To find the quarterly payments for a loan with compound interest, we can use the amortization formula:

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

where:

• $R$ is the quarterly payment,
• $P$ is the principal (initial loan amount),
• $r$ is the quarterly interest rate,
• $n$ is the total number of payments.

Given dataTo find the quarterly payments for a loan with compound interest, we can use the amortization formula:

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

where:

• $R$ is the quarterly payment,
• $P$ is the principal (initial loan amount),
• $r$ is the quarterly interest rate,
• $n$ is the total number of payments.

Given

• Principal, $P = 13,000$
• Annual interest rate = 4.4%, compounded quarterly
• Loan term = 12 years

Step 1: Convert the interest rate to a quarterly rate

The quarterly interest rate is:

$r = \frac{4.4\%}{4} = \frac{0.044}{4} = 0.011$

Step 2: Calculate the total number of payments

Since the payments are quarterly over 12 years:

$n = 12 \times 4 = 48$

Step 3: Substitute values into the formula

Now we can substitute $P = 13000$, $r = 0.011$, and $n = 48$:

$R = \frac{13000 \times 0.011}{1 - (1 + 0.011)^{-48}}$

Calculating this:

1. $1 + 0.011 = 1.011$
2. $(1.011)^{-48} \approx 0.5884$
3. $1 - 0.5884 = 0.4116$
4. $13000 \times 0.011 = 143$
5. $R = \frac{143}{0.4116} \approx 347.31$

Answer

The quarterly payment is approximately:

$\boxed{347.31}$

Would you like further explanation or have any questions?

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Here are five related questions:

1. How would the quarterly payment change if the loan term was only 10 years?
2. What would the quarterly payment be if the interest rate increased to 5%?
3. How does compounding frequency affect the total interest paid over the loan term?
4. How much total interest is paid over the course of the loan with these parameters?
5. How would an increase in the principal amount to $15,000 affect the quarterly payment?

Tip: Always check if interest compounding matches the payment frequency.