Question

Mr. Brown wants to buy a Tesla Model S car, whose price is $100, 848. The dealer offers a loan plan: $30, 000 down payment, $X at the end of year 1, year 2, year 3, and year 4. Assume the constant annual interest rate is 25%.

(a) What is X? [Hint: you may want to use calculators.]

(b) Suppose Mr. Brown accepts the offer. How much does Mr. Brown finance?

(c) What is the principle payment at the end of year 1?

(d) After taking the auto loan, Mr. Brown gets a refinance opportunity at the end of year 1 (right after he makes the first payment). That is, Mr. Brown can borrow from the bank to pay all remaining principle at the end of year 1. In this refinance plan, Mr. Brown needs to pay the bank $25, 000 at the end of year 2, year 3, year 4, and year 5. Will Mr. Brown choose to refinance? [Hint: the interest rate is still 25%. Mr. Brown will choose to refinance if and only if the NPV of the refinance plan is positive.]

Please focus on (d).

Answer 1

Price of car = $100,848

Down payment = $ 30,000

Loan amount = $ 70.848 (payable in 4 annual installments 'X'), Rate of interest = 25%

Installment = Principal * Rate of interest [(1+ rate of interest)^number of installments) / ( (1+rate of interest)^number of installments -1) ]

X = 70,848 * 0.25 * [ ( 1+0.25)^4 / ((1+0.25)^4 - 1) ]

X = 17,712 * [ (2.44140625) / (2.44140625-1) ]

X = 17712 * 2.44140625 / 1.44140625

=> X (Intallment amount) = 30,000

(b) If Mr brown accepts the offer, he will accept $70,848 as financing and pay $30,000 for 4 annual installments

(c) Payment at the end of year 1 = X = $30,000

Interest for year 1 = Principal of loan * interest rate = 70,848 * 25% = 17,712

Principal paid in year 1 = Payment of year 1 - Interest of year 1 = 30,000 - 17,712 = $12,288

(d) If Mr Brown chooses to refinance, he will pay $25,000 for the next 4 years after year 1. i.e. in years 2,3,4&5.

Amount of loan refinanced at the end of year 1 = 70,848 - 12,288 = $58,560

Present value at the end of year 1 for the future installments @ 25% rate of interest

Annuity factor = [ 1 - (1+rate of interest) ^ (-)number of installments ] / rate of interest

= [ 1 - (1.25)^-4 ] / 0.25

= (1 - 0.4096 ) / 0.25 = 2.3616

Present value at the end of year 1 for the future installments @ 25% rate of interest = annuity factor * yearly payments

= $ 59,050.

Since the present value of the payments under the refinancing option is higher than the amount acutally due at the end of year one (of the in service loan), Mr. Brown should not opt for the refinancing option.