Question

2: Applications of finance (20 Mark)

Please answer the following questions. Show all your workings when calculations are required and round off your FINAL result to TWO decimal places.

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.

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?

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.

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).

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?

Answer 1

a) Calculate the present value of each alternative :

Option 1 for purchasing the Toyota Landcruiser 4WD is to make an upfront payment today of $100,000.

Since this payment is to be made today, the present value of this option is the upfront amount itself = $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.

The applicable market interest rate is 6% p.a. Since payments are to be made monthly, the monthly interest rate will be 6% / 12 = 0.50% per month

We can calculate the present value of this payment plan by either using the formula for the Present Value of an annuity due (the first payment starts today so this is an annuity due), or by using the PV function in Excel.

Let's solve this in Excel :

Inputs for the PV function are :

Rate : the interest rate per period = 6%/12

Nper : the number of payment periods = 24 (the final payment to be made 23 months from today will be the 24th payment, hence there will be a total of 24 payments)

Pmt : the monthly payment = $4,500

Fv : future value : not applicable here, to be left blank

Type : indicates whether payments start today or at the end of the first period. For payments starting today, the required input is 1

Present Value of option 2 = $102,040.56 (ignore the negative sign in formula result)

Since the upfront payment of $100,000 is less than the present value of the monthly payment plan in option 2, hence Option 1 is the cheaper alternative.

b) Monthly payment amount for option 2 that would make you indifferent to the two options :

You would be indifferent between the two options if the Present Value of option 2 was to be equal to that of Option 1 = $100,000

Hence, we will calculate the Monthly payment amount required under option 2 to make its present value = $100,000

We use the PMT function in Excel for this.

Inputs : Rate = 6%/12, Nper = 24, Pv (present value) = 100,000, Fv = blank, and Type = 1 (monthly payments starting today)

The monthly payment would be $4,410.01 to be indifferent between option 1 and option 2.

c) If the monthly payments of option 2 were to be made at the end of each month : In this case the input for 'Type' in the PMT function used above, would be 0 (zero) (Type = 0) which would indicate that payments are at the end of each period. All the other inputs in the function would remain the same as in part (b) above. The final answer would be different.

If you were to solve using the mathematical formula, then you would use the formula for the Present Value of an ordinary annuity, instead of the formula for the Present Value of an annuity due. You would plug in the same inputs as above (except Type), and solve for the periodic payment - P.

d) The effective annual interest rate (EAR) that would make you indifferent to the two options :

The present value will be $100,000, monthly payments will be $4,500 for 24 months.

We need to solve for the interest rate that will make the present value of option 2 = $100,000, with the following inputs, using the RATE function in Excel :

Inputs : Nper = 24, Pmt = 4500, Pv = -100,000 (this has to be entered as a negative figure else the function won't work, Fv = blank, Type = 1, Guess (asks for a guess of the interest rate, can be left blank) = blank

The formula result of 0.68% is the periodic (monthly) interest rate. We need the effective annual interest rate (EAR) for which we have to multiply the periodic rate by 12 in the excel sheet.

Press enter, and we get the result 0.081736 or 8.17% p.a. which is the effective annual interest rate (EAR).