Question

In order to buy a new car, you finance $23,000 with no down payment for a term of five years at an APR of 6%. After you have the car for one year, you are in an accident. No one is injured, but the car is totaled. The insurance company says that before the accident, the value of the car had decreased by 25% over the time you owned it, and the company pays you that depreciated amount after subtracting your $500 deductible. How much equity have you built up after one year? Suggestion: Use the following formula for the equity built up after k monthly payments. (Round your answer to the nearest cent.) Equity = Amount borrowed × ((1 + r)k − 1) ((1 + r)t − 1)

Answer 1

Loan Amount = $23,000

APR = 6% or 0.50% per month (6%/12)

Tenure = 5 years or 60 months (5*12 months)

a. Calculation of equity after 12 months

Equity after 12 months = Amount borrowed * (((1+r)^k-1)/((1+r)^t-1))

Where Amount borrowed = $23,000 (no down payment, hence total car value is the loan amount);

r = monthly rate of 0.50%;

k = number of payments made = 12 months;

t = total numbers of payments = 60 months

=$23000*(((1+0.5%)^12-1)/((1+0.5%)^60-1 = $4,066.47

Thus, this denotes that out of the loan amount of $23,000, $4,066.47 of loan has been paid in 12 months. Thus, equity built-up by you after one year (before accident) is $4,066.47

Lets find the net loan outstanding after 12 months and total equity built-up after the accident:

b. Amount of insurance payment

Decrease in value of car before accident = 25% (in 12 months).

Thus, value of the car after 12 months = $23,000*(100%-25%) = $23,000*75% = $17,250

Deductibe = $500

Net payment by Insurance company = $17,250-$500 = $16,750

c: Net loan outstanding after 12 months after accident = Car/Loan Value - Equity paid by you - Payment from Insurance = $23,000-$4,066.47-$16,750=$2,183.53

Thus, after 12 months and after the accident, total equity paid by you and insurance company together is $20,816.47 ($4,066.47+$16,750)