7.84 A&B&C Some lenders charge an up-front fee on a loan, which is subtracted fr
ID: 2763559 • Letter: 7
Question
7.84
A&B&C
Some lenders charge an up-front fee on a loan, which is subtracted from what the borrower receives. This is typically described as "points" (where one point equals 1% of the loan amount). The federal government requires that this be accounted for in the APR that discloses the loan's cost. A 5-year auto loan for $18,000 has monthly payments at a 9% nominal annual rate. If the borrower must pay a loan origination fee of 2 points, what is the true effective cost of the loan? What would the APR be? If the car is sold after 2 years and the loan is paid off, what is the effective interest rate and the APR? Graph the effective interest rate as the time to sell the car and pay off loan varies from 1 to 5 years.Explanation / Answer
a) The PMT for the loan = 18000/pvifa(0.09/12,60) = 18000/48.1734 = $373.65
For the PMT of 373.65 the borrower receives only 18000*.98 = $17,640
So the effective interest rate per month will be higher than 0.75%.
The 17640 would be the PV of the PMT of 373.65 with n=60. We have to find 'i' from the following equation:
17640/373.65 = pvifa(i,60) = 47.21
pvifa = 46.8481 for 0.85%
= 48.1734 for 0.75%
for 47.21, i = 0.75 + (.9634/1.3253)*0.1 = 0.8227%
APR = 0.8227*12 = 9.87%
Effective cost = 1.008227^12 - 1 = 10.33%
b) If the loan is closed after two years the amount that would have been o/s = 373.65*pvifa(0.75,36)
=373.65*31.4468=11750
the monthly interest rate would be the value of i in the following equation
17640 = 11750*pvif(i,24) + 373.65(i,24)
This can be solved by trial and error, or by an irr calculator.
i = 0.86% (using an IRR calculator)
APR = 0.86*12 = 10.32%
Effective interest rate = 1.0086^12 - 1 = 10.82%
c)
Time of sale/pay-off
Effective interest rate
1
11.88
2
10.82
3
10.48
4
10.35
5
10.33
Calculation for 5th year payoff of loan is given in answer for question (a)
For other years, workings are given below:
If the loan is closed after two years the amount that would have been o/s = 373.65*pvifa (0.75,36)
=373.65*31.4468=11750
the monthly interest rate would be the value of i in the following equation
17640 = 11750*pvif (i,24) + 373.65(i,24)
This can be solved by trial and error, or by an irr calculator.
i = 0.86% (using an IRR calculator)
APR = 0.86*12 = 10.32%
Effective interest rate = 1.0086^12 - 1 = 10.82%
If the loan is closed after one year the amount that would have been o/s = 373.65*pvifa (0.75,48)
=373.65*40.1848 = 15015
the monthly interest rate would be the value of i in the following equation
17640 = 15015*pvif(i,12) + 373.65(i,12)
This can be solved by trial and error, or by an irr calculator.
i = 0.94% (using an IRR calculator)
APR = 0.94*12 = 11.28%
Effective interest rate = 1.0094^12 - 1 = 11.88%
If the loan is closed after three years the amount that would have been o/s = 373.65*pvifa (0.75,24)
=373.65*21.8891 = 8179
the monthly interest rate would be the value of i in the following equation
17640 = 8179*pvif(i,36) + 373.65(i,36)
This can be solved by trial and error, or by an irr calculator.
i = 0.94% (using an IRR calculator)
APR = 0.834*12 = 10.01%
Effective interest rate = 1.00834^12 - 1 = 10.48%
If the loan is closed after four years the amount that would have been o/s = 373.65*pvifa (0.75,12)
=373.65*11.4349 = 4273
the monthly interest rate would be the value of i in the following equation
17640 = 4273*pvif(i,48) + 373.65(i,48)
This can be solved by trial and error, or by an irr calculator.
i = 0.824% (using an IRR calculator)
APR = 0.824*12 = 9.89%
Effective interest rate = 1.00834^12 - 1 = 10.35%
Time of sale/pay-off
Effective interest rate
1
11.88
2
10.82
3
10.48
4
10.35
5
10.33
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