Final answer:
The final payment on a student loan will be less than the regular payment after making an extra $100 payment, and the effective rate of return is the interest saved by this $100 payment compared to earning interest elsewhere at the loan's interest rate.
Step-by-step explanation:
The student's question pertains to determining both the amount of the final payment on a student loan after making an extra payment of $100 today as well as the effective annual return rate (APR) that the $100 extra payment yields from the savings in interest. This question involves understanding of amortization, interest calculation, and the time value of money, which are fundamental concepts in financial mathematics.
Firstly, to find the final payment after the additional $100 payment, one would need to recalculate the amortization schedule of the loan, taking into account the reduced principal balance due to the extra payment. This process consists of applying the $100 payment immediately to the principal balance, then continuing with the regular $500 monthly payments. The amortization schedule will now show a reduced balance that needs to be cleared with the final payment, which will be less than the regular $500 payment.
For the second part, finding the effective rate of return on the $100 extra payment, we compare the interest saved by this $100 extra payment against what the same $100 could have earned if invested elsewhere at the same 9% APR. The savings constitute a 'return' on the investment of the extra payment. Calculating the exact rate of return would require a detailed analysis, usually involving financial calculators or spreadsheet software, to find the value of interest saved over the remainder of the loan as a result of the $100 reduction in principal.