Analyzing Car Buying Options: Rebate vs. 0% Financing at 1% Interest

You can buy a car that is advertised for $24,000 on the following terms: 

• (a) pay $24,000 and receive a $2,000 rebate from the manufacturer; 

• (b) pay $500 a month for four years for total payments of $24,000, implying zero percent financing. Which is the better deal if the interest rate is 1% per month?

Solution:

To determine which option is better, we compare the present value (PV) of the total cost under each scenario, given a monthly interest rate of 1% (0.01 per month).

Option A: Pay $24,000 upfront and receive a $2,000 rebate

• Net cost = $24,000 - $2,000 = $22,000
• Since this is a lump-sum payment today, its present value is simply $22,000.

Option B: Pay $500 per month for 4 years

• This is a 48-month annuity where each payment is $500.

• Using the present value formula for an annuity:

  $$PV = P \times \left( 1 - \frac{1}{(1 + r)^n} \right) \div r$$
  
  

  where:

  • $P=\mathrm{<div\; style="text-align:\; justify;"><span\; style="font-family:\; arial;\; font-size:\; large;">500(monthly\; payment)</span></div><span\; style="font-family:\; arial;\; font-size:\; large;\; text-align:\; justify;"><br></span>}$
  • $r=\mathrm{<div\; style="text-align:\; justify;"><span\; style="font-family:\; arial;\; font-size:\; large;">0.01(</span><span\; style="font-family:\; arial;\; font-size:\; large;">monthly\; interest\; rate)</span></div><span\; style="font-family:\; arial;\; text-align:\; justify;"><br></span>}$
  • $n=\mathrm{48(number\; of\; months) <span\; style="font-family:\; arial;\; font-size:\; large;"><br></span>}$

  Plugging in the values:

  $$PV = 500 \times \left( 1 - \frac{1}{(1.01)^{48}} \right) \div 0.01$$
  
  
  $$PV = 500 \times \left( 1 - \frac{1}{1.6084} \right) \div 0.01$$
  
  
  $$PV = 500 \times \left( 1 - 0.6218 \right) \div 0.01$$
  
  
  $$PV = 500 \times 0.3782 \div 0.01$$
  
  
  $$PV = 500 \times 37.82 = 18,910$$
  
  

  So, the present value of payments under the financing option is $18,910.

Conclusion

• Option A (lump sum payment) has a present value of $22,000.
• Option B (zero-percent financing) has a present value of $18,910.
• Since the present value of payments is lower in Option B, it is the better deal financially.

Thus, taking the zero-percent financing and making $500 monthly payments is the better choice.