# interest rate not constant with constant installments

Do you know a formulae which would calculate a constant installment withe non constant interest rate.

For example let's say that Mr A take a loan of 1000 and pays a monthly interest rate of 6% the first month then 5% for the next 5 months.

Is it possible to find a constant monthly installment that would fit this contract?

Generally speaking, if one pays an interest rate of $i_k$ in the $k$th month on a principal $P$ over $n$ months, then using the same derivation as that for monthly payments at a constant interest rate, I get the following general formula for the monthly payment $m$:

$$m = \frac{\displaystyle P \prod_{k=1}^n (1+i_k)}{\displaystyle 1+\sum_{\ell=2}^n \prod_{k=\ell}^n (1+i_k)}$$

In general, when only the first interest rate $i_1$ differs, the above formula may be vastly simplified using a geometric series in the denominator:

$$m = \frac{P i_2}{1-\left(1+i_2\right)^{-n}} \frac{1+i_1}{1+i_2}$$

In your case, $n=6$, $i_1=0.06$, $i_2=0.05$, and $P=1000$, so that $m \approx 198.89$. Note that this differs from the case in which all of the interest rates are at $5 \%$, in which the monthly payment becomes about $197.02$, by $1.87$ per month.

First, draw a diagram.

This helps you pick the best point in time; the point to which you bring all debts and payments so you can equate them and solve the problem.

In this case, the best (IMO) point is the moment of the first payment. At that moment, the debt $D$ is given by:$$D=1000\times (1.06)-P$$where $P$ is the regular payment needed to discharge the debt. The debt has grown over one 6% interest period, and then you made the first payment of $P$. You can simply subtract them, because they are at the same moment in time.

The value of the five future payments at that same moment, $V$, is given by the present value of an annuity formula:$$V=P\times\frac{(1-1.05^{-5})}{0.05}$$Set $D=V$, evaluate the ugly looking numerical expressions and solve; $P$ turns out to be $\$198.89\$, as already shown.