Evaluating principal and interest at different points in time [closed]

Question

Consider simple interest and suppose we have a certain principal and interest at t=7 months, we want to find the value of that amount of money when t=3 months. I would like to do it in two different ways: 1)first we go to t=0, then to t=3; 2) we go directly to t=3 from t=7 Now, in the first case, I multiply my amount of money (M) by $\frac{1}{1+r\frac{7}{12}} \cdot (1+r \frac{3}{12})$, where r is the interest rate, whereas in case 2), I simply multiply M by $\frac{1}{1+r\frac{4}{12}}$. The two procedures should give the same results in theory, so clearly I'm doing something wrong, but I can't figure it out

Answer 1

I think the problem stems from the incorrectly combined application of the concepts of compound and simple interests.

Let's assume $M$ is the principal plus interest at the end of the 7th month, $M_3$ is the principal plus interest at the end of the 3rd month (what you are trying to calculate), and $M_0$ is the principal.

If $r$ represents a monthly compounded annual interest rate, you should divide $M$ by $(1+r)^{7/12}$ and multiply it $(1+r)^{3/12}$ to get $M_3$ in the first option. In the second option, should simply divide $M$ by $(1+r)^{4/12}$ which will give you exactly the same result as in the first option for the compound interest.

If $r$ represents a simple annual interest rate, then the formula for $M_3$ in your first option, $\frac{M}{(1+\frac{7}{12}r)} \times (1+ \frac{3}{12}r)$, is correct but the one in the second is not.

For the second option in the case of the simple annual interest rate, you should subtract the last four months of interest from $M$, not divide it by the number you suggest. That is, you should calculate $M - M_0 \times \frac4{12} r$ where $M_0$ can be calculated as $\frac{M}{(1 + \frac7{12} r)}$. Then the formula for the second option becomes

$ \begin{align} M_3 &= M - M_0 \times \frac4{12} r \\ &= M - \frac{M}{(1 + \frac7{12} r)} \times \frac4{12}r, \end{align} $

and if you do the algebra, this is equal to the formula in the first option for $M_3$ in the case of the simple annual interest rate.