Question

You have decided to purchase a new automobile with a hybrid-fueled engine and a six-speed transmission. After the trade-in of your present car, the purchase price of the new automobile is $35,000. This balance can be financed by the auto dealer at 2.3% APR (compounded monthly) and repaid over 60 monthly payments. Alternatively, you can get an instant rebate on the purchase price if you finance the loan balance at an APR of 9.1% (compounded monthly) over 60 months. Determine the amount of the instant rebate that would make you indifferent between the financing plans The amount of the instant rebate that would make you indifferent between the financing plans is (Round to the nearest dollar)

Answer 1

If the monthly installment of both the scenarios are equal, then we would be indifferent between both these cases

In first case, Loan is taken on entire payable amount of the automobile i.e. loan amount = $35,000

Financing rate = 2.3% APR compounded monthly

So, monthly rate = 2.3/12 = 0.19%

Payable period = 60 months

Lets assume monthly installment be 'm'Sum of present value of all the monthly installments should be equal to the loan amount i.e. $35,000

To calculate the present value (PV) of a cash flow, we need to discount it by the required return.

PV = cash flow_n/(1+r)ⁿ,

where cash flow_n is cash flow in period n, r is required rate of return, n is time period

Sum of present values = CF₁/(1+r)¹ + CF₂/(1+r)² + ................... + CF_n-1/(1+r)^n-1 + CF_n/(1+r)ⁿ

In this case, all cash flows are monthly installment i.e. 'm'

Now, putting all the values in the equation

35,000 = m/(1+0.0019)¹ + m/(1+0.0019)² + ........................... + m/(1+0.0019)⁵⁹ + m/(1+0.0019)⁶⁰

35,000 = m*56.63

Solving this, we get m = $618.08

i.e. Monthly installment = $618.08

In second case, a rebate is offered on the loan

Lets assume the rebate amount = $x

So, the remaining loan amount = $(35000-x)

Financing rate = 9.1% APR compounded monthly.

So, monthly rate = 9.1/12 = 0.76%

Time period = 60 months

To be indifferent between both the scenarios, monthly installment should be same

Therefore, monthly installment in this case = $618.08

The loan amount would be the sum of present value of all the monthly payments

So, Loan amount = 618.08/(1+0.0076)¹ + 618.08/(1+0.0076)¹ + ............................. + 618.08/(1+0.0076)⁵⁹ + 618.08/(1+0.0076)⁶⁰

Solving this we get loan amount equal to $29,705.31

So the rebate amount = 35000 - loan amount = 35000 - 39705.31 = $5294.69

Instant rebate = $5,295