Finding the monthly payment for fixed-rate mortgage, but with first month interest free.

Question

I'm trying to calculate the monthly payment of a fixed-rate (annuity) loan, but with the twist that the first month is interest free.

I.e., I have a principal $P_0$ - the total sum that I've loaned - and I want to pay it off completely in $N$ months. The monthly interest rate is $r$, except for the first month, where it is zero.

I want to find the annuity $c$. The formula I've found helps me calculate this in the situation without the first month exception:

$$c = \frac{r}{1 - (1+r)^{-N}}P_0$$

How can I modify it?

Answer 1

My friend helped me come up with this solution:

$$c = \frac{P_0 (1+r)^{N-1}r}{(1+r)^N-1} $$

Answer 2

Let's write the Loan $P_0$ as the present value of the $N$ payments \begin{align} P_0&=c+cv^2+cv^3+\cdots+cv^N=c+c\left(\sum_{k=2}^N v^k\right)=c+c\left(a_{\overline{N}|r}-v\right)=c\left(1-v+a_{\overline{N}|r}\right)\\ &=c+cv\left(v+v^2+\cdots+v^{N-1}\right)=c+cv\left(\sum_{k=1}^{N-1} v^k\right)=c+cva_{\overline{N-1}|r}=c\left(1+va_{\overline{N-1}|r}\right) \end{align} So we have $$ c=\frac{P_0}{1-v+a_{\overline{N}|r}}=\frac{P_0}{1+v\,a_{\overline{N-1}|r}} $$ where $ a_{\overline{n}|r}=\frac{1-v^n}{r} $ and $v=\frac{1}{1+r}$.