Question

Question 2: Applications of finance 

You are offered two options by the Waverley Toyota dealer for purchasing a Toyota Landcruiser 4WD.

Option 1: Upfront where you pay $100,000 now.

Option 2: 2- year monthly payment plan of $4500/month, starting today, with a final payment to be made 23 months from today.

a) If the market interest rate is 6% p.a., calculate the present value of each alternative and identify which alternative is cheaper. (4 marks)

b) If you could decide on the monthly payment amount for option 2, at 6% p.a. interest rate what would that amount be so that you are indifferent to the two options? (3 marks)

c) If the monthly payments of option 2 were to be made at the end of each month, how would your answer to part b) on the monthly payment amount be different? Explain briefly. NO CALCULATIONS ARE REQUIRED HERE. (3 marks)

d) What is the effective annual interest rate (EAR) that would make you indifferent to the two options? (Note that the market interest rate is not given, you need to solve for the interest rate that equates the present value of the two options). (6 marks)

e) Assume that you have chosen option 2 to purchase the car. Immediately after the 12th monthly payment, you encounter some temporary financial difficulty and are only able to afford monthly payment of $4000. After negotiating with the dealer, they are happy for you to pay $4000 per month for the rest of the contract period and a final lump sum payment right at the end of the contract. If the market interest rate is still 6% p.a., how much would this final lump sum payment be so that you will not be worse off by this new arrangement? (4 marks)

Answer 1

Question Summary : You are offered two options by the Waverley Toyota dealer for purchasing a Toyota Landcruiser 4WD.

Option 1: Upfront payment $100,000

Option 2: 2- year monthly payment plan of $4500/month, starting today, with a final payment to be made 23 months from today.

Analyze the options

A1) Solution :

Present value of Option 1: Upfront payment is $100,000

Present value of Option 2 :  $ 102,040.56

So upfront payment is cheaper than monthly payments

It is annuity due as immediate payments are to be made and remaining 23 payments

Pv of annuity = Annuity * 1 - (1+r/t )^-nt * (1 +r/t)

r/t

where r = rate of interest = 6%

n= number of years = 2 years

t = times paid in a year = monthly = 12 times

Pv of annuity =4500 * 1 - (1+0.06 /12  )^-2*12 * (1 +0.06 /12)

0.06 /12

= $ 102,040.56

A2) what monthly payment at 6% makes the 2 options indifferent.

Solution : If monthly payment si $ 4410.10 , it makes the 2 options indifferent

So, finding the amount of annuity which makes PV = 100000 $

1000000 =Annuity * 1 - (1+0.06 /12  )^-2*12 * (1 +0.06 /12)

0.06 /12

Annuity = $ 4410.10

A3) If monthly payemts are made at the end of each month, it is a case of annuity ordinary

PV of an Annuity Due = PV of Ordinary Annuity * (1+i)

PV of Ordinary Annuity= $ 102,040.56 / ( 1+0.06/12) = 1010532. 90 $

So even then upfront payment is cheaper.

A4) what interest rate makes the 2 option indifferent

Solution 8 .17%

Pv of annuity = Annuity * 1 - (1+r/t )^-nt * (1 +r/t)

r/t

100000 =4500 * 1 - (1+ r /12  )^-2*12 * (1 + r /12)

r /12

r = 8.17%