Richard purchased a car for $39,905. He made a downpayment of $15,000 and paid $614 monthly for 4 years. Find the APR.
Richard was able to payoff the loan at the end of 30 months. Using the Actuarial method find he unearned interest, and payoff amount
let the monthly rate be i
614( 1 - (1+i)^-48)/i = 24905
(1 - (1+i)^-48)/i = 40.56189
that's toughie to solve, I will use Wolfram
https://www.wolframalpha.com/input/?i=solve+614(+1+-+(1%2Bx)%5E-48)%2Fx+%3D+24905
i = .00709
so the annual rate compounded monthly
= 12(.00709) = appr.085 or 8.5%
for the second part, I am not familiar with US actuarial methods
To find the Annual Percentage Rate (APR), we need to calculate the interest rate based on the given information.
First, let's find the total amount financed. To do that, we subtract the down payment from the purchase price:
Total amount financed = Purchase price - Down payment
Total amount financed = $39,905 - $15,000
Total amount financed = $24,905
Next, we need to calculate the total interest paid over the 4 years. We know that Richard paid $614 per month for a total of 4 years, which means 48 payments in total:
Total interest paid = (Monthly payment * Number of payments) - Total amount financed
Total interest paid = ($614 * 48) - $24,905
Total interest paid = $29,472 - $24,905
Total interest paid = $4,567
Now, to calculate the APR, we use the following formula:
APR = (Total interest paid / Total amount financed) * (12 / Number of months) * 100
APR = ($4,567 / $24,905) * (12 / 48) * 100
APR = 0.18337 * 0.25 * 100
APR = 4.59%
Therefore, the APR for Richard's car loan is approximately 4.59%.
Now, let's move on to calculating the unearned interest and payoff amount after 30 months, using the Actuarial method.
To find the unearned interest, we need to calculate the interest paid up to 30 months and subtract it from the total interest paid over 48 months:
Interest paid up to 30 months = (Monthly payment * Number of payments) - (Monthly payment * Number of payments remaining)
Interest paid up to 30 months = ($614 * 30) - ($614 * 18)
Interest paid up to 30 months = $18,420 - $11,052
Interest paid up to 30 months = $7,368
Unearned interest = Total interest paid - Interest paid up to 30 months
Unearned interest = $4,567 - $7,368
Unearned interest = -$2,801
The negative value indicates that there is no unearned interest, meaning Richard has paid more than the interest he should have paid up to 30 months.
To find the payoff amount after 30 months, we add the total amount financed to the interest paid up to 30 months:
Payoff amount = Total amount financed + Interest paid up to 30 months
Payoff amount = $24,905 + $7,368
Payoff amount = $32,273
Therefore, the payoff amount after 30 months is $32,273.