TI-30Xa Algorithms: Annuity Factors

TI-30Xa Algorithms:   Annuity Factors

Introduction

Even when a calculator isn't (technically) programmable, algorithms can be applied to scientific and financial calculations.

The calculations take numerical arguments that are stored in the TI-30Xa's three memory slots:  M1, M2, and M3.  Store amounts into the memory registers by the [ STO ] key.  

Careful:  For the solar versions of the TI-30Xa, do not press the [ON/AC] button as doing so clears the memory registers.

Today's list of routines deals with annuity factors.

SPFV:  Future Value of a Single Present Value Factor

SPFV = (1 + i%)^n

FV = PV × SPFV

SPPV:  Present Value of a Single Future Value Factor

SPPV = (1 + i%)^(-n)

PV = FV × SPPV

USFV:  Future Value of a Payment Stream Factor

USFV = ((1 + i%)^n - 1) ÷ i%

FV = PMT × USFV

USPV:  Present Value of a Payment Stream Factor

USPV = (1 - (1 + i%)^(-n)) ÷ i%

PV = PMT × USPV

where:

PV = present value

PMT = payment

FV = future value

n = number of periods

i% = periodic interest rate

For monthly payments per year:

n = number of years × 12

i% = annual interest rate ÷ 12

For quarterly payments per year:  

n = number of years × 4

i% = annual interest rate ÷ 4

For the routines, n is stored in memory register 1 and i% is stored in memory register 2. 

n [ STO ] 1 

i% [ STO ] 2

SPFV:  Future Value of a Single Present Value Factor

 [ ( ] 1 [ + ] [ RCL ] 2 [ 2nd ] { % } [ ) ] [ y^x ] [ RCL ] 1 [ = ]

SPPV:  Present Value of a Single Future Value Factor

 [ ( ] 1 [ + ] [ RCL ] 2 [ 2nd ] { % } [ ) ] [ y^x ] [ RCL ] 1 [ ± ] [ = ]

USFV:  Future Value of a Payment Stream Factor

[ ( ] [ ( ] 1 [ + ] [ RCL ] 2 [ 2nd ] { % } [ ) ] [ y^x ] [ RCL ] 1 [ - ] 1 [ ) ] 

[ ÷ ] [ RCL ] 2 [ 2nd ] { % } [ = ]

USPV:  Present Value of a Payment Stream Factor

[ ( ] 1 [ - ] [ ( ] 1 [ + ] [ RCL ] 2 [ 2nd ] { % } [ ) ] [ y^x ] [ RCL ] 1 [ ± ] [ ) ]

[ ÷ ] [ RCL ] 2 [ 2nd ] { % } [ = ]

Example

Let n = 60, i% = 5  

60 [ STO ] 1

5 [ STO ] 2

SPFV:  18.67918589

SPPV:  0.053535524

USFV:  353.5837179

USPV:  18.92928953

Eddie

All original content copyright, © 2011-2024.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

HP 17BII+ and Numworks: Payment of a Loan with an Unusual First Period

 HP 17BII+ and Numworks: Payment of a Loan with an Unusual First Period

The First Payment Is Not Due in a Month

In finance, the first payment of a loan is not due at the end of a month from signing. This happens when all payments are due at a certain date, such as the 30th/31st or 1st of each month, regardless of when the loan papers are signed.  Loans of this type are referred to odd period loans or partial period loans.  Every payment after the first follows the conventional month time line.

HP 17BII+ Formula:  Partial Period Loan 

The following HP 17BII+ formula deals with calculating odd period loans.  This solver formula is directly from the HP 17BII+ Financial Calculator User's Guide, pg. 196 (see source):

ODD:  PV×(I%÷100×FP(DAYS÷30)+1)=-IF(DAYS<30:(1+I%÷100)×PMT:PMT)×UPSV(I%:N)-FV×SPPV(I%:N)

In the solver, the percent sign is a character used in variables, not a function.  PV, FV, PMT, I%, N are all valid variables in the HP 17BII+ solver.  The variables:

N:  number of payments

DAYS:  number of days in the first period, from 0 to 59.

I%:  the periodic interest rate.   For monthly payments, divide the annual interest rate by 12.

PV:  the loan amount

FV:  the balloon amount

PMT:  payment

This formula follows the cash flow convention:  negative amounts for cash outflows, positive amounts for cash inflows.

Numworks Script:  Partial Period Loan

oddperiod.py

My Numworks script:  https://my.numworks.com/python/ews31415/oddperiod

# 2021-11-14 EWS

# odd period

from math import *

print("Loan with partial period")

print("0 - 59 days")

print("Monthly Payments")

n=float(input("N: n? "))

rate=float(input("I/YR:  rate? "))

pv=float(input("PV: loan amt? "))

fv=float(input("FV: balooon pmt? "))

days=float(input("DAYS: odd period? "))

r=rate/1200

if days<30:

  j=1+r

else:

  j=1

f=days/30-int(days/30)

w=r*f+1

  

# uspv

u=(1-(1+r)**(-n))/r

# sppv

s=(1+r)**(-n)

pmt=(pv*w+fv*s)/(j*u)

# rounding

pmt=-int(pmt*100+.5)/100

# result

print("Payment: "+str(pmt))

Note:   the cash flow convention is followed

Examples

Loan 1:

Inputs:

N = 60

DAYS = 10

PV = $58,425.10

I%/YR = 4.48%  (annual rate), for the HP 17BII+, divide this by 12

FV = $0.00 (no balloon payment)

Result:  PMT:  -1,085.99  

Loan 2:

Inputs:

N = 240

DAYS = 40

PV = $16,956.00

I%/YR = 7.02%  (annual rate), for the HP 17BII+, divide this by 12

FV = $500.00 (enter -500.00 for HP 17BII+)

Result:  PMT:   -130.96

Source

HP 17BII+ Financial Calculator User's Guide.  Ed. 2.  Hewlett Packard.  San Diego, CA  2004

http://h10032.www1.hp.com/ctg/Manual/c00363348

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

HP Prime CHOOSE for Repeated Use - Present Value

 HP Prime CHOOSE for Repeated Use - Present Value

Introduction:  Using the CHOOSE Command to Allow for Repeated Calculations

The use of REPEAT and CHOOSE will allow the programmer to set up a choose from a set of calculations and allows for additional calculations until the user exits the program.  

Here is a sample syntax, with comments followed by double slashes (//):

// Set up local variables:  a choice variable, a list of options, the size of the list

LOCAL ch, lst, s;

// Use a list of options.  Be sure to have the last option as "Exit", "Quit", etc.

lst:={"opiton 1", "option 2", ... , "Exit"};

s:=SIZE(lst);

// Set a choice variable to a starting value. Use a "dummy" value to begin with, like -1.  The choice variable will take on a positive integer value after the CHOOSE command has been executed.

ch:=-1;

// The REPEAT loop

REPEAT

CHOOSE(ch,"title",lst);

IF ch==1 THEN

[choice 1 - calculation]

END;

IF ch==2 THEN

[choice 2 - calculation]

END;

...

IF ch==n-1 THEN

[choice n-1 - calculation]

END;

UNTIL ch==s;   // This is the Exit option

Let's look at an example program.

HP Prime Program: PVACT

*  This is an example program that uses CHOOSE to allow for repeated calculations.   The application is for several types of present value for several types of annuities.

*  As a bonus, you can save some settings, such as format and number of digits, to variables before switching settings that the program requires.  This allows for the programmer to restore settings at the end of the program.  

EXPORT PVACT()

BEGIN

// EWS 2021-01-10

// CHOOSE demonstration

// loop until user exits

// Grab the settings

LOCAL h1,h2;

h1:=HFormat;

h2:=HDigits;

// ch choice varaible

LOCAL ch,n,p,r,m,w,v;

LOCAL lst,s,g;

// Set a "dummy" initial choice

// for ch

ch:=−1;

// Set a list of choices

lst:={"Constant Annuity",

"Growing Annuity",

"Perpetual Annuity",

"Exit"};

s:=SIZE(lst);

// Set decimal fix to 2 places

HFormat:=1;

HDigits:=2;

// Use a Repeat loop to allow

// more than one calculation

REPEAT

CHOOSE(ch,"Present Value",lst);

 

// Constant Annuity - Choice 1

IF ch==1 THEN

INPUT({n,r,m},"Constant Annuity",

{"n?  ","i%? ","PMT? "},

{"number of payments",

"periodic interest rate",

"payment"});

v:=Finance.TvmPV(n,r,−m,0,1);

PRINT();

PRINT("n = "+n);

PRINT("i% = "+r);

PRINT("PMT = "+m);

PRINT("PV = "+v);

// show the screen

// WAIT is needed in a loop

WAIT();

END;

// Growing Annuity - choice 2

IF ch==2 THEN

INPUT({n,r,g,m},"Growing Annuity",

{"n?  ","i%? ","g%? ","PMT? "},

{"number of payments",

"periodic interest rate",

"payment growth rate",

"base payment"});

w:=(1+0.01*g)/(1+0.01*r);

v:=m/(1+0.01*r)*(1-w^n)/(1-w);

PRINT();

PRINT("n = "+n);

PRINT("i% = "+r);

PRINT("g% = "+g);

PRINT("Base Payment = "+m);

PRINT("PV = "+v);

// show the screen

// WAIT is needed in a loop

WAIT();

END;

// Perpetual Annuity - choice 3

IF ch==3 THEN

INPUT({r,m},"Perpetual Annuity",

{"i%? ","PMT? "},

{"interest rate",

"payment"});

v:=m/(0.01*r);

PRINT();

PRINT("r% = "+r);

PRINT("PMT = "+m);

PRINT("PV = "+v);

// show the screen

// WAIT is needed in a loop

WAIT();

END;

// if Exit is selected

UNTIL ch==s;

// Restore the setting

HFormat:=h1;

HDigits:=h2;

END;

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Annuity Due in Terms of Ordinary Annuity

Annuity Due in Terms of Ordinary Annuity

Annuity:  Ordinary vs. Due

The distinction is determined when the first payment is made:

Ordinary Annuity:   the first payment is made at the end of the first period, with no payment made at the beginning (period 0) of the annuity. 

Annuity Due:  the first payment is made at the very beginning of the annuity, with no payment made at the end of the annuity.  

For today's exercise, assume all payments in annuity are all equal.  To simplify, I am going to set payments to 1 monetary unit (Dollars, Euros, Lira, Yen, etc).

Present Value

The present value of an annuity is today's value of the annuity, with future payments and cash flows are discounted back to today using the periodic interest rate.  On financial calculators, the present value is symbolized by the PV key.  

Let i be the interest rate of the annuity, in decimal format.  

Example, for 5% periodic rate, i=0.05.

Let v = 1/(1 + i).   

Present Value of an Ordinary Annuity

Given the payment of 1 monetary unit, an annuity of n periods, equally spaced out, and interest rate i, the present value of this ordinary annuity is the sum:

PVAF = v + v^2 + v^3 + ... + v^(n-1) + v^n = Σ(v^x for x=1 to n)

The above is known as the present value annuity factor, or PVAF.  

A closed formula for PVAF is:

PVAF = (1 - (1 + i)^-n) / i

Present Value of an Annuity Due

Likewise, the present value of the annuity due is the sum:

PVAFD = 1 + v + v^2 + v^3 + ... + v^(n-1) = Σ(v^x for x=0 to n-1)

A closed formula for PVAFD is:

PVAFD = (1 - (1 + i)^-n) / (1 - 1/(1 + i))

We can express PVAFD in terms of PVAF by:

PVAFD = 1 + v + v^2 + v^3 + ... + v^(n-1)

PVAFD + v^n = 1 + v + v^2 + v^3 + ... + v^(n-1) + v^n

PAVFD + v^n = 1 + PVAF

PAVFD = 1 + PVAF - v^n

PAVFD = 1 + (1 - (1 + i)^-n) / i - (1 + i)^-n

Example:  i = 0.04, n = 36

PAVFD = 1 + (1 - 1.04^-36) / 0.04 - 1.04^-36 ≈ 19.6646

This is the same as setting up TVM (time value of money) keys on most financial calculators or any calculator with a TVM solver as:

n = 36, I = 4, PMT = -1, FV = 0, (if needed, P/Y = 1),  BEGIN mode

Solve for PV:  19.6646...

Future Value

The future value of an annuity is the final value of the annuity, including the accumulated interest of payments and cash flows from the date the payment to the end of the annuity.  On financial calculators, the future value is symbolized by the FV key.  

Let w = 1 + i

Future Value of an Ordinary Annuity

Given the payment of 1 monetary unit, an annuity of n periods, equally spaced out, and interest rate i, the future value of this ordinary annuity is the sum:

FVAF = 1 + w + ... + w^(n-3) + w^(n-2) + w^(n-1) = Σ(w^x for x=0 to n-1)

The above is known as the future value annuity factor, or FVAR.  

A closed formula for FVAF is:

FVAF = ( (1 + i)^n - 1 ) / i

Future Value of an Annuity Due

Likewise, the future value of the annuity due is the sum:

FVAFD = w + ... + w^(n-3) + w^(n-2) + w^(n-1) + w^n = Σ(w^x for x=1 to n)

A closed formula for FVAFD is:

FVAFD = (1 + i) * ( (1 + i)^n - 1 ) / i 

We can express FVAFD in terms of FVAF by:

FAVFD = w + ... + w^(n-3) + w^(n-2) + w^(n-1) + w^n 

1 + FAVFD = 1 + w + ... + w^(n-3) + w^(n-2) + w^(n-1) + w^n 

1 + FAVFD = FAVD + w^n

FAVFD = FAVD + w^n - 1

FAVFD = ( (1 + i)^n - 1 ) / i + (1 + i)^n - 1

Example:  i = 0.04, n = 36

FAVFD = (1.04^36 - 1) / 0.04 + 1.04^36 - 1 ≈ 80.7022

Using the TVM keys or solver:

n = 36, I = 4, PMT = -1, PV = 0, (if needed, P/Y = 1),  BEGIN mode

Solve for FV:  80.7022...

On tomorrow's blog, January 24, I will cover two actuarial problems. 

Source:

Finan, Marcel B.  A Basic Course in the Theory of Interest and Derivatives Markets:  A Preparation for the Actuarial Exam FM/2   Arkansas Tech University, 2017.  

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Swiss Micros DM42 and HP71B: Present Value of a Growing Annuity

Swiss Micros DM42 and HP71B: Present Value of a Growing Annuity

Introduction

Today we are going to calculate the present value of a growing annuity.  Unlike most annuities where the payment is constant, in a growing annuity, the payment increases each period.  For this particular blog, we are working with annuities that payments increase by a growth percent (g%) each period.  The annuity has an different interest rate (r%) in which payments are discounted.  

Since the payments are not constant, the time value of money (TVM) keys on a financial calculator are not going to be used.  If your calculator has the the net present value (NPV) function, this can assist you in these calculations.

I am going to use a different approach.

Derivation

Variables:

P = base payment (the first payment)

g = growth rate per period 

r = interest rate per period

n = number of periods 

PV = present value

Ordinary Growing Annuity

In an ordinary growing annuity, the first payment will be received after one period (typically a year or a month) has passed.  Discounting all the payments to calculate present value:

PV 

=  P/(1+r) + P * (1+g)/(1+r)^2 + P * (1+g)^2/(1+r)^3 + ... + P * (1+g)^(n-1)/(1+r)^n

=  P/(1+r) * [ 1 + (1+g)/(1+r) + (1+g)^2/(1+r)^2 + ... + (1+g)^(n-1)/(1+r)^(n-1)

Let w = (1+g)/(1+r), then:

PV

= P/(1+r) * [ 1 + w + w^2 + ... + w^(n-1) ]

The result is a geometric series.  In a general geometric series:

a + a*r + a*r^2 + ... + a*r^(n-1) = Σ(a*r^k, k=0 to n-1) = a * (1 - r^n)/(1 - r)

Then:

PV

= P/(1+r) * [ 1 + w + w^2 + ... + w^(n-1) ]

= P/(1+r) * Σ(w^k, k=0 to n-1) 

= P/(1+r) * (1 - w^n)/(1 - w)

Alternatively, change w back to (1+g)/(1+r):

PV

= P/(1+r) * (1 - (1+g)^n/(1+r)^n) / (1 - (1+g)/(1+r))

= [ P/(1+r) * (1 - (1+g)^n/(1+r)^n) ] / [ 1 - (1+g)/(1+r) ]

= [ P/(1+r) * P/(1+r) * (1+g)^n/(1+r)^n ] / [ 1 - (1+g)/(1+r) ]

The article from finaceformulas.net (see source) suggests multiplying by (1+r) / (1+r):

= [ P/(1+r) - P/(1+r) * (1+g)^n/(1+r)^n ] / [ 1 - (1+g)/(1+r) ] * (1 + r) / (1 + r)

= [ P - P * (1+g)^n/(1+r)^n ] / [ 1 + r - (1 + g) ]

= [ P - P * (1+g)^n/(1+r)^n ] / [ r - g ]

= P / (r - g) * (1 - (1+g)^n/(1+r)^n )

Growing Annuity Due

On an annuity due, the first payment takes place immediately.  The present value is calculated as:

PV 

=  P +  P * (1+ g)/(1+r) + P * (1+g)^2/(1+r)^2 + P * (1+g)^3/(1+r)^3 + ... + P * (1+g)^n/(1+r)^n

=  P * [ 1 + (1+ g)/(1+r) + (1+g)^2/(1+r)^2 + (1+g)^3/(1+r)^3 + ... + (1+g)^n/(1+r)^n ]

Let w = (1+g)/(1+r), then:

PV 

= P * [1 + w + w^2 + w^3 + ... + w^n ]

We have another geometric progression:

PV 

= P  * (1 - w^(n+1))/(1 - w)

Summary:

Present Value of a Growing Annuity - Ordinary

PV = P/(1+r) * (1 - w^n)/(1 - w)

Present Value of a Growing Annuity - Due

PV = P  * (1 - w^(n+1))/(1 - w)

HP 42S/DM42 Program:  PVGROW

Both PVGROW and PVGDUE use only one register, R01.

00  {79-Byte Prgm}

01  LBL "PVGROW"

02  "BASE PMT?"

03  PROMPT

04  "INTEREST?"

05  PROMPT

06  1

07  X<>Y

08  %

09  +

10  STO 01

11  ÷

12  1

13  "GROWTH?"

14  PROMPT

15  %

16  +

17  RCL÷ 01

18  STO 01

19  "N?"

20  PROMPT

21  Y↑X

22  1

23  X<>Y

24  -

25  1

26  RCL- 01

27  ÷

28  ×

29  "PV="

30  ARCL ST X

31  AVIEW

32  END

HP 42S/DM42 Program:  PVGDUE

00  {79-Byte Prgm}

01  LBL "PVGDUE"

02  "BASE PMT?"

03  PROMPT

04  "INTEREST?"

05  PROMPT

06  1

07  X<>Y

08  %

09  +

10  1

11  "GROWTH?"

12  PROMPT

13  %

14  +

15  ÷

16  1/X

17  STO 01

18  "N?"

19  PROMPT

20  1

21  +

22  Y↑X

23  1

24  X<>Y

25  -

26  1

27  RCL- 01

28  ÷

29  ×

30  "PV="

31  ARCL ST X

32  AVIEW

33  END

HP 71B Program: PVGROW

Note:  This is for both ordinary and growing annuities due.

100  DESTROY P,G,R,W,A,C

105  INPUT "PAYMENT? "; P

110  INPUT "INTEREST? "; R

115  R=.01*R

120  INPUT "GROWTH? "; G

125  G=.01*G

130  W=(1+G)/(1+R)

135  INPUT "N? "; N

140  INPUT "DUE?(Y=1,N=0) ";C

145  IF C=1 THEN 200

150  IF C=0 THEN 300 ELSE 140 

200  A=P*(1-W^(N+1))/(1-W) @ ! DUE

205  GOTO 400

300  A=P*(1-W^N)/((1+R)*(1-W)) @ ! ORD

305 GOTO 400

400 PRINT "PV ="; A

  

Examples:

Base Payment:  P = 20.00

Interest Rate:  r = 4%

Growth Rate:  g = 5%

n = 5

Ordinary Growing Annuity

Result:  PV = 98.02

Timeline of Payments:

Period 0:  0.00

Period 1:  20.00

Period 2:  21.00

Period 3:  22.05

Period 4:  23.15

Period 5:  24.31

Growing Annuity Due

Result:  PV = 122.92

Period 0:  20.00

Period 1:  21.00

Period 2:  22.05

Period 3:  23.15

Period 4:  24.31

Period 5:  25.53

Source:

"Present Value of a Growing Annuity" financeformulas.net  https://financeformulas.net/Present_Value_of_Growing_Annuity.html   Retrieved December 13, 2020.  

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Time Value of Money: Using Present Value Factors

Time Value of Money:  Using Present Value Factors

Introduction

The time value of money equation is:

0 = PV + PMT * PVAF + FV * PVSF

where

PVAF = present value of annuity factor = Σ( (1 + i%)^(-k), k, 1, n )
PVSF = present value of single factor = (1 + i%)^-n
n = number of payments
i% = periodic interest rate (as a decimal)

The cash flow convention is maintained with this model.  Cash outflows (payments) are negative, cash inflows (receipts) are positive.

Solving for Payment (PMT)

0 = PV + PMT * PVAF + FV * PVSF
-PMT * PVAF = PV + FV * PVSF
-PMT = (PV + FV * PVSF) / PVAF
PMT = -(PV + FV * PVSF) / PVAF

Example

PV = $50,000.00
FV = -$10,000.00
n = 60  (5 years of monthly payments)
i% = 0.05/12

PVAF = 52.99071
PVSF = 0.77921

PMT = -(50000 - 10000 * 0.77921) / 52.99071 = -796.5150873
(cash payment of $796.52)

Solving for Future Value (FV)

0 = PV + PMT * PVAF + FV * PVSF
-FV * PVSF = PV + PMT * PVAF
FV = -( PV + PMT * PVAF ) / PVSF

Example

PV = -$1,000.00
PMT = -$250.00
n = 60  (5 years of monthly payments)
i% = 0.05/12

PVAF = 52.99071
PVSF = 0.77921

FV = -( -1000 - 250 * 52.99071) / 0.77921 = 18284.7724
(future value of deposit $18,284.77)

We can use this method with any calculator.  A partial present value factor table, one for single payment, and one for annuity, is presented below.

Source:

"HP 17BII Financial Calculator Owner's Manual"  Edition 1  Hewlett Packard.  Corvallis, OR.  December 1989.

Have fun,

Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Four Function and Desktop Calculators: The [ TAX+ ] and [ TAX- ] Keys

Four Function and Desktop Calculators: The [ TAX+ ] and [ TAX- ] Keys

A good subject for a Monday (start off the work week for most of us, me included)

The TAX+ and TAX- keys up close and personal  (Casio SL-300VC)

What is needed:  a calculator with [ TAX+ ] and [ TAX- ]

Introduction

On a four function or desktop calculator with the [ TAX+ ] and [ TAX- ] keys, you can perform tax calculations and financial calculations.  The primary key purpose of these keys is for sales tax.  The procedure for these calculations are (almost) universal.  However, the procedure to set such tax varies by manufacturer.

Casio:  Hold [AC] until the display blanks and refreshes.  Enter the tax rate and press [ % ] (RATE SET). 

Texas Instruments (TI-1795SV):   Enter the tax rate, press [ rate ], [ tax+ ] (store)

Canon:  Press [AC], [ TAX+ ] (SET), enter the rate, and press [ TAX+ ]

Sharp: Press [ CE/C ], [ CE/C ] (twice), [ TAX+ ], enter the rate, press [ TAX+ ]

Printing Calculators (various):  Set tax setting switch to SET, enter the rate, set the tax setting switch back to calculator mode.  Check your manual because printing calculators may vary.

My focus will be on four function and desktop calculators.  Procedures for printing calculators will probably vary.

Note:  I did the procedures listed in today's blog entry on both the Casio SL-300VC and Casio LS-123K.  For using independent memory, check to see if you have either a combined memory recall/clear key [MRC] (or [RM/CM on the LS-123K) or two separate keys ([MR] for recall and [MC] for clear).  For illustration purposes I will use [MRC]  (once for recall, twice for clear).   Remember that you can press [AC] to clear memory. 

Left:  Canon LS-123K (Green), Right:  Casio SL-300VC (Blue)

Note:  Pressing [AC] or [MRC] [MRC] will only clear the memory register, not the tax rate.

Contents

1.  Testing What Tax Rate is Stored in Memory
2. Price After Sales Tax and the Amount of Tax
3. Adding Taxable and Non-taxable Amounts
4. Calculating Use Tax
5.  Find the Taxable Amount Given the Grand Total (Total Plus Tax)
6.  Finance: Using [ TAX+ ] to calculate Future Value
7.  Finance: Using [ TAX- ] to calculate Present Value

For all the examples today, I have the tax rate set to 9.5%.  Please see the Introduction section above on how to set the tax rate.

1.  Testing What Tax Rate is Stored in Memory

This procedure is for any calculator that does not have a recall tax feature, although it works for these calculators too.

Procedure:

100 [ TAX+ ] [ - ] 100 [ = ]

Example:  (remember I'm using the tax rate set at 9.5% for all examples on this blog entry)

100 [ TAX+ ] [ - ] 100 [ = ]
Result:  9.5

2. Price After Sales Tax and the Amount of Tax

The [ TAX+ ] key performs two operations:

*  Press [ TAX+ ] once to calculate the total plus tax
*  Press [ TAX+ ] again to calculate the tax amount

Example:

An invoice shows of a purchase of equipment totaling $250.00.  Find the total after tax and the sales tax.

250 [ TAX+ ]
Display:  273.75   (total invoice:  $273.75)

[ TAX+ ]
Display:  23.75   (sales tax:  $23.75)


3. Adding Taxable and Non-taxable Amounts

Sometimes an invoice has both taxable amount and non-taxable amounts (such as most services, installation fee, sometimes freight, some food).  We can calculate the total invoice accurately with both the TAX and memory keys.  Start by clearing out the memory. 

Procedure:

[MRC] [MRC] taxable amount [ TAX+ ] [ M+ ] n
non-taxable amount [ M+ ] 
[MRC]  (recall memory)

Example:

Suppose has the invoice has:

A scanner that subject to sales tax:  $69.99
Computer services not subject to sales tax: $34.95
Sales Tax Rate:  9.5%

[ MRC ] [ MRC ] 69.99 [ TAX+ ] [ M+ ] 34.95 [ M+ ] [MRC]

The total invoice is $111.59   (8-Digit display shows 111.58905)


4. Calculating Use Tax

In certain states, such as California, use tax on a purchase can occur.  Where you take delivery determines what sales tax rate you would pay.  If a vendor does not charge the required tax, the use tax kicks in, which constitutes the remaining amount required from the buyer.  How to pay the use tax is beyond the scope of this blog entry. 

Procedure:

taxable amount [ TAX+ ] [ TAX+ ] [ - ] sales tax charged [ = ]

Example:

A purchase of a printer from an online store has a retail price of $164.79.  Sales tax of 8% was charged on the printer in the amount of $13.18.  The purchaser lives in a tax district that has a sales tax rate of 9.5%.  What is the use tax?

164.79 [ TAX+ ] [ TAX+ ] [ - ] 13.18 [ = ]  

The use tax is $2.48  (8-Digit display shows 2.47505)


5.  Find the Taxable Amount Given the Grand Total (Total Plus Tax) 

Sometime you know only the grand total of the invoice, but you need to find what was the taxable amount and the tax applied to that amount.  That is what the key [ TAX- ] is for.

*  Press [ TAX- ] once to calculate the taxable amount
*  Press [ TAX- ] again to calculate the tax amount

This is the inverse of the [ TAX+ ] key.

Example:

During an audit, we find an invoice from an electronics retail store for a purchase of a video projector.  However, only the total invoice is readable, in the amount of $187.44.  What is the retail price and what was the sales tax charged?

187.44 [ TAX- ]
Display:  171.17809   (taxable amount:  $171.18)

[ TAX- ]
Display:  16.261917 (sales tax:  $16.26)


6.  Finance: Using [ TAX+ ] to calculate Future Value

Time for a little unorthodox use of the TAX keys to calculate simple compound interest problems.  If you have an investment and you want to know how much your account will be in n periods (usually year), you can use the [ TAX+ ] [ = ] combination.

FV = PV * (1 + r%)^n

FV = future value
PV = present value
r% = interest rate, stored as the TAX rate
n = number of periods

Procedure:

[ MRC ] [ MRC ] 
present value [ M- ]
Loop:  [ TAX+ ] [ = ]   (do this n times for n periods)
(display future value)
[ M+ ] [ MRC ] 
(display interest earned)

Example:

You deposit $1,000.00 in a moderate to aggressive investment account that pays an average of 9.5% per year.  What is the balance after 5 years?  How much interest is earned in those five years?

[ MRC] [ MRC ] 
1000 [ M- ]
[ TAX+ ] [ = ]
[ TAX+ ] [ = ]
[ TAX+ ] [ = ]
[ TAX+ ] [ = ]
[ TAX+ ] [ = ]    (loop the last two keys 5 times)
Future value:  $1,574.24  (8-Digit display: 1574.2385)

[ M+ ] [ MRC ]
Interest earned:  $574.24  (8-Digit display:  574.2385)


7.  Finance: Using [ TAX- ] to calculate Present Value

Similarly, we can use the [ TAX- ] [ = ] combination to calculate the present value of a discounted note. 

PV = FV / (1 + r%)^n

FV = future value
PV = present value
r% = interest rate, stored as the TAX rate
n = number of periods

Procedure:

[ MRC ] [ MRC ] 
future value [ M+ ]
Loop:  [ TAX- ] [ = ]   (do this n times for n periods)
(display future value)
[ M- ] [ MRC ] 
(display interest earned)

Example:

You want to save $10,000.00 in five years.  You find an account that pays 9.5% annual interest.  How much will you need to deposit today to get that $10,000.00 goal in five years?

[ MRC] [ MRC ] 
10000 [ M+ ]
[ TAX- ] [ = ]
[ TAX- ] [ = ]
[ TAX- ] [ = ]
[ TAX- ] [ = ]
[ TAX- ] [ = ]    (loop the last two keys 5 times)
Present value:  $6,352.28  (8-Digit display: 6352.2775)

[ M- ] [ MRC ]
Interest earned:  $3,647.72  (8-Digit display:  3647.723)

What the [ TAX+ ] and [ TAX- ] Keys Calculate

With the tax rate r set:

[ TAX+ ] calculates:   number in the display * ( 1 + r/100)

[ TAX- ] calculates:  number in the display / (1 + r/100 )

Happy calculating and may all your calculations, and work weeks, be successful,

Eddie


All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.