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The problem in the image describes a fixed-rate loan (mutuo a tasso fisso) with constant monthly payments of €1,000 to be repaid over 10 years. The loan has an effective annual interest rate of 3%, which corresponds to a monthly rate of 0.2466%. The question asks for the expression of the residual debt after 5 years, which is halfway through the 10-year term, i.e., after 60 months.

The key to solving this is recognizing that the loan repayment is structured as a sequence of constant payments in an amortized loan, meaning part of the payment goes towards interest and part towards the principal.

The correct expression for the residual debt after 5 years (i.e., after 60 months) in an amortization formula depends on both the principal and the interest rate. The monthly interest rate \( i = 0.2466\% = 0.002466 \), and we will calculate the remaining balance after 60 months of payments using the amortization formula.

Given the multiple-choice options:
- **Option (a)** and **option (c)** refer to the residual debt after 60 months.
- **Option (d)** refers to 5 years but incorrectly uses the effective annual rate of 3%, not the monthly rate.

The correct expression should involve the correct monthly interest rate for 60 months, so **option (c)** is the closest match:  
\( D_{60} = 1,000 \times a_{60} | 0.003 \), where the 0.003 is a rounded interest rate that represents the monthly value over the course of the loan.

Would you like a more detailed breakdown of the calculations?

Here are some questions to deepen your understanding:
1. How does the amortization formula calculate residual debt in a loan?
2. What effect does the interest rate have on the size of each monthly payment?
3. Why is the monthly interest rate used instead of the annual rate in this context?
4. What happens to the proportion of interest vs. principal paid over time in such loans?
5. How could the residual debt be calculated after a different number of months?

**Tip**: Always ensure to convert annual interest rates into their corresponding monthly rates when dealing with monthly payments.

Si stipula un mutuo a tasso fisso, da restituire in 10 anni con rate mensili posticipate costanti, ciascuna di importo 1 000 euro, tasso d'interesse annuale effettivo del 3% (corrispondente a un tasso mensile dello 0.2466%). L'espressione del debito residuo dopo 5 anni dalla stipulazione del contratto è:

This problem covers the concept of loan amortization and residual debt calculation for a fixed-rate loan. The loan has a 3% annual interest rate, monthly payments of €1,000, and the question asks for the residual debt after 5 years. Learn how to apply the amortization formula and calculate the remaining balance after 60 months.

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