Reconciling Simple and Compound Interest Across Time Periods
Summary
The document explains why moving a balance from a later date to an earlier date can produce inconsistent results when formulas for simple and compound interest are mixed. Under compound interest, discounting from the later date to the present and then accumulating to the target date gives the same result as discounting directly across the time difference, provided the same compounding convention is used throughout.
Under simple interest, dividing by a factor for the elapsed four months is not the correct direct adjustment. The response first recovers the original principal from the seven-month balance, then subtracts the interest accrued during the final four months. This matches the result from discounting to the starting date and accumulating to month three. The central lesson is that simple interest does not compound across subperiods like compound interest. The explanation assumes a consistent annual rate and specified time fractions; it does not discuss alternative day-count conventions or other compounding frequencies.
Key ideas
- Compound interest permits direct discounting across a time interval when the same compounding convention is used.
- For simple interest, the later balance must be adjusted by removing interest accrued during the final subperiod.
- Recovering the original principal makes the simple-interest calculation consistent across both paths.
- Mixing simple-interest and compound-interest formulas causes the apparent discrepancy.
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Full text
# Evaluating principal and interest at different points in time
# Evaluating principal and interest at different points in time
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 by Alper (score 0, accepted)
https://quant.stackexchange.com/a/68391
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.