Savings & Loans

To compare savings accounts with different compounding frequencies, the Annual Equivalent Rate (AER) is used. It standardizes the interest to a single annual figure using the formula: $AER = (1 + \frac{i}{n})^n - 1$ where $i$ is the nominal annual rate and $n$ is the number of compounding periods per year.

For loans, the Annual Percentage Rate (APR) provides a standardized comparison by including interest and mandatory fees. It represents the total cost of borrowing over a year.

Monthly loan repayments ($M$) are calculated using an amortization formula that accounts for the reducing balance of the loan: $M = \frac{r \cdot L}{1 - (1 + r)^{-n}}$ where $L$ is the loan amount, $r$ is the monthly interest rate (APR divided by 12), and $n$ is the total number of monthly payments.

Unit Consistency: Always ensure the interest rate ($r$) and the time period ($n$) match. If payments are monthly, the annual rate must be divided by 12, and the term in years must be multiplied by 12.

Sanity Checks: For loan problems, calculate the total amount repaid ($M \times n$). This value must always be significantly higher than the original loan amount ($L$). If it is lower, the formula was likely applied incorrectly.

Decimal Precision: When performing intermediate steps in compound interest or amortization formulas, keep at least 4-6 decimal places. Rounding too early can lead to significant errors in the final currency amount.

Confusing $i$ and $r$: Students often use the annual rate in a monthly repayment formula without dividing by 12. This results in an impossibly high monthly payment.

Total Interest vs. Total Repayment: Remember that the total interest paid is the difference between the total of all repayments and the original principal ($Total Interest = (M \times n) - L$).

Compounding Frequency: A common misconception is that a higher nominal rate is always better for savings. However, a slightly lower nominal rate with more frequent compounding can sometimes result in a higher AER.