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An accurate TVM equation written as a program for the HP42S Solver.

n   number of periods
i interest rate
PMT periodic payment
PV present value
FV future value
Solve equation:

where (i1-1) = e1^(n*ln1(i))
and i1 = e^(n*ln1(i))
and ln1(x) = ln(1+x)
and e1(x) = e^x-1
p=0 for END, p=1 for BEGIN

The use of ln1+x and e^x-1 prevents the loss of significant digits for small values of i. Consider these extreme examples for accuracy checking:

Example 1:
END Mode (set B/E=0 in Solver)
Solving for FV:

HP 42S: 63,000,031.4434 (correct is 63,000,031.4433)

Example 2:

END Mode
Solving for FV:

HP 42S: 331,667.006689 (correct is 331,667.006691)

You might compare these results using the "normal" TVM equation, found in many manuals ;)

E.g. using the TVM equation in the 42S manual, we get
62,921,744.4422 and 311,565.594018 respectively.

The program:

00 {91-Byte Prgm }
01 LBL "TVM"
02 MVAR "N"
03 MVAR "I%"
04 MVAR "PV"
06 MVAR "FV"
07 MVAR "B/E"
08 RCL "I%"
09 100
10 /
11 LN1+X
12 RCL* "N"
13 E^X-1
14 RCL* "PMT"
15 RCL "I%"
16 100
17 /
18 1/X
19 RCL+ "B/E"
20 *
21 RCL "PV"
22 RCL "I%"
23 100
24 /
25 LN1+X
26 RCL* "N"
27 E^X
28 *
29 +
30 RCL+ "FV"
31 END

Edited: 23 Jan 2005, 12:13 p.m.

hi tommi,

yes indeed. that is a good way to tackle the problem. i wrote a page on this problem (also with the same test case)


there's some stuff about how to fix it when you dont have the ln1 and exp1 functions (see the 15c case). also solving for `i' is interesting. checkout the references at the end too.


Yes, the second test case is well-known. Also known as "A Penny for your Thoughts", see Mathematics Written in Sand

The first test case is from Hewlett-Packard Journal, October 1977.

In fact, while lnx+1 and e^x-1 are useful indeed, even better would be the super accurate


found in many HP financial calculators. At least in HP-37E, n can also be fractional, while HP-12C only works with INT(n), of much less use. Don't know how other HP financial calculators behave in this respect.


Edited: 23 Jan 2005, 5:02 p.m.