To solve this problem, we will use the formulas for amortized loans and compound interest. Let's break it down step by step.

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a. Calculate the size of quarterly payments

The formula for an amortized loan payment is:

$PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}}$

Where:

• $P = 30,000$ (loan amount),
• $r = \frac{\text{annual interest rate}}{\text{number of payments per year}} = \frac{3.50\%}{4} = 0.035 / 4 = 0.00875$,
• $n = \text{total number of payments} = 14 \cdot 4 = 56$.

Substituting:

$PMT = \frac{30,000 \cdot 0.00875}{1 - (1 + 0.00875)^{-56}}$

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b. Calculate the principal portion of payment 19

To find the principal portion of a specific payment, we first calculate the remaining balance after 18 payments, then determine how much of payment 19 goes toward reducing the principal.

Remaining Balance Formula:

The remaining balance after $k$ payments is:

$B_k = P \cdot (1 + r)^k - \frac{PMT}{r} \cdot \left[ (1 + r)^k - 1 \right]$

Substitute $k = 18$ into this formula.

Principal Portion of Payment:

$\text{Principal Portion} = PMT - (\text{Interest on Remaining Balance})$

Interest for the payment is:

$\text{Interest} = B_k \cdot r$

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c. Calculate the size of the final payment

In an amortized loan, the last payment is adjusted slightly to fully clear the remaining balance (if there’s rounding). Use:

$\text{Final Payment} = \text{Remaining Balance after 55 payments} + \text{Interest on Remaining Balance}$

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Let me compute these values. Constants P = 30000 # Loan amount annual_rate = 0.035 # Annual interest rate n_years = 14 # Loan duration in years payments_per_year = 4 # Quarterly payments total_payments = n_years * payments_per_year # Total number of payments

Quarterly interest rate

r = annual_rate / payments_per_year

a. Calculate the quarterly payment (PMT)

PMT = P * r / (1 - (1 + r)**-total_payments)

b. Principal portion of payment 19

Remaining balance formula after k payments

def remaining_balance(P, r, PMT, k): return P * (1 + r)**k - PMT / r * ((1 + r)**k - 1)

Balance after 18 payments

k = 18 B_k = remaining_balance(P, r, PMT, k)

Interest portion of payment 19

interest_19 = B_k * r

Principal portion of payment 19

principal_19 = PMT - interest_19

c. Final payment (size)

Balance after 55 payments

B_final = remaining_balance(P, r, PMT, total_payments - 1) final_payment = B_final * (1 + r) # Add final interest

PMT, principal_19, final_payment### Solutions:

a. Size of Alexander's payments at the end of every quarter:

$PMT = 679.93$

b. Principal portion of payment 19:

$\text{Principal Portion} = 488.31$

c. Size of Alexander's final payment:

$\text{Final Payment} = 679.93$

Explanation:

• In part a, we calculated the fixed quarterly payment using the amortization formula.
• For b, the principal portion of payment 19 was determined by subtracting the interest from the total payment.
• For c, the size of the final payment matches the regular payment since rounding effects are minimal in this case.

Let me know if you want further breakdowns or explanations! Here are some related questions:

1. How is the formula for amortized payments derived?
2. How does the principal portion change with each payment?
3. What happens if the loan term or interest rate changes?
4. How do semi-annual payments affect the calculations?
5. What if the loan was compounded monthly instead of quarterly?

Tip: Understanding the distinction between principal and interest in payments can help with budgeting for loans.