
Financial Mathematics
$\displaystyle P=A(((1+re)^1/12 1) / (1  (1+re)^n)))$
Consider a €100,000 mortgage borrowed for a period of 40 years.
How many years is half the oroiginal mortgage paid?
Interest rate = 4%
n = 40 years
re= .04 therefore r = .039 as $\displaystyle (1+ r/12)^12 = 1 + re$
the problem i have is, P is meant to equal €413.50 but i just cant seem to get it to work out like that, the P formula is giving me minus answers and stuff.
By the way in the P formula it is 1+re to the power of 1/12 and the other 1+re is to the power of n
Please help!!

1) You must define your terms. Notation is not always the same. Various authors and teachers use various notations. What are 'r' and 're'? Have they names or relationships?
2) The problem statement is insufficient. We do not know the terms of the loan. What it the interest crediting methodology? What is the payment frequency? No lumps?
3) You have some rounding problems. If you are using the effective monthly interest to only two decimals, you should have some problems. 100000*0.039 = 3900. There are still two zeros to the LEFT of the decimal! How are you going to get exact pennies out of that? You need at least four more decimal places. Of course, this still won't help for the reasons in the next note.
4) Please see note #1. $\displaystyle (1+0.04)^{\frac{1}{12}}  1 = 0.0032737397822$ Notice how this has a zero you didn't have before digits are nonzero. Your version is off by an order of magnitude. 0.039 just won't do.
5) Do NOT memorize a formula for 1/2 the remaining balance. Just learn how to calculate values. Once you've the exact payment, you can calculate the value for ANY number of remaining payments.
If i = 0.04, then $\displaystyle i^{(12)} = 0.0032737397822$ and $\displaystyle v^{12}=\frac{1}{1+i^{(12)}} = 0.9967369426186$. $\displaystyle 100,000*\frac{i^{(12)}}{1(v^{(12)})^{12*40}} = 413.50189234$, so p = 413.51.
Just solve for 'n'
$\displaystyle 50,000*\frac{i^{(12)}}{1(v^{(12)})^{n}} = 413.50189234$