Threaded Mode | Linear Mode

Thomas Klemm · (This post was last modified: 04-09-2022 08:00 AM by Thomas Klemm.)

There is a bug in the listing.
When calculating

$
- n (i + 1)^{n - 1}
$

the multiplication by $ n $ is missing.

In the third column we find:

Code:

rcl 9           2

aˣ rcl 4 =      3

sto - chs       1

Instead it should be:

Code:

rcl 9           2

aˣ rcl 4 =      3

rcl × 7         2

sto - chs       1

Quote:The machine should have a capacity of 98 keystrokes to perform this routine.

With this fix we are at 100 keystrokes.

Instead of using this suggestion I would rather follow how this is done in the HP-80.

Initial Value

\[i_0 = \frac{2(n-P)}{n(n+1)}\]

Newton Iteration

\[
\begin{align}
Z &= (1 + i)^{-n} \\
\\
i_\text{next} &= i \left[ 1 - \frac{iP - (1 - Z)}{1 - Z - \frac{niZ}{1 + i}} \right]
\end{align}
\]

Register

R0: n
R1: P = PV / PMT
R2: i
R3: u = 1 + i
R4: Z = u^-n
R5: v = Z - 1

Program

This program is for the HP-25, but can easily be adapted for other models:

Code:

 24 00    : RCL 0            ; n

 24 01    : RCL 1            ; P

 41       : -                ;

 02       : 2                ;

 61       : *                ;

 24 00    : RCL 0            ;

 31       : ENTER            ;

 15 02    : g x^2            ;

 51       : +                ;

 71       : /                ; 2 * (n - P) / (n^2 + n)

 23 02    : STO 2            ; i

 01       : 1                ;

 51       : +                ;

 23 03    : STO 3            ; u = 1 + i

 24 00    : RCL 0            ; n

 32       : CHS              ; -n

 14 03    : f y^x            ;

 23 04    : STO 4            ; Z = u^(-n)

 01       : 1                ;

 41       : -                ;

 23 05    : STO 5            ; v = Z - 1

 24 02    : RCL 2            ; i

 24 01    : RCL 1            ; P

 61       : *                ;

 51       : +                ; i * P + v

 24 00    : RCL 0            ; n

 24 02    : RCL 2            ; i

 61       : *                ; n * i

 24 04    : RCL 4            ; Z

 61       : *                ; n * i * Z

 24 03    : RCL 3            ; u

 71       : /                ; n * i * Z / u

 24 05    : RCL 5            ; v

 51       : +                ; v + n * i * Z / u

 71       : /                ; (i * P + v)/(v + n * i * Z / u)

 01       : 1                ;                                     

 51       : +                ; 1 + (i * P + v)/(v + n * i * Z / u)

 24 02    : RCL 2            ; i

 61       : *                ;

 74       : R/S              ; i_next

 13 11    : GTO 11           ;

Quote:The procedure is terminated when two successive values differ by less than a pre-set amount. This test can easily be built-in to the program by a conditional if-then-else command sequence if desired and if sufficient capacity is available. A round-off command should be programmed just prior to the test, of course.

Example: True Interest Rate of a Loan

What is the true interest rate on a 36 month loan for $3000 having monthly payments of $100?

FIX 6

36
STO 0

3000
ENTER
100
/
STO 1

CLEAR PRGM
R/S
0.010190

R/S
0.010207

R/S
0.010207

Thus, after only three iterations, we get the result.
We can now calculate the annual percentage rate:

FIX 2

1200
×
12.25

References