The TVM formula

I have a son studying engineering and who is now having some finance subjects as part of his major. Today he asked me about a finance problem, which involved, at first sight, two Time Value of Money steps. He was getting a wrong result and he asked me how to do it.

I solved it with a HP12c quite fast - he was surprised how a such small calculator could do it so simply. Then I remembered that the HP Prime that my son has, there is a Finance app! I guided him through it and he was surprised that it could be so fast and practical. I am still interested on his delving deep into the formulae and understanding the math behind - and not just use the solver (which solved his problem in two steps).

From hpmuseum.org:

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: