

Key in the total number of payments made in advance and press SHIFT, then STO, 0, then N.

Key in the annual interest rate as a percentage and press I/YR

Press 1,[+/], PMT, 0, FV, PV,[+], RCL, 0,[=]then [÷]

Key in the initial loan amount and press SHIFT, then SWAP,then [=] to display the periodic payment received by the lessor.
Example of calculating payment without residual
Equipment worth $7,500 is leased for 24 months. The equipment is assumed to have no salvage value at the end of the lease. The lessee has agreed to make three payments at the time of closing. What monthly payment is necessary to yield the lessor 10 percent annually?
Set to END mode. Press SHIFT, then BEG/END if BEGIN annunciator is displayed.
Keys

Display

Description

Press 12, SHIFT, then P/YR

12.00

Sets payments per year

Press 24, [], 3, SHIFT, STO,0, then N

21.00

Stores number of even payments

Press 10, then I/YR

10.00

Stores annual interest

Press 1, [+/], then PMT

1.00

Calculates PV factor

Press 0, then FV

0.00
 
Press PV

19.19
 
Press [+], RCL, then 0=

22.19
 
Press [(], 7500, SHIFT, SWAP, then [=]

337.96

Calculates monthly payment

Calculating payment (with residual)

Set to END mode (Press SHIFT, then BEG/END if BEGIN annunciator is displayed) and store the number of payments per year.

Calculate the present value of the residual, add the initial loan amount, and store the result in register one.

Perform steps 2 through 5 of the Calculating Payment (Without Residual) routine above.

Press RCL, 1, SHIFT, SWAP, then [=] to calculate the periodic payment received by the lessor.
Example of calculating payment with residual
A copier worth $22,000 is to be leased for 48 months. The lessee has agreed to make 4 payments in advance, with a purchase option at the end of 48 months enabling them to buy the copier for 30 percent of the purchase price. What monthly payment is necessary to yield the lessor 15 percent annually?
Set to END mode. Press SHIFT, then BEG/END if BEGIN annunciator is displayed.
Keys

Display

Description

Press 12, SHIFT, then P/YR

12.00

Sets payments per year

Press 22000, [x], 30, then [%]

0.30
 
Press FV

6,600.00
 
Press 15, then I/YR

15.00

Calculates PV of residual

Press 0, then PMT

0.00
 
Press PV

6,600.00
 
Press [+], 22000, [=], SHIFT, STO, then 1

15,400.00

Calculates and stores "new" lease amount

Press 48, [], 4, SHIFT, STO, 0, then N

44.00

Stores number of even payments

Press 1 [+/] then PMT

1.00

Calculates PV factor

Press 0, then FV

0.00
 
Press PV

33.69
 
Press [+], RCL, 0 Then [=]

37.69
 
Press [(], RCL, then 1

15,400.00

Calculates monthly payment

Press SHIFT, SWAP, then [=]

408.64

Calculating yield
Solving for yield is essentially the same as solving for the internal rate of return of uneven cash flows on the HP 10b. Organize the cash flows and use the cash flow application to calculate the annual yield (the IRR).
Example of calculating yield
Equipment worth $5,000 is leased for 36 months at $145 per month. The lessee has agreed to pay the first and last month's payments in advance. At the end of the lease, the equipment may be purchased for $1,500. What is the annual yield to the lessor if the equipment is purchased? The net cash flows are as follows:
Flow #

Amount

# of Times

Initial

4,710.00

1

1

145.00

34

2

0.00

1

3

1,500.00

1

note:
Initial cash flow = (5000 + (2 x 145))
How to calculate yield:
Keys

Displays

Description

Press SHIFT, then CLEAR ALL

0.00

Clears all registers

Press 12, SHIFT, then P/YR

12.00

Sets periods per year

Press 4710, [+/], then CFj

4,710.00

Stores initial cash flow

Press 145, then CFj

145.00

Enters first cash flow

Press 34, SHIFT, then Nj

34.00

Enters number of occurrences

Press 0, then CFj

0.00

Enters second cash flow

Press 1500, then CFj

1,500.00

Enters final cash flow

Press SHIFT, then IRR/YR

18.10

Calculates annual yield to lesson

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