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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?

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My friend helped me come up with this solution:

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

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    $\begingroup$ This would work better, as currently phrased, as an Edit to the Question. One-liners rarely make good Answers, and while you may be happy with the progress made, Math.SE aims to collect questions and answers that address problems with reasoned mathematical arguments. Possibly you could elaborate on the expression above with some details about how it addresses the "first month, where [interest] is zero." $\endgroup$ – hardmath Apr 22 '16 at 14:08
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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 $v=\frac{1}{1+r}$.

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