Mathematics of Leasing - The Alta Group · Mathematics of Leasing Use of the HP-12C/17B Preliminary...

Mathematics of Leasing

Use of the HP-12C/17B

Preliminary Study Material

To Accompany

Carpediem International Learning Services Courses

Carpediem International Learning Services 124 S. 400 E., Suite 310 Salt Lake City, UT 84111

(801) 322-4499 phone (801) 322-5454 fax

Foreword

This introductory material provides a working knowledge of the HP-12C and HP-17B/18C/19B financial calculators. Numerous examples are provided to familiarize you with your calculator. For each example, the HP-12C solution is given, immediately followed by the keystrokes for the HP-17B/18C/19B.

You should read this material carefully and work through the examples prior to attending the seminar. It is essential that you do so, since numerous concepts and leasing issues are illustrated with the calculator. By coming prepared to use the calculator, you can focus more effectively on learning the material presented during the seminar.

Mathematics of Leasing

Basic Functions Page

HP-12C …………………………………………………………………………….. 5

HP-17B …………………………………………………………………………….. 8

HP-18C/19B Conversions ………………………………………………………. 11

Present Value ………………………………………………………..…………………… 13

Internal Rate of Return …………………………………………………………………. 32

Amortization of Lease Payments ………………………………………………………. 38

Note: For additional clarification, participants may wish to review similar material contained in Sections 1,3 and 4 of the HP-12C Owner’s Handbook and Problem Solving Guide; Sections 1,2,4,5 and 6 of the HP-17B Owner’s Manual, or Sections 1,2,4 and 5 of the HP-18C and HP-19B Owner’s Manual.

HP-12C Basic Functions

Power On and Off

To turn the calculator on, press the ON key, which is located at the lower, left-hand corner of the keyboard.

To turn the calculator off, push the ON key again.

The Keyboard

Most keys on the HP-12C perform two or even three functions.

The White Keys - the primary function of a key is indicated by the characters printed in white on the upper flat face of the key. To specify the white function, press the key alone.

The Gold Keys - one alternate function of a key is indicated by the characters printed in gold above the key. To specify the gold function press the gold prefix key f, then press the function key.

The Blue Keys - another alternate function of a key is indicated by the characters printed in blue on the lower face of the key. To specify the blue function, press the blue prefix key g, then press the function key.

For example, the key in the upper left-hand corner of the keyboard contains the n (# of periods) function in white, the AMORT (amortization) function in gold, and the 12x (changing periods from years to months) function in blue.

To specify n, press n.

To specify AMORT, press f AMORT.

To specify 12x, press g 12x.

Setting the Number of Decimal Places

Press f then the number of decimal places desired. The calculator rounds to this number. You can round the answer up to 9 decimal places.

Negative Numbers

To make a displayed number negative, press the CHS (change sign) key.

To make a negative number positive, press the CHS key.

Simple Arithmetic Calculations: +, -, ÷, x

To multiply 10 x 6, proceed as follows:

Key in 10, the first number.

Press ENTER to separate the first number from the second.

Key in 6, the second number

Press X, the desired operation, and 60, the product of 10 X 6 appears. Other arithmetic operations are performed by pushing the appropriate function key: +, -, or ÷. Note there is no equal sign on the HP-12C.

To divide 182 by 13:

Key in 182, the first number

Press ENTER.

Key in 13.

Press ÷ and the answer, 14, appears.

Clearing the Calculator

To CLEAR THE DISPLAY ONLY - before having performed a function, press CLX.

To CLEAR THE CALCULATOR - after performing a financial or arithmetic function, press f REG (same as f CLX).

HP-17B Basic Functions

Power On and Off

To turn the calculator on, press the CLR key, which is located at the lower left hand corner of the keyboard.

To turn the calculator off, press the GOLD shift key and then the CLR key.

The Keyboard

The HP-17B features a two-line display.

Main Labels - the bottom line of the display shows each of the five major menu labels.

Calculator Line - the top line of the display shows the numbers entered and the results of calculations.

Many keys on the HP-17B perform two functions.

Alternate functions are specified by first pressing the GOLD key.

Setting the Number of Decimal Places

Press the DSP key, which displays two alternative menus.

FIX - allows you to set the decimal places you desire (from 0 to 11).

ALL - the calculator displays a number as precisely as possible. (12 digits maximum)

To select FIX or ALL, press the ^ key below the desired options.

Negative Numbers

The +/- key changes the sign of a number.

To make a displayed number negative, press the +/- key.

To make a negative number positive, press the +/- key.

Simple Arithmetic Calculations: +, -, x, ÷

To multiply 10 x 6, proceed as follows:

Key in 10, the first number.

Press the x key.

Key in 6, the second number.

Press the = key.

Note: The entire calculation as input, appears in the display. When a calculation has been completed, pressing another DIGIT key starts a new calculation. Pressing an OPERATOR key (+, -, x, ÷) continues the calculation.

Clearing the Calculator

The HP-17B has three different means of clearing information.

← Erases characters before the cursor.

CLR - Clears the calculator line.

CLEAR DATA - Gold key that clears all information in the current work area.

Other Features

The HP-17B allows the user to complete long calculations without pressing = after each intermediate calculation. Compare:

36 + 86.3

and 36 + 86.3

The keystrokes are as follows;

36 + 8 ÷ 6.3 = 6.98 and 36 + (8 ÷ 6.3) = 37.27

Notice that parentheses are used to designate the order of calculation.

The Main Menu

This menu is the initial menu that leads to other options and can be retrieved by pressing the Gold Shift key and MAIN. The menus guide you through the calculation.

The EXIT key is used to leave a current menu and go back to the previously displayed menu. The GOLD CLEAR DATA key clears all the data in the currently displayed menu, giving a clean slate for new calculations.

HP-18C/19B Conversions

The HP-18C and HP-19B (18C/19B) calculators are very similar in format and function to the HP-17B (17B). The keyboard layouts, however, differ slightly. Instead of the alphabetic functions being built into the menu branch system, as in the 17B, the alphabetic functions are contained in the left-hand side of the 18C/19B calculators when they are open. The 17BII is the same as the 17B except you can choose between RPN and algebraic logic for the arithmetical functions.

Keyboard

The location of some of the keys on the keyboard of the 18C/19B also differs slightly from the 17B. For instance, the gold key is located at the lower left hand side of the 17B. The gold key for the 18C/19B calculators is located on the upper left-hand side, just below the display.

Turning the 18C/19B on and off requires a process different from the 17B. To turn the 18C/19B on or off, use the On key in the lower left hand side of the calculator. Setting the number of decimal places in the 18C/19B calculator, on the other hand, is the same as in the 17B. the key for doing this is labeled DISP on the 18C/19B, instead of DSP, as on the 17B.

The major difference between the 18C/19B and the 17B calculator is the display area. The 18C/19B has a three-line display versus a two-line display in the 17B. This feature allows you to visually trace more information.

TVM

When using the TVM branch menu some of the variables used in calculations are shown in the display area of the 18C/19B. This eliminates the need to utilize the RCL function to review variables used in performing TVM calculations. The 17B is able to only display the most recently entered variable or the answer to a particular query at any one time.

Amortization information is entered into the 18C/19B in the same manner as in the 17B. However, the display of the 18C/19B provides the user with amortization data. This saves time when writing down the amounts for interest, principal and remaining balance.

CFLO

When using the CFLO branch menu, the additional display space allows the use of an arrow in the left-hand side of the display to move back and forth between the FLOW and #TIMES elements. The arrow prompts you as to the information currently required for entry. This eliminates the need for use of the #TIMES PROMPTING: ON or OFF used in the 17B calculator.

The cash flow amount at time zero also is labeled differently in the 18C/19B. The 17B asks for FLOW (0) for any cash flows occurring at inception. The 18C/19B asks for INITIAL FLOW. When entering a series of cash flows in the 17B, you press the EXIT key to calculate IRR or NPV values. When using the 18C/19B, this EXIT step is not necessary. Press the menu option CALC instead.

Conclusion

A last difference is the ability of the 18C/19B calculator to perform a PLOT function in the CFLO branch menu. This function plots a graph of NPV versus I% for the cash flows in the current list. The 17B does not have this capability. To further understand the usage of this function in the 18C/19B calculator, please refer to Section V of the 19B owner’s manual.

Present Value Computations

Introduction

Present value is used throughout leasing. For instance, one cannot complete a lease versus buy analysis, or structure a lease without familiarity with, and use of, present value computations. Present value concepts also are utilized when determining whether a lease is finance versus operating.

The following information is essential in calculating present values:

The number and amount(s) of the future cash flow(s).

The number of periods over which the cash flow(s) will be discounted.

The discount rate to be used in present valuing the cash flows.

This section presents 12C (followed by 17B) solutions to the following four present value problems:

PV1 Present value of one.

Pva Present value of an ordinary annuity with payments in arrears - annuity in arrears.

Pva Due Present value of an annuity due with payments in advance - annuity in advance.

PV Uneven Payments Multiple uneven cash flows.

Present Value of One-PV1

The present value of one (PV1) refers to the present value of a single cash flow, such as a purchase option or a nonrecurring expense.

In order to compute PV1, the following information is keyed into the calculator:

17B 12C

N n the number of periods over which the cash flow is to be discounted.

FV FV the amount of the future cash flow (inflow or outflow).

I% YR i the discount rate.

With this information entered into the calculator, the present value of the future cash flow is calculated by pressing the PV key. The proceeding data can be entered into the calculator in any sequence.

Example

Assumption:

A lessor has a residual of $100,000 at the end of a 48-month lease that needs to be present valued. The discount rate to be used in the analysis is 12% per year or 1% per month.

Solution HP-12C

Keystrokes Display Explanations

f REG 0.0000 Clears all registers

f 2 0.00 Rounds answers to 2 decimal places

100,000 CHS FV -100,000.00 Enters purchase option cost as a negative amount

48 n 48.00 Enters number of discount periods

1 i 1.00 Enters monthly discount rate

PV 62,026.04 Calculates present value of the purchase option.

Note: ALWAYS CLEAR THE CALCULATOR PRIOR TO BEGINNING A NEW CALCULATION. The HP-12C has a continuous memory and will mix old data with new unless the calculator is cleared first by pressing f REG.

Solution HP-17B


CLEAR DATA 0 Clears all data from menus

DSP SELECT DISPLAY FORMAT Initiates decimal point selection menu

FIX TYPE # DIGITS (0-11); Prompts selection of # of decimal PRESS [INPUT] points

2 INPUT 0.00 Rounds answers to 2 decimal places

FIN SELECT A MENU Prompts selection of a menu

TVM 12 P/YR BEGIN MODE Indicates 12 payments per year and the payment mode

100,000 +/- FV FV = -100,000.00 Enters purchase option cost as a negative amount

48 N N = 48.00 Enters number of discount periods

12 I%YR I%YR = 12.00 Enters annual discount rate

PV PV = 62,026.04 Calculates present value of the purchase option

Explanation:

62,026.04 represents the present value of 100,000 discounted over 48 months at 1.0% per month.

Example:

Assumptions:

Assume that a lessor will receive a guaranteed residual of 125,000 from a lessee at the end of a 60 month lease. What is the present value of the guaranteed residual using an annual discount rate of 13% (or 1.08% per month)?

Solution HP-12C



125,000 CHS FV -125,000.00 Enters amount of future cash flow

13 g 12÷ 1.08 Converts discount rate to a monthly rate; 12÷ is the blue function for the i key as explained below.

60 n 60.00 Enters numbers of discount periods

PV 65,484.23 Calculates present value of the purchase option.

Note: An annual discount rate is converted to its monthly equivalent by keying in the annual discount rate, and then pressing g 12÷. The monthly discount rate is automatically stored in the i register.

Solution HP-17B



TVM 12P/YR BEGIN MODE Indicates 12 payments per year and the payment mode

125,000 +/- FV FV = -125,000.00 Enters amount of future cash flow


60 N N = 60.00 Enters number of discount periods

PV PV = 65,484.23 Solves for present value

Explanation

The present value of 125,000 discounted 60 periods at 13% is 65,484.23.

Present Value of an Ordinary Annuity (PVa) - Annuity in Arrears

An annuity in arrears refers to a stream of future cash flows, such as lease payments, that are due at the end of each period; in other words, the first payment or cash flow is received at the end of the first period, and each subsequent payment arrives at the end of each succeeding period.

Any calculation of an annuity in arrears must be preceded by an indication to the calculator that the cash flows will be received at the end of each period. On the HP-12C, this is achieved by pressing g end (same as g 8). It is not necessary to press g end if the calculator is already in the END mode from a previous computation.

On the HP-17B, this is accomplished as follows:



TVM 12P/YR BEGIN MODE Displays whether in BEGIN or END payment mode

OTHER 12P/YR BEGIN MODE Initiates new menu

END 12P/YR END MODE Changes payment mode to arrears

EXIT 12P/YR END MODE Returns to TVM menu to continue present value computations

The HP can calculate the present value of future cash flows whether the payments are even (the same amount each period) or uneven. The examples in this subsection demonstrate even cash flows.

In order to compute PVa, the calculator must be given the following information:

17B 12C

I% YR i discount rate

N n the number of cash flows (not periods).

PMT PMT The amount of the annuity cash flows.

END g END The ordinary annuity (or annuity in arrears) designation

With this information entered into the calculator the present value can be calculated by pressing the PV key.

Example:

Assumptions:

A four-year lease has 16 quarterly lease payments in arrears of 34,400 each. Present value the stream of even (or equal) payments using a 14% annual discount rate (3.50% quarterly).

Solution HP-12C



g END 0.00 Indicates that cash flows are in arrears

34,400 CHS PMT -34,400.00 Enters amount of each payment

14 ENTER 14.00

4 ÷ i 3.50 Enters quarterly discount rate

16 n 16.00 Enters number of cash flows

PV 416,037.62 Solves for present value

Solution HP-17B



TVM 12P/YR BEGIN MODE Selects menu necessary to do present value calculation

OTHER 12P/YR BEGIN MODE Selects menu necessary to change payment mode and frequency

4 P/YR 4 P/YR BEGIN MODE Changes payment frequency

END 4 P/YR END MODE Changes payment mode to END

EXIT 4 P/YR END MODE Exits to previous menu

34,400 +/- PMT PMT = -34,400.00 Enters amount of each payment


16 N N = 16.00 Enters number of cash flows


Explanation:

The present value of 16 even quarterly payments of 34,400 discounted at 14% is 416,037.62.

Note: Converting an annual discount rate to its quarterly equivalent is accomplished by keying in the annual discount rate, pressing ENTER, then pressing 4 ÷ and finally, entering the value shown by pressing the i key. This stores the quarterly discount rate in the i register.

Example:

Assumptions:

A lessee must find the present value of 60 lease payments of 14,000 due at the end of each period. The lessee’s discount rate is 13.5%.

Solution HP-12C



g END 0.00 Indicates that cash flows are in arrears

14,000 CHS PMT -14,000.00 Enters amount of each payment

13.5 g 12÷ 1.13 Enters monthly discount rate



Solution HP-17B



TVM 12P/YR END MODE Selects menu necessary to do present value calculation


13.5 I%YR I%YR = 13.50 Enters annual discount rate



Present Value of an Annuity Due (PVa Due)

An annuity due (annuity in advance) is similar to an ordinary annuity except that payments are received at the beginning of each period (the first one being due at time “0”). Annuities due frequently occur in leasing since many leases begin with a payment due in advance before the lessee has the right to use the equipment.

Prior to an annuity due calculation, the calculator must receive an indication that cash flows will be received at the beginning of each period. On the HP-12C, this is done by pressing g BEG. The display indicates the BEGIN mode.

On the 17B, this is accomplished as follows:



TVM 12P/YR END MODE Displays whether in BEGIN or END payment mode

OTHER 12P/YR END MODE Initiates new menu

BEGIN 12P/YR BEGIN MODE Changes payment mode to advance

EXIT 12P/YR BEGIN MODE Returns to TVM menu to continue present value computation

Example:

Assumptions:

Calculate the present value of 20 quarterly lease payments of 7,000 each, due in advance. The lease payments are to be discounted at an annual rate of 11.5%.

Solution HP-12C



g BEG 0.00 Sets calculator so that the annuity cash flows are received in advance-at the beginning of each period

7,000 CHS PMT -7,000.00 Enters amount of monthly cash flow

11.5 ENTER 11.50

4 ÷ i 2.88 Enters quarterly discount rate



BEGIN

Solution HP-17B



TVM 12P/YR END MODE Selects menu necessary to do present value calculation

OTHER 12P/YR END MODE Selects menu necessary to change payment mode and frequency

4 P/YR 4 P/YR END MODE Changes payment frequency

BEGIN 4 P/YR BEGIN MODE Changes payment mode to BEGIN

EXIT 4 P/YR BEGIN MODE Exits to previous menu


11.5 I%YR I%YR = 11.50 Enters annual discount rate



Explanation:

The present value of 20 even quarterly payments in advance of 7,000, discounted at 11.5%, is 108,385.13.

Example:

Assumptions:

A lessee desires to compute the present value of 48 monthly lease payments of 5,000, in advance. Use 13% to discount the lease payments.

Solution HP-12C



g BEG 0.00 Sets calculator so that the annuity cash flows are received in advance-at the beginning of each period

5,000 CHS PMT -5,000.00 Enters amount of monthly cash flow

13 g 12÷ 1.08 Enters monthly discount rate



Solution HP-17B



TVM 12P/YR BEGIN MODE Selects menu necessary to do present value calculation





BEGIN

Present Value of Multiple Uneven Cash Flows

Many lease contracts involve uneven rents. If cash flows are uneven, use of the blue keyboard is necessary on the HP-12C. The blue keyboard accommodates leases requiring an initial cash flow amount that differs from subsequent cash flows, or lease payments.

Three blue keys are used to determine the present value of uneven cash flows. Remember that g must be pressed to activate the blue keyboard followed by pressing any of the three function keys CFo, CFj, or Nj which appear on the top row of the calculator keys.

g CFo is used for any cash flows occurring at the inception of an investment, lease, or loan.

g CFj is used for cash flows occurring subsequent to the inception (i.e., lease payments, purchase options). Such cash flows are entered in chronological order.

g Nj is used to designate the number of times a particular cash flow CFj repeats. This key is not required if a cash flow occurs only once.

The annuity due or ordinary annuity keystroke (g BEG, g END) is ignored when using the blue keyboard since the net amount of all cash flows occurring at the inception of the lease is input into the CFo register, and subsequent cash flows are entered into the CFj register chronologically.

On the HP-17B, if uneven cash flows occur, use of the cash flow (CFLO) menu is required. This menu is accessed as follows:



CFLO FLOW (0) = ? Prompts for initial cash flow at time

0

Note: Flow (0) = ? appears if the current list is empty. If the list is not empty FLOW (1 or more) = ? will appear. To clear the list press CLEAR DATA YES.

The HP-17B will prompt as to the amount of the cash flow and the number of times the cash flow occurs. Both numbers are entered by using the input key.

Example:

Display Input Explanations FLOW (0) = ? 1,000 Cash flow at time 0

FLOW (1) = ? 300 Amount of 1st cash flow

# TIMES (1) = 1 3 Number of times 1st cash flow occurs

FLOW (2) = ? 400 Amount of 2nd cash flow

# TIMES (2) = 1 2 Number of times 2nd cash flow occurs

The # TIMES prompting may be deactivated when all the cash flows are ungrouped (that is, they are all different). The HP-17B assumes each cash flow occurs only once. This is accomplished as follows:



CFLO FLOW (0) = ? Prompts for initial cash flow

#T? Times prompting OFF Deactivates prompting for number of cash flows

FLOW (1) = ?

Note: Times prompting ON can be activated by pressing #T? again.

Example

Assumptions:

A lease requires 48 payments of 2,400 per month with three advance payments due at the inception of the lease. What is the present value of the lease payments discounted at 12.75%?

Solution HP-12C



12.75 g 12÷ 1.06 Divides the annual interest rate by 12

7,200 g CFo 7,200.00 Places 3 advance payments in the CFO register

2,400 g CFj 2,400.00 Places the next cash flow in the CFj register

45 g Nj 45.00 Places the number of times the immediately preceding CFj repeats

f NPV 92,694.15 Computes the present value of the cash flows

Solution HP-17B




7,200 INPUT FLOW (1) = ? Prompts for next cash flow and 7,200.00 shows input of initial cash flow

2,400 INPUT # TIMES (1) = 1 Prompts for number of times 1st cash 1.00 flow occurs

45 INPUT FLOW (2) = ? Prompts for next cash flow and 45.00 shows number of times cash flow (1) occurs

EXIT FLOW (2) = ? Exits from cash flow input and displays next menu

CALC NPV, NUS, NFV, NEED I% Indicates that NPV, NUS and NFV calculations need an interest rate entered

12.75 ÷ 12 I% I% = 1.06 Enters monthly discount rate since these are monthly cash flows

NPV NPV = 92,694.15 Computes the present value

Example

Assumptions:

Compute the present value of a 48-month lease in advance. The lease is a step-down lease such that the first 36 payments are 84,000 each, and the remaining 12 payments are 21,000 each. At the end of the lease term, a purchase option of 50,000 is expected to be exercised. The cash flows should be discounted at the monthly equivalent of a 14% annual rate.

Solution HP-12C



14 g 12÷ 1.17 Converts annual discount rate to a monthly rate

84,000 g CFo 84,000.00 Enters amount of cash flow in advance

84,000 g CFj 84,000.00 Enters amount of 1st series of cash flows

35 g Nj 35.00 Enters number of cash flows

21,000 g CFj 21,000.00 Enters amount of 2nd series of cash flows

12 g Nj 12.00 Enters number of cash flows

50,000 g CFj 50,000.00 Enters final cash flow

f NPV 2,670,920.58 Solves for present value

Solution HP-17B



84,000 INPUT FLOW (1) = ? Prompts for next cash flow and 84,000.00 shows input of advance payment



21,000 INPUT # TIMES (2) = 1 Prompts for number of times 2nd 1.00 cash flow occurs


50,000 INPUT # TIMES (3) = 1 Prompts for number of times 3rd 1.00 cash flow occurs

INPUT FLOW (4) = ? Prompts for next cash flow and 1.00 shows number of times cash flow (3) occurs



14 ÷ 12 I% I% = 1.17 Enters monthly discount rate since these are monthly cash flows

NPV NPV = 2,670,920.58 Computes the present value

Internal Rates of Return (IRR)

Introduction

Leases are commonly analyzed in terms of the internal rate of return (IRR) or yield they provide the lessor. An IRR is the discount rate that equates the present value of a series of cash inflows (i.e., lease payments, purchase option) to the present value of the cash outflows (equipment or investment cost).

The HP computes yields on leases that have either even or uneven payment streams. This section presents HP solutions to the following two types of problems:

IRRs on leases with even cash flows (annuities in arrears or advance).

IRRs on leases with uneven cash flows

IRR - Even Cash Flows

When computing lease yields, investments are considered as cash outflows (negative) and lease payments as cash inflows (positive). An outflow is designated by pressing CHS (change sign) on the HP-12C or the +/- key on the HP-17B, prior to entering the outflow amount into the appropriate register. Cash inflows are keyed in as positive amounts and, therefore, do not require any sign designation.

Example:

Assumptions;

A lease contract with a 10-year term requires monthly payments in arrears of 4,000 based on an original investment or equipment cost of 250,000. What is the IRR in this lease investment?

Solution HP-12C



g END 0.00 Payments in arrears setting

120 n 120.00 Enters number of payments

4,000 PMT 4,000.00 Enters amount of payments

250,000 CHS PV -250,000.00 Enters original investment

i 1.23 Solves for monthly IRR yield

12 x 14.78 Multiplies the monthly IRR by 12 to convert it to an annual rate

Solution HP-17B



TVM 12 P/YR END MODE Selects menu necessary to do IRR calculation

120 N N = 120.00 Enters number of payments

4,000 PMT PMT = 4,000.00 Enters amount of payments

250,000 +/- PV PV = -250,000.00 Enters original investment

I% YR I% YR = 14.78 Annual IRR

Explanation:

The annual yield in this lease is 14.78%

Example:

Assumptions:

A lessor invests 5,500,000 in a new piece of equipment that will generate net cash returns of 575,000 per quarter, in arrears, for three years (12 quarters). What is the annual IRR in this investment?

Solution HP-12C



g END 0.00 Payments in arrears setting

12 n 12.00 Enters number of payments

575,000 PMT 575,000.00 Enters amount of each cash flow

5,500,00 CHS PV -5,500,000.00 Enters original investment

i 3.67 Solves for quarterly IRR

4 x 14.70 Converts quarterly rate to an annual IRR

Solution HP-17B


FIN Prompts for selection of a menu


5,500,000 +/- INPUT FLOW (1) = ? Prompts for next cash flow and -5,500,000.00 displays input of original investment


12 INPUT FLOW (2) = ? Prompts for next cash flow and 12.00 displays number of times cash flow (1) occurs



IRR % IRR = 3.67% Solves for quarterly IRR

x 4 14.70 Converts quarterly rate to an annual IRR

Explanation

This investment’s annual yield or IRR is 14.70%

IRR - Uneven Cash Flows

The HP can be used to analyze a lease having uneven cash flows. Uneven cash flows occur in step-up, step-down, and skipped-payment leases. Also, the after-tax analysis of a lease having even monthly lease payments will invariably give rise to uneven cash flow amounts.

Example:

Assumptions:

A lease is generating net, after-tax annual cash flows in arrears as follows:

Year 1 400,000 2 550,000 3 476,250 4 382,500

Compute the IRR of this lease based on an original net investment of 1,300,000.

Solution HP-12C



1,300,000 CHS g CFo -1,300,000.00 Enters original outflow

400,000 g CFj 400,000.00 Enters 1st inflow

550,000 g CFj 550,000.00 Enters 2nd inflow

476,250 g CFj 476,250.00 Enters 3rd inflow

382,500 g CFj 382,500.00 Enters 4th inflow

f IRR 14.83 Solves for IRR

Solution HP-17B


FIN SELECT A MENU Prompts for selection of a menu


#T? # TIMES PROMPTING: OFF Turn off # Times prompting since FLOW (0) =? cash flows are ungrouped

1,300,000 +/- INPUT FLOW (1) = ? Prompts for 1st cash inflow -1,300,000.00

400,000 INPUT FLOW (2) = ? Prompts for 2nd cash inflow 400,000.00

550,000 INPUT FLOW (3) = ? Prompts for 3rd cash inflow 550,000.00

476,250 INPUT FLOW (4) = ? Prompts for 4th cash inflow 476,250.00

382,500 INPUT Flow (5) = ? Prompts for 5th cash inflow 382,500.00



IRR % IRR% = 14.83 Solves for IRR

Explanation

14.83% represents the IRR of this investment assuming that the four net cash flows are received exactly at the end of each of the four respective periods.

Amortization of Lease Payments

Introduction

Amortization is the process of separating payments into their principal and interest components. The HP enables one to calculate the principal and interest portion of each lease payment, as well as keep track of the remaining balance after each payment (or set of payments) is amortized.

To obtain an amortization schedule on the HP-12C:

Press f CLEAR FIN to clear the financial registers.

Enter the periodic interest rate using i or g 12÷.

Enter the amount of the investment plus or minus any other investment cash flows at time zero, such as advance payments, into the PV register.

Key in the periodic payment, then press CHS PMT. The sign of PMT must be negative, in accordance with the cash flow sign convention.

Press g END to set the payment mode. (This assumes all cash flows at time zero, including any advance payments, are entered as one net cash flow such that remaining payments are in arrears.)

Key in the number of payments to be amortized.

Press f AMORT to display the amount from those payments applied toward interest.

Press X><Y to display the amount from those payments applied toward the principal.

To display the number of payments just amortized, press R↓ R↓.

To display the remaining balance of the loan, press RCL PV.

To display the total number of payments amortized, press RCL n.

To obtain an amortization schedule on the HP-17B:

Press FIN.

Press TVM.

Press OTHER.

Press AMRT - This step initiates the amortization menu which includes the following selections:

#P - Stores the number of payments to be amortized. #P can be an integer from 1 through 1,200.

INT - Displays the interest portion of the payment.

PRIN - Displays the principal portion of the payment.

BAL - Displays the unamortized loan balance.

NEXT - Calculates the next amortization schedule of #P payments. This set of payments start where the last set left off.

TABLE -- Initiates a new menu for printing an amortization table.

In order to calculate an amortization schedule the PV, I%YR and PMT must be entered. This is done in the TVM menu. The calculator must also be in the proper payment mode (advance or arrears).

Example:

Assumptions:

A lease investment of 1,000,000 provides 48 payments in arrears of 26,827.50 yielding 13.00% to the lessor. Neither tax benefits nor residual are to be considered.

Solution HP-12C



f 2 0.00 Rounds to 2 decimal places

13 g 12÷ 1.08 Monthly yield rate

1,000,000 PV 1,000,000.00 Amount of investment

26,827.50 CHS PMT -26,827.50 Monthly payment

0 n 0.00 Sets n register to 0

12 f AMORT -118,140.97 First 12 months’ interest

X><Y -203,789.03 First 12 months’ principal

R↓ R↓ 12.00 Number payments just amortized

RCL PV 796,210.97 Remaining principal balance

RCL n 12.00 Payments cumulatively amortized

Note: The number of payments keyed in just before f AMORT is pressed is taken to be the payments following any that have already been amortized. If 12 f AMORT is pressed now, the HP-12C calculates the amounts applied to interest and to principal from the second year’s payments (that is, the second 12 months):

Solution HP-12C


12 f AMORT -90,011.48 Second 12 months’ interest

X><Y -231,918.52Second 12 months’ principal

RCL PV 564,292.45 Remaining balance after 24 payments

12 f AMORT -57,999.18 Third 12 months’ interest

X><Y -263,930.82 Third 12 months’ principal

RCL PV 300,361.63 Remaining balance after 36 payments

12 f AMORT -21,568.16 Last 12 months’ interest

X><Y -300,361.84 Last 12 months’ principal

RCL PV -0.21 Remaining balance which should be zero, except for rounding errors

Note: The values in the PV and n registers change after each amortization, allowing one to quickly check the remaining unamortized balance and the total number of payments amortized. If a new amortization schedule is desired, the PV register must be reset to its original value and the n register reset to 0.

Solution HP-17B Keystrokes Display Explanations

FIN SELECT A MENU Prompts for selection of a menu

TVM 12 P/YR END MODE Displays TVM menu

13 I%YR I%YR = 13.00 Annual yield rate

26,827.50 +/- PMT PMT = -26,827.50Monthly payment

1,000,000 PV PV = 1,000,000.00 Amount of investment

OTHER 12 P/YR END MODE Displays additional menu options

AMRT KEY # PMTS; PRESS [#P] Displays AMRT menu

12 #P #P = 12 PMTS: 1-12 Calculates amortization schedule for 1st 12 months

INT INTEREST = -118,140.97 First 12 months’ interest

PRIN PRINCIPAL = -203,789.03 First 12 months’ principal

BAL BALANCE = 796,210.97 Remaining principal balance

NEXT #P = 12 PMTS: 13-24 Calculates amortization for next 12 months

INT INTEREST = -90,011.48 Second 12 months’ interest

PRIN PRINCIPAL = -231,918.52 Second 12 months’ principal



INT INTEREST = -57,999.18 Third 12 months’ interest

PRIN PRINCIPAL = -263,930.82 Third 12 months’ principal



INT INTEREST = -21,568.16 Last 12 months’ interest

PRIN PRINCIPAL = -300,361.84 Last 12 months’ principal

BAL BALANCE = -0.21 Should be zero except for rounding errors

The results of the above amortization are summarized in the following matrix:

Remaining Year Months Interest Principal balance

1 12 118,140.97 203,789.03 796,210.97

2 12 90,011.48 231,918.52 564,292.45

3 12 57,999.18 263,930.82 300,361.63

4 12 21,568.16 300,361.84 -0.21

Totals 48 287,719.79* 1,000,000.21 -0.21

*Rounding errors

Date post:	25-May-2020
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