Loan Fallacy

You are about to pay $10,000 cash for a new car. The finance manager, wanting to loan you money, asks you what interest rate you could earn on that money if you didn't spend it. You answer 6%, compounded monthly. The finance manager then says that he will give you a loan at 7.5% interest, compounded monthly, with a five year term. He then shows you on his computer that if you invested the $10,000 on your own you would earn $3,488.50 in interest in five years. Then he shows you that your payments on the car would be $200.38 a month, making total interest paid on the car $2,022.77. You do the math on your own and you realize it is correct. He then argues that you would come out ahead by taking out a loan since you would make more in interest than you would pay. Is this argument valid?


Answer:


One way of looking at the flaw in his logic is that he is assuming that you don't withdraw money from your investment to make the car payments. If you did you would have only $110.47 in the account after the 57th payment, with three full payments to go.
However, if you were willing to not touch your investment and make the payments out of your pocket there is still a problem. You have to consider the time value of money. Let's assume inflation equal to the 6% interest rate, compounded monthly. The time value of the car payments made would be $10,364.77, however the time value of your $13,488.50 after five years would only be $10,000. In other words after five years your initial investment of $10,000 has gone down in value more than the interest savings.
Note: The payments on a loan of $x, at intrest rate i, over n periods is $xi/(1-vn), where v=1/(1+i). In this case x=$10,000, i=.075/12=.00625, and n=60.