I have the following formula which calculates the initial payment for a graduated mortgage:
$P =\frac{PV}{\left(1+\sum_{t=1}^{N-1}\frac{(1+g)^t}{(1+i)^{12t}}\right)\cdot\left(\frac{1}{i}-\frac{1}{i\cdot(1+i)^{12}}\right)+\left(\frac{(1+g)^{N}}{(1+i)^{12N}}\right)\cdot\left(\frac{1}{i}-\frac{1}{i\cdot(1+i)^{12M-12N}}\right)}$
Where $PV$ represents the loan present value, $g$ is the constant growth rate, $i$ is the periodic (monthly) interest rate, $M$ is the number of years in the term, and $N$ is the number of years of constant growth. Example:
$PV = 1000000$
$N = 4$
$M = 30$
$i = 0.01$
$g = 0.075$
$P = $$$$8255.76$$
How do I rewrite the equation to have a variable number of periods between growth periods? For instance, the above assumes annual growth but what if I want 7.5% every 2 years?
I think I may have figured it out...please review my answer and improve!
$${P}=\frac{{{PV}\cdot{r}\cdot(-{1}-{g{+}}{{\left({1}+{r}\right)}}^{{s}})}}{{{\left(-{1}+{{\left({1}+{r}\right)}}^{{-{s}}}\right)}\cdot{{\left({1}+{r}\right)}}^{{s}}\cdot{\left({{\left(\frac{{{1}+{g}}}{{{\left({1}+{r}\right)}}^{{s}}}\right)}}^{{\frac{{T}}{{s}}}}-{1}\right)}}}$$
where $PV$ is loan value, $r$ is the periodic (monthly) interest rate, $g$ is the growth rate, $s$ is number of periods between "step-ups", and $T$ is total number of periods (months).
Thanks