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.
