Question

3) Tian buys a car that costs $35,000.

a) He pays $5,000 down (i.e. immediately), and he pays off the rest of the loan with 26 bi-weekly payments per year of $250 for 5 years. What is the effective annual interest rate i?

b) Instead, he pays no money down but increases his monthly payments to $290, except for the last one which is exactly enough to pay off the loan. The interest rate is the same as in part a). Is the last payment a balloon or a drop payment? How much is it?

Answer 1

Answer:-

The cost of car is = $ 35000
Down payment = $ 5000
Remaining payment = $ 35000 - $ 5000 = $ 30000

The rest of loan is paid bi weekly ie. each payment every 14 days. ie. 26 payments of $ 250 for a duration of 5 years.

Therefore the total payments amount to $ 250 x 26 x 5 = $ 32500

Therefore the PV= $ 30000 and the FV = $ 32500

FV = PV [1+ R]^N
FV = $ 32500 and PV = $ 30000

a)

Since the payments are made 26 times an year the interest rate would be R= (r/26) and the value of N would be 5 x 26= 130

$ 32500 = $ 30000 [ 1 + (r/26) ]¹³⁰
[ 1 + (r/26) ]¹³⁰ = ($ 32500 / $ 30000)

[ 1 + (r/26) ] = ( $ 32500 / $ 30000) ^(1/130)
[ 1+ (r/26) ] = (1.0833)^(1/130)
[ 1 + (r/26) ] = ( 1.0833)^0.0077
1+ (r/26) = 1.00062
(r/26) = 1.0062 - 1
r/26 = 0.0062
r = 26 x 0.0062
r = 0.1612
r = 16.12 %, which is the effective annual rate.

b)

Given the monthly payments of $ 290 and the interest rate is same as calculated in the part (a) which is 16.12%

The amount is $ 35000 and the payments are made 26 times an year N= 26 x 5 = 130
FV = PV ( 1+ R)^N
FV = $ 35000 x [1 + (0.1612 / 26)]¹³⁰
FV = $ 35000 x [ 1+ .0062 ]¹³⁰
FV = $ 35000 x ( 1.0062)¹³⁰
FV = $ 35000 x 2.23
FV = $ 78050

The total payments are 26 x 5 - 1= 130 - 1 = 129 ( Since the last payment is paid as closing the final amount

So the total amount = 129 x $ 290 = $ 37410
Therefore the last installment = FV - total amount paid in 129 installments
= $ 78050 - $ 37410
= $ 40640

The payment of $ 40640 is a lump sum payment to clear of the loan amount and is significantly higher which is a balloon payment. The drop payment is smaller than the payments paid before. Hence this is a balloon payment and is equal to $ 40640.