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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
B/E BEGIN/END```
```Solve equation:
PMT*(i1-1)*(1/i+p)+PV*i1+FV=0
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:
```n=63
i=0.00000161
PV=0
PMT=-1,000,000
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:

```n=31536000
i=10/n
PV=0
PMT=-0.01
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"
05 MVAR "PMT"
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)

http://www.voidware.com/tvm.htm

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.

cheers,

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

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

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.

Cheers!

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