Question

You have a loan outstanding. It requires making six annual payments of $ 4 comma 000 each at the end of the next six years. Your bank has offered to allow you to skip making the next two payments in lieu of making one large payment at the end of the​ loan's term in six years. If the interest rate on the loan is 10 %​, what final payment will the bank require you to make so that it is indifferent to the two forms of​ payment? The final payment the bank will require you to make is ​___? ​(Round to the nearest​ dollar.)

Answer 1

Compute the present value annuity factor, using the equation as shown below:

PVIFA = {1 – (1 + Rate)-Number of periods}/ Rate

                   = {1 – (1 + 0.10)-6}/ 10%

             = 4.35526069943

Hence, the present value annuity factor is 4.35526069943.

Compute the present value of the loan, using the equation as shown below:

Present value = Annual payment*Present value annuity factor

                       = $4,000*4.35526069943

                       = $17,421.0427977

Hence, the present value of the loan is $17,421.0427977.

Compute the value of loan after 2 years, using the equation as shown below:

Value of loan = Present value*(1 + Rate)Time

                       = $17,421.0427977*(1 + 0.10)2 years

                       = $17,421.0427977*1.21

                       = $21,079.4617852

Hence, the value of the loan after 2 years is $21,079.4617852.

Compute the present value annuity factor, using the equation as shown below:

PVIFA = {1 – (1 + Rate)-Number of periods}/ Rate

                   = {1 – (1 + 0.10)-4}/ 10%

             = 3.16986544634

Hence, the present value annuity factor is 3.16986544634.

Compute the final payment the bank will require to make, using the equation as shown below:

Value of loan after two years = (Annual payment*Present value annuity factor) + {Final payment/ (1 + Rate)Time}

$21,079.4617852 = ($4,000*3.16986544634) + {Final payment/ (1 + 0.10)4}

$21,079.4617852 = $12,679.4617853 + {Final payment/ 1.4641}

Rearrange the above-mentioned equation to determine the final loan payment as follows:

Final payment = ($21,079.4617852 - $12,679.4617853)*1.4641

                        = $12,298.4399998

Hence, the final payment of the loan is $12,298.4399998.