12c Solving for n

[Dieter]

(07-25-2017 02:38 AM)Zac Bruce Wrote:  It was late last night when I posted, so yes the figure you gave for the first question is correct and I have edited the post. The second figure I meant 214.87 I.e the amount over what we expected or wanted.

OK, then this is the additional amount you get after full 9 periods. You can also enter n=8 and get the (lower) amount after 8 periods.

(07-25-2017 02:38 AM)Zac Bruce Wrote:  I'm guessing that there is not a simple solution similar to the first problem.

You mean a solution for the PV=1250 and FV=2500 problem? There is a very simple solution. I even posted the steps to calculate n as a fractional result. ;-)

(07-25-2017 02:38 AM)Zac Bruce Wrote:  Can you explain a little more about it being only an approximate answer?

Note: in the following formulas all values are unsigned, e.g. PMT=325 and not –325 due to the HP sign convention.

I assume you mean the first problem here, the one with a given PMT.
The basic annuity formula for this case is
\[PV=PMT \cdot \frac{1-(1+i)^{-n}}{i}\]
Your approximate approach in effect calculates PV for n=327 (=34991,78) and n=328 (=35010,44) and then interpolates the "correct" n for PV=35000 between these two (your %T method is mathematically equivalent). But PV is not a linear function of n. For instance, the PV for n=11 is not the mean of the PVs for 10 resp. 12 periods. The exact value can be calculated with the given formula:
\[n=- \frac{\ln(1 - \frac{i \cdot PV}{PMT}} {\ln(1+i)}\]

(07-25-2017 02:38 AM)Zac Bruce Wrote:  Is this still the case using the formula you presented as a program as a solution to the second problem?

No, this calculation is exact.
The basic formula is  \[FV=PV \cdot (1+i)^{n}\]
Solve for n and get  \[n=\frac{\ln \frac{FV}{PV}}{\ln(1+i)}\]
That's what these steps calculate.
Here is a short program that does these two calculations, depending on whether PMT=0 or not.

Code:

01 RCL PMT
02 x=0?
03 GTO 15
04 1
05 RCL i
06 RCL PV
07 x
08 RCL PMT
09 /
10 %
11 +
12 LN
13 CHS
14 GTO 20
15 RCL FV
16 RCL PV
17 /
18 CHS
19 LN
20 1
21 RCL i
22 %
23 +
24 LN
25 /
26 GTO 00

(07-25-2017 02:38 AM)Zac Bruce Wrote:  Is it just beyond the capabilities of the calculator to give an exact answer? Or is further information needed? I know it works with certain assumptions e.g. periods of equal length, payments always made on either exactly the end or start of a period, all of which would effect real world accuracy.

The point is: there are two "exact" answers. One of them is mathematically exact. That's the n that solves the TVM equation, resp. the FV resp. PV equations given above. Here the answers are n=8,0432 and n=327,4393.

But these answers are not valid in the real world. They assume than n is a continuous variable, which is not true. There are no 0,0432 or 0,4393 periods. You can have either 8 or 9 periods at the end of which interest is accumulated. You can make either 327 or 328 payments to get a little less or a little more than $35000.

And this is the approach of the 12C. It's a tool for real world applications, and less so for students that want a textbook answer for a mathematical formula. The 12C tells you that, if you deposit $1250 at 9% it will take 9 years until you have $2500. After 8 years it's only 2490,70 and after 9 years it's 2714,87. After 8 years your bank will not pay the missing $9,30 for another 0,0432 years, so you have to wait one more year before the next interest payment gets you beyond $2500.

So the 12C handles fractional periods different from the theoretical textbook solution. That's not less exact, it's simply a real-life approach for real-life problems.

Dieter