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A successful businessman is selling one of his fast food franchises to a close friend. He is selling the business today for $2,005,600.00. 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 11.00 years. The deal calls for a 6.04% APR “loan” rate with quarterly compounding. What quarterly payment will the friend make on the loan?

5 answers

To calculate the quarterly payment that the friend will make on the loan, we first need to break down the problem into a few clear steps:

  1. Determine the future value of the loan after the initial delay.

    • The initial loan amount is $2,005,600.
    • The payments will start after 4 years, during which interest will accrue.

  2. Calculate the total amount owed after the 4-year delay.

    • The interest rate is 6.04% annually, which needs to be converted into a quarterly rate.
    • The quarterly interest rate can be calculated as: \[ r = \frac{6.04%}{4} = \frac{0.0604}{4} = 0.0151 \]
    • The number of compounding periods over the 4 years is: \[ n = 4 \text{ years} \times 4 \text{ quarters/year} = 16 \text{ quarters} \]

    The future value can be calculated using the formula for compound interest: \[ FV = PV \times (1 + r)^n \] Substituting in the values: \[ FV = 2,005,600 \times (1 + 0.0151)^{16} \]

    First, calculate \( (1 + 0.0151)^{16} \): \[ (1 + 0.0151)^{16} \approx 1.2725 \] Now substitute this back into the future value equation: \[ FV = 2,005,600 \times 1.2725 \approx 2,549,463.20 \]

  3. Calculate the quarterly payments over the 11 years following the 4-year delay.

    • There will be quarterly payments for 11 years, which means: \[ N = 11 \text{ years} \times 4 \text{ quarters/year} = 44 \text{ quarters} \]

    The formula to calculate the payment (PMT) for an annuity is: \[ PMT = \frac{PV \times r}{1 - (1 + r)^{-N}} \] Given that \( PV = 2,549,463.20 \), \( r = 0.0151 \), and \( N = 44 \): \[ PMT = \frac{2,549,463.20 \times 0.0151}{1 - (1 + 0.0151)^{-44}} \]

    First, calculate \( (1 + 0.0151)^{-44} \): \[ (1 + 0.0151)^{-44} \approx 0.4846 \] Now, substitute this value back into the PMT formula: \[ PMT = \frac{2,549,463.20 \times 0.0151}{1 - 0.4846} \] Compute \( 1 - 0.4846 \): \[ 1 - 0.4846 \approx 0.5154 \] Now compute PMT: \[ PMT = \frac{2,549,463.20 \times 0.0151}{0.5154} \approx \frac{38,566.25}{0.5154} \approx 74,798.67 \]

Thus, the quarterly payment will be approximately $74,798.67.

at values do i enter for N, i/y,PV, PMT and FV to solve on a finance calculator?

To solve for the quarterly payment using a finance calculator, you can identify the parameters as follows:

  1. N (Number of periods): This is the total number of payments. Since the friend will make quarterly payments for 11 years, you calculate: \[ N = 11 \text{ years} \times 4 \text{ quarters/year} = 44 \text{ quarters} \]

  2. i/y (Interest rate per period): This is the interest rate for each period. The annual interest rate is 6.04%, which is compounded quarterly, so: \[ i/y = \frac{6.04%}{4} = 1.51% \]

  3. PV (Present Value): This is the amount financed or the future value that needs to be paid off after the initial delay. We previously calculated the future value after 4 years of compounding, which is approximately: \[ PV \approx 2,549,463.20 \] Make sure to enter this as a negative value (e.g., -2,549,463.20) because it represents an outgoing payment.

  4. PMT (Payment): This is what you are trying to solve for. You will leave this blank as the calculator will compute this value.

  5. FV (Future Value): Since you are calculating the payment on a loan that will be completely paid off after the series of payments, set: \[ FV = 0 \]

Summary of inputs for the finance calculator:

  • N = 44
  • i/y = 1.51
  • PV = -2,549,463.20 (enter as negative)
  • PMT = ? (leave this blank while solving)
  • FV = 0

Once you input these values into your financial calculator, you can compute the PMT, which will give you the quarterly payment amount your friend will need to make.

what values do i enter for N, i/y,PV, PMT and FV to solve on a finance calculator for the total amount owed after the 4-year delay?

To calculate the total amount owed after the 4-year delay (the future value of the loan after 4 years of accumulating interest) using a finance calculator, you will set up the following:

  1. N (Number of periods): This is the total number of compounding periods during the 4-year delay. Since the interest is compounded quarterly, you calculate: \[ N = 4 \text{ years} \times 4 \text{ quarters/year} = 16 \text{ quarters} \]

  2. i/y (Interest rate per period): This is the quarterly interest rate. The annual interest rate is 6.04%, so (compounded quarterly): \[ i/y = \frac{6.04%}{4} = 1.51% \]

  3. PV (Present Value): This is the initial amount being financed (the principal). In this case, it is: \[ PV = -2,005,600 \] (Again, enter this as a negative value because it represents an outgoing cash flow.)

  4. PMT (Payment): Since you're calculating the future value and not making any payments during this period, you can leave this as: \[ PMT = 0 \]

  5. FV (Future Value): This is what you are solving for, which will be the total amount owed after the 4-year period due to interest accumulation.

Summary of inputs for the finance calculator:

  • N = 16 (the number of quarters in 4 years)
  • i/y = 1.51 (the quarterly interest rate)
  • PV = -2,005,600 (enter as negative)
  • PMT = 0 (no payments are made during this period)
  • FV = ? (leave this blank while solving)

Once you input these values into your financial calculator and calculate the FV, you will find the total amount owed after the 4-year delay.