Looking for TVM formulas

[Jeff_Kearns]

(04-01-2014 01:16 PM)Dave Britten Wrote:  it would look something like this, right?

PV * (1 + i)^n + PMT * (((1 + i)^n) - 1)) * ((1 / i) + B) + FV = 0

Where B = 0 for end-period payments, and 1 for begin-period payments.

Only if you are content with inevitable rounding errors. Take a look at p. 180 of the HP-15C Advanced Functions Handbook for an example:

Susan tried to calculate FV=PMT x (((1+1/n)^n)-1)/(i/n), where
payment = $0.01,
i = 0.1125, and
n = 60 X 60 X 24 X 365 = 31,536,000.

She calculated $376,877.67 on her HP-15C, but the bank's total was $333,783.35, and this latter total agrees with the results calculated on good, modern financial calculators like the HP-12C, HP-37E, HP-38E/ 38C, and HP-92. Where did Susan's calculation go awry? No severe cancellation, no vast accumulation of errors; just one rounding error that grew insidiously caused the damage:

i/n = 0.000000003567351598
1 + i/n = 1.000000004

when rounded to 10 significant digits. There is the rounding error that hurts. Subsequently attempting to calculate (1 + i/n)^n, Susan must get instead (1.000000004)^31,536,000 = 1.134445516, which is wrong in its second decimal place.
How can the correct value be calculated? Only by not throwing away so many digits of i/n. Observe that (1 + i/n)^n = e^(n x ln(1 + i/n), so we might try to calculate the logarithm in some way that does not discard those precious digits.