HP 12C Platinum: Present Value of a Fractional Year

 HP 12C Platinum: Present Value of a Fractional Year

This blog features the HP 12C Platinum, HP 10BII+, and HP 22S calculators.

Short Term Transactions

Here is the scenario: A bank offers a short term bond, which last less than one year, which pays $100.00 at maturity date. The interest rate stated is an annual interest rate. While determining a pricing schedule, one banker uses an HP 12C Platinum calculator while another uses the HP 10BII+ calculator. They both use the TVM (time value of money) keys. A 365-day year is used.

FV = -100, I% (see table), N (see table), PMT = 0, Solve for PV, P/Y = 1

Term (days)	N = term ÷ 365 (to five decimal places) (for reference)	I%	HP 12C Platinum (to 5 decimal places)	HP 10BII+ (to 5 decimal places)
89	0.24384	5	98.79551	98.81737
141	0.38630	5	97.99973	98.02800
181	0.49589	5	97.58054	97.60958
365	1	5	95.23810	95.23810
89	0.24384	8	98.08664	98.14091
141	0.38630	8	96.83753	96.90714
181	0.49589	8	96.18425	96.25548
365	1	8	92.59259	92.59259

As you can see, the results are different! Why?

According to HP-12C Solutions Handbook (see the Source section), when it comes to fractional periods, simple interest is used instead of compound interest in the TVM solver. Most financial calculators, such as HP 10BII+ always uses compound interest.

Cash flow convention states that:

1. Cash inflows, such as deposits, are positive.

2. Cash outflows, such as payments, are negative.

3. In most problems, the present value and future value have opposite signs.

Respecting cash flow convention, the formulas for present value are:

Simple Interest:

P = -F ÷ (1 + D ÷ 365 × I ÷ 100)

Compound Interest:

P = -F ÷ (1 + I ÷ 100) ^ (D ÷ 365)

where:

P = present value (PV)

F = future value (FV)

I = annual interest rate

D = number of days

If leap years, substitute 366 for 365. If we are working with 30/360 day years, substitute 360 for 365.

These formulas are set up to be entered in calculators with equation solvers such as the HP 22S. I have used the HP 22S to verify each of the results above.

Now why is the results the say when the term exactly 365? It’s pretty simple to prove:

Simple Interest:

P_simple = -F ÷ (1 + 365 ÷ 365 × I ÷ 100) = -F ÷ (1 + I ÷ 100)

Compound Interest:

P_compound = -F ÷ (1 + I ÷ 100) ^ (365 ÷ 365) = -F ÷ (1 + I ÷ 100) = P_simple

When the Term Exceeds One Year

Let’s say the $100.00 bond lasts for 545 days, about one year and a half. This time the interest rate is 7%.

On the HP 12C, any fractional period is treated with simple interest. The HP 12C’s TVM solver (and the HP 12C Platinum) treats the timeline as such.

365 days: full period, compound interest	180 days: partial year, simple interest
PV	FV = -$100.00

To break it down, the HP 12C starts determining the value after 365 days.

N = 180 ÷ 365

I = 7

FV = -100

PV ≈ 96.66314

P = -(-100) ÷ (1 + 180 ÷ 365 × 7 ÷ 100) ≈ 96.66314

365 days: full period, compound interest	180 days: partial year, simple interest
PV	FV = -$100.00
	96.66314

From here, the HP 12C uses that value to calculate final present value. Since we are now working with a full period (one year in this case), compound interest is used with n = 1:

N = 1

I = 7

FV ≈ -96.66314 (treated as an outflow and becoming the acting future value)

PV ≈ -(-96.6314 ÷ (1 + 7 ÷ 100) ^ (1) ≈ 90.33938

The final present value (and price) of this bond is 90.33938.

If we enter following the HP 12C Platinum:

N: 545 [ ENTER ] 365 [ ÷ ] [ N ] (≈ 1.49315)

I: 7 [ i ]

FV: 100 [ CHS ] [ FV ]

PMT: 0 [ PMT ]

[ PV ] → PV ≈ 90.33938

Enter the same problem on most other financial calculators, like the HP 10BII+, will result in a final present value of 90.39108. (P/Y = 1) This is because compounding interest is used for the entire time:

P = -(-100) ÷ (1 + 7 ÷ 100) ^ (545 ÷ 365) ≈ 90.39108

HP 12C Program: Present Value Using Compounding Interest Including Fractional Periods

The program calculates present value given the future value, interest, and the number of days using compounding interest for the entire period. A 365 day year is assumed.

Code: Key; Key Code

ENTER; 36

3; 3

6; 6

5; 5

÷; 10

1; 1

RCL i; 45, 12

%; 25

+; 40

x<>y; 34

y^x; 21

RCL FV; 45, 15

x<>y; 34

÷; 10

CHS; 16

GTO 000; 43,33,000 (GTO 00; 43, 33,00 for HP 12C Classic)

Future value is stored in FV and interest rate is stored in i. The number of days is on the X stack.

Source

Hewlett Packard. HP-12C Solutions Handbook. 2004. pg. 45 https://literature.hpcalc.org/official/hp12c-sh-en.pdf

Eddie

All original content copyright, © 2011-2026. 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.

BA-54: Real Estate Programs

BA-54: Real Estate Programs

BA-54:  Chris won this calculator at HHC 2024 and donated it to me.   Much appreciation as always. 

Background: The Texas Instruments BA-54 is a 1986 update of the BA-55 (not a typo) financial calculator. The BA-54 is a keystroke programming calculator which can hold up to 40 programming steps. However, it’s up to 40 programming steps or up to 5 memory registers (register 1 through 5). Programming mode only occurs in finance mode, though the BA-54 also has a cash flow mode and a statistics (linear regression) mode.

Because steps can add up quickly, limited memory registers are really available. Going beyond step 32 will leave no memory register to work with.

My review of the BA-55 from 2018: http://edspi31415.blogspot.com/2018/02/retro-review-texas-instruments-ba-55.html

APR (Annual Percentage Rate) with Fees

Inputs:

number of payments [ N ]

annual interest rate ÷ 12 = [ %i ]

gross loan amount [ PV ]

points (in %) [ STO ] 1

Operation:

[ 2^nd ] (RST) [ R/S ] payment of the loan (PMT, shown as a positive number)

[ R/S ] APR of the rate when fees are considered

APR is based off the net loan amount (gross loan amount – points%)

Code:

KEY	STEP #	KEY CODE
CPT	00	12
PMT	01	23
R/S	02	13
1	03	01
-	04	75
RCL	05	71
1	06	01
%	07	52
=	08	95
×	09	65
RCL	10	71
PV	11	24
=	12	95
PV	13	24
CPT	14	12
%i	15	22
×	16	65
1	17	01
2	18	02
=	19	95
R/S	20	13
RST	21	37

Example

What is true APR on a 20 year loan when the loan amount is 58000 at 6.8% when the loan carries the cost of 3.5 points?

Inputs:

20 [ × ] 12 [ = ] [ N ]

6.8 [ ÷ ] 12 [ = ] [ %i ]

58000 [ PV ]

3.5 [ STO ] 1 (points)

Operation:

[ 2^nd ] (RST) [ R/S ] 442.74 (payment)

[ R/S ] 7.26% (APR with points)

The true APR is 7.26%.

Source

Smith, Jon M. Financial Analysis & Business Decisions on the Pocket Calculator. John Wiley & Sons: New York. 1976. pp. 177-178. ISBN 0-471-80184-4

Time and Interest Saved by Paying off a Loan Early

Inputs:

number of payments [ N ]

annual interest rate ÷ 12 = [ %i ]

gross loan amount [ PV ]

points (in %) [ STO ] 1

Operation:

Compute the payment by pressing [ PMT ]. Press [ 2^nd ] (RST) [ R/S ].

Result: estimated original interest paid

Enter the new payment [ R/S ]

Result: number of payments with the higher payment [ R/S ]

Result: estimated interest saved

It is assumed that there are no penalties for early payment.

Code:

KEY	STEP #	KEY CODE
RCL	00	71
PMT	01	23
×	02	65
RCL	03	71
N	04	21
-	05	75
RCL	06	71
PV	07	24
=	08	95
STO	09	61
1	10	01
R/S	11	13
PMT	12	23
CPT	13	12
N	14	21
R/S	15	13
×	16	65
RCL	17	71
PMT	18	23
-	19	75
RCL	20	71
PV	21	24
=	22	95
+/-	23	94
SUM	24	81
1	25	01
RCL	26	71
1	27	01
R/S	28	13
RST	29	37

Example

We have a 30-year loan (360 payments) of 176000. The interest rate is 9.6%. If the user pays 1500 a month, how much time and interest are saved?

Inputs:

360 [ N ]

8 ÷ 12 = [ %i ]

176000 [ PV ]

Operation:

[ PMT ] Result: 1291.42

[ R/S ] Total original interest paid: 288913.24

Enter 1500 [ R/S ]

Result: 229.40 (n ≈ 230 payments) [ R/S ]

Result: 120808.27 (interest saved)

By paying 1500 instead of 1291.42, the loan will be paid about 230 payments (19 years, 2 months). Interest saved will be 120808.27.

The Front End Qualification Test

A common method to test whether a potential buyer of a house qualifies is the 28/36 Rule. Simply:

28% of the buyer’s monthly gross income is the maximum amount that the payment with their mortgage with insurance and property taxes (PITI) can be.

36% of the buyer’s monthly gross income is the maximum amount that PITI plus regular monthly debts can be.

The 28% is referred to as the front end test, while the 36% is referred to as the back end test.

The program returns two results: the PITI and difference between 28% of the gross income and PITI. If the difference is positive, the buyer pass the 28% test.

Due to the structure of how memory is set up, I could only fit the front end while being allowed to use 1 additional memory register.

Inputs:

number of payments [ N ]

annual interest rate ÷ 12 = [ %i ]

gross loan amount [ PV ]

monthly tax and property insurance [ STO ] 1

monthly gross income [ STO ] 2

Operation:

Compute the payment by pressing [ PMT ]. Press [ 2^nd ] (RST) [ R/S ].

Result: PITI. (principal, interest, taxes, interest).

Press [ R/S ].

Result: Difference between 28% of income minus PITI.

KEY	STEP #	KEY CODE
CPT	00	12
PMT	01	23
+	02	85
RCL	03	71
1	04	01
=	05	95
R/S	06	13
RCL	07	71
2	08	02
×	09	65
2	10	02
8	11	08
%	12	52
-	13	75
RCL	14	71
1	15	01
-	16	75
RCL	17	71
PMT	18	23
=	19	95
R/S	20	13
RST	21	37

Example

A potential buyer wants to finance a house with a 30 year term at 5%. The purchase price of 375000. The estimated monthly property and insurance is 250. The buyer earns a monthly income of 12000. Does the buyer qualify (at least pass the front test (28%))?

Inputs:

30 [ × ] 12 [ = ] [ N ]

5 ÷ 12 = [ %i ]

375000 [ PV ]

250 [ STO ] 1

12000 [ STO ] 2

Operation:

Compute the payment by pressing [ PMT ]. Press [ 2^nd ] (RST) [ R/S ].

Result: 2263.08 (principal, interest, taxes, interest).

Press [ R/S ].

Result: 1096.92 (it is positive, so the buyer passes the 28% test)

Source

Probasco, Jim. “What is the 28/36 rule for home affordability?” Bankrate. 09 October 2023, https://www.bankrate.com/real-estate/what-is-the-28-36-rule/ Retrieved October 16, 2024.

Enjoy!

Eddie

All original content copyright, © 2011-2025. 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.