ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
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![**Problem: Amortization and Compound Interest Calculation**
A debt of ₱35,000 is to be paid off in eight equal yearly payments. Each payment combines an amortization installment and interest at 10% compounded quarterly. What should be the amount of each payment?
**Solution Overview:**
To solve this problem, we will use the formula for calculating the amount of an annuity payment with compound interest. Given that the interest is compounded quarterly, the formula to determine the amount of each payment (A) is:
\[ A = \frac{P \times r}{1 - (1 + r)^{-n}} \]
Where:
- \( P \) = Principal amount (₱35,000)
- \( r \) = Quarterly interest rate (annual rate divided by number of compounding periods per year)
- \( n \) = Total number of payments
**Steps:**
1. **Calculate the Quarterly Interest Rate:**
- Annual interest rate = 10%
- Quarterly interest rate \( r = \frac{10\%}{4} = 2.5\% = 0.025 \)
2. **Determine Total Number of Payments:**
- Since payments are yearly and the duration is eight years, \( n = 8 \).
3. **Calculate Each Payment:**
- Substitute \( P = 35,000 \), \( r = 0.025 \), and \( n = 8 \) into the formula to find the amount of each payment.
This will give the annual payment amount needed to amortize the debt over the specified period with the given compound interest rate.](https://content.bartleby.com/qna-images/question/6432dfce-8649-42dd-95b4-a18a04ece3e7/f68f1eb7-3267-4c32-a2fa-aa0e11f02795/oglawqm_processed.jpeg)
Transcribed Image Text:**Problem: Amortization and Compound Interest Calculation**
A debt of ₱35,000 is to be paid off in eight equal yearly payments. Each payment combines an amortization installment and interest at 10% compounded quarterly. What should be the amount of each payment?
**Solution Overview:**
To solve this problem, we will use the formula for calculating the amount of an annuity payment with compound interest. Given that the interest is compounded quarterly, the formula to determine the amount of each payment (A) is:
\[ A = \frac{P \times r}{1 - (1 + r)^{-n}} \]
Where:
- \( P \) = Principal amount (₱35,000)
- \( r \) = Quarterly interest rate (annual rate divided by number of compounding periods per year)
- \( n \) = Total number of payments
**Steps:**
1. **Calculate the Quarterly Interest Rate:**
- Annual interest rate = 10%
- Quarterly interest rate \( r = \frac{10\%}{4} = 2.5\% = 0.025 \)
2. **Determine Total Number of Payments:**
- Since payments are yearly and the duration is eight years, \( n = 8 \).
3. **Calculate Each Payment:**
- Substitute \( P = 35,000 \), \( r = 0.025 \), and \( n = 8 \) into the formula to find the amount of each payment.
This will give the annual payment amount needed to amortize the debt over the specified period with the given compound interest rate.
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