1
$\begingroup$

Assuming we can afford to pay 1000 E/month and we subscribe to a 20-year mortgage plan at 5% annual rate. I first want to know what the maximal mortgage can be.
I have the following formula:
$C=\cfrac{P}{\cfrac{1-(1+r/12)^{-n}}{r/12}}$,
where $n$ is the number of monthly instalments of $C$ euros we have to pay in order to pay off our loan of $P$ euros with $r$ the annual interest rate.
I know that $n=20*12=240$ months, $ r=0.05$ and $ C=1000 $ thus I calculate the maximal mortgage ($P$) using the formula this gives $P=151525$.
Know assume that after 5 years we readjust the installments to 1200 E/month. I want to know how much this shortens the repayment time. Again I have a formula that calculates the amount we have to pay after k payments:
$P\cfrac{(1+r/12)^{n}-(1+r/12)^{k}}{(1+r/12)^{12}-1}$
with this I calculate the amount we have to pay after 5 years i.e. $5*12=60$ payments thus if we use $k=60$, $r=0.05$ and $n=20*12=240$ we get that $P=126454.98$. Now that we have the amount we still have to pay I can use the initial formula $C=\cfrac{P}{\cfrac{1-(1+r/12)^{-n}}{r/12}}$ with $P=126454.98$, $ r=0.05$ and to calculate $n$ when I do this I get that $n=-138.473$ in other words I still have to pay $-138.473/12=-11.5$ years. This doesn't seem right can anyone tell me what I'm doing wrong?

$\endgroup$
2
$\begingroup$

Loan balance equals the NPV of future cash-flows.

Under level payments:

$P = C\sum_\limits{i=1}^{n} (\frac {1}{1+r})^i$

$P = C\frac {(1- (\frac {1}{1+r})^n)}{r}$

with stepped up payments after 5 years

$1000\sum_\limits{i=1}^{240} (\frac {1}{1.004167})^i = 1000\sum_\limits{i=1}^{60} (\frac {1}{1.004167})^i+1200\sum_\limits{i=61}^{x} (\frac {1}{1.004167})^i$

And we need to find x

$1000\sum_\limits{i=61}^{240} (\frac {1}{1+r})^i = 1200\sum_\limits{i=61}^{x} (\frac {1}{1+r})^i$

$1000\frac {(1+r)^{60} - (1+r)^{240}}{r} = 1200\frac {(1+r)^{60} - (1+r)^x}{r}\\ 200(1+r)^{60} + 1000(1+r)^{240} = 1200(1+r)^x\\ \frac {2969}{1200} = (1.004167)^x$

take the log of both sides

$x = \frac {\log 2.4744}{\log 1.04167} = 218$ total payments or 22 months shorter.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.