Question

A loan is being repaid with annual payments of $600 made at the beginning of each year for 16 years. Find the loan amount if the effective annual rate of interest is 6% for the first 4 years and 8% for the last 12 years.

Answer 1

To find the loan amount, calculate the Present Value (PV) of the annual payments for each interest rate separately. For the 6% interest rate, the PV is $2,224.62. For the 8% interest rate, the PV is $4,332.49. The total loan amount is $6,557.11.

Step-by-step explanation:

To find the loan amount, we need to calculate the Present Value (PV) of the annual payments made over 16 years. The effective annual rate of interest changes after 4 years, so we need to calculate the PV for each interest rate separately.

For the first 4 years (6% interest rate):

1. 
   
3. Calculate the Present Value of the annual payments using the formula: PV = R * (1 - (1 + i)^(-n)) / i, where R is the annual payment, i is the annual interest rate, and n is the number of years.
4. 
   
6. Substitute the given values: R = $600, i = 6% = 0.06, n = 4.
7. 
   
9. Calculate PV: PV = $600 * (1 - (1 + 0.06)^(-4)) / 0.06 = $2,224.62.

For the last 12 years (8% interest rate):

1. 
   
3. Calculate the Present Value of the annual payments using the same formula, but with the new interest rate: R = $600, i = 8% = 0.08, n = 12.
4. 
   
6. Calculate PV: PV = $600 * (1 - (1 + 0.08)^(-12)) / 0.08 = $4,332.49.

The total loan amount is the sum of the PV for each interest rate: Loan Amount = $2,224.62 + $4,332.49 = $6,557.11.

Answer 2

The loan amount is calculated by summing the present value of two separate annuities due to the changing interest rate: one for the first 4 years at 6% and another for the last 12 years at 8%, with both discounted according to their respective rates before adding them together.

Step-by-step explanation:

To solve for the loan amount with annual payments made at the beginning of each year, we need to consider the loan as two separate annuities due to the change in the interest rate after the first 4 years. For the first annuity (first 4 years at 6%), we must discount each payment at 6%, and for the second annuity (last 12 years at 8%), we discount at 8% but also account for the initial 4 years during which a different interest rate was applied.

First Annuity (Years 1-4)

The present value of an annuity due is calculated using the formula: PV = Pmt x [(1 - (1 + i)^-n) / i] x (1 + i), where Pmt is the annual payment, n is the number of periods, and i is the period interest rate.

For the first 4 years:

PV1 = $600 x [(1 - (1 + 0.06)^-4) / 0.06] x (1 + 0.06)

We calculate PV1, which is the present value of the $600 annual payments at the end of 4 years.

Second Annuity (Years 5-16)

For the next 12 years, since payments are at the beginning of the year, we must first calculate the present value of these payments at the beginning of year 5 and then discount it back to the present value at the beginning of year 1. The formula for these payments is: PV2 = Pmt x [(1 - (1 + i)^-n) / i] x (1 + i).

For the last 12 years:

PV2 = $600 x [(1 - (1 + 0.08)^-12) / 0.08] x (1 + 0.08)

Then we discount PV2 back to year 1 by dividing it by (1 + initial rate of 0.06)^4.

Finally, we sum PV1 and the discounted PV2 to obtain the total loan amount. We then solve for these values to find out the loan amount.