A successful businessman is selling one of his fast food franchises to a close friend. He is selling the business today for $2,441,400.00. However, his friend is short on capital and would like to delay payment on the business. After negotiation, they agree to delay 4.00 years before the first payment. At that point, the friend will make quarterly payments for 16.00 years. The deal calls for a 7.36% APR “loan” rate with quarterly compounding. What quarterly payment will the friend make on the loan?

This problem involves finding the periodic payment required to amortize a loan with a specific interest rate, compounding frequency, and payment schedule. The given situation describes a delayed payment with an agreed upon annual percentage rate (APR) compounded quarterly.

Let's break down the problem:

1. **Loan amount (Present Value, $ PV $)**: \$2,441,400.00
2. **Interest rate (APR)**: 7.36% compounded quarterly. This means the quarterly interest rate is:
   \[
   i = \frac{7.36\%}{4} = \frac{0.0736}{4} = 0.0184 \, \text{or} \, 1.84\% \, \text{per quarter}.
   \]
3. **Loan term after deferral**: 16 years of quarterly payments, so the total number of payments is:
   \[
   n = 16 \times 4 = 64 \, \text{quarters}.
   \]
4. **Deferral period**: Payment starts after 4 years, but the present value is still calculated for today. The present value will be deferred over 4 years at the same quarterly compounding rate, so it must be discounted.

### Steps to Solve:

#### Step 1: Discount the Present Value
We must first discount the loan amount (\$2,441,400.00) back to the point when the first payment is made after 4 years.

The formula for discounting is:
\[
PV_{\text{deferred}} = PV \times (1 + i)^{-n_{\text{deferral}}}
\]
Where:
- $ PV_{\text{deferred}} $ is the loan amount deferred for 4 years.
- $ i = 0.0184 $ (quarterly interest rate).
- $ n_{\text{deferral}} = 4 \times 4 = 16 $ quarters (4 years deferral).

#### Step 2: Use the amortization formula to find the quarterly payment
Once we have the deferred present value, we use the amortization formula:
\[
P = \frac{PV_{\text{deferred}} \times i}{1 - (1 + i)^{-n}}
\]
Where:
- $ P $ is the quarterly payment.
- $ i = 0.0184 $ (quarterly interest rate).
- $ n = 64 $ (number of quarterly payments).

I'll now compute this step by step.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can walk you through the calculations step-by-step manually or help with another aspect of this problem. Would you like to proceed with that or do you have any other questions?

Here are 5 related questions that could help deepen your understanding:
1. How does the quarterly compounding affect the loan calculations compared to annual compounding?
2. How would the payments differ if the interest rate was compounded monthly instead of quarterly?
3. What would happen to the payment amount if the loan term was extended by 4 more years?
4. How is the deferral period impacting the overall cost of the loan?
5. How can the businessman and his friend negotiate a different payment structure with variable interest rates?

**Tip**: When dealing with compound interest, always be mindful of the compounding frequency, as it significantly impacts the overall amount payable.

A successful businessman is selling one of his fast food franchises to a close friend. He is selling the business today for $2,441,400.00. However, his friend is short on capital and would like to delay payment on the business. After negotiation, they agree to delay 4.00 years before the first payment. At that point, the friend will make quarterly payments for 16.00 years. The deal calls for a 7.36% APR 'loan' rate with quarterly compounding. What quarterly payment will the friend make on the loan?

Calculate the quarterly payment required to amortize a $2,441,400 loan for a fast food franchise after a 4-year deferral period with a 7.36% APR and quarterly compounding. This problem involves concepts like loan amortization, compound interest, and time value of money. Suitable for students in advanced high school mathematics or undergraduate finance courses.

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