What is the Monthly Payment on a Car Loan with 6% Interest?

[rmiller70015]

Homework Statement

This problem is for a senior level math class, but since the problem involves finance and exponentials I posted it here. There is some Taylor series stuff, but that is all of the higher math involved.

When you finance a car in equal monthly payments of 6 years at 6% annual interest, the monthly interest rate is 0.06/12 = 0.005. The present value of these monthly payments equals the initial loan ammount. If you borrow 20,000 today, then what is the monthly payment that you must make?

Hint: let x = 1/1+i be the monthly discounting factor. Equate 20000 -= M + Mx + Mx^2 + ... +M^71 and solve for M.

Homework Equations

$M + Mx + Mx^2 + Mx^3 + ... + Mx^{71} = \frac{1-x^n}{1-x}$ This is a terminating geometric series so it equals that fraction.

The Attempt at a Solution

The formula for future value is: $$F = P(1 + i)^n$$

This is the monthly payment being equal to the principal times compounding factor: $$M = \sum^{71}_{0}P(1 + i)^n$$

Solving for P gives: $$P = \sum^{71}_{0}\frac{M}{(1 + i)^n}$$

72 iterations and setting $x = \frac{1}{1.005}$ gives: $$P = Mx[1 + x + x^2 + ... + x^{71}]$$

This is a terminating series: $$P = Mx(\frac{1-x^n}{1-x})$$

Solving for $M$ gives: $$M = \frac{Px(1-x)}{1-x^n}$$

Plugging in values of x and n gives: $$M = \frac{20,000\cdot \frac{1}{1.005}(1-\frac{1}{1.005})}{1-\frac{1}{1.005}^{71}} = \mathbf{\$335.34}$$

 

[MarkFL]

Check the step where you solve for $M$...you have an algebraic error there.

 

[Ray Vickson]

Homework Statement

This problem is for a senior level math class, but since the problem involves finance and exponentials I posted it here. There is some Taylor series stuff, but that is all of the higher math involved.

When you finance a car in equal monthly payments of 6 years at 6% annual interest, the monthly interest rate is 0.06/12 = 0.005. The present value of these monthly payments equals the initial loan ammount. If you borrow 20,000 today, then what is the monthly payment that you must make?

Hint: let x = 1/1+i be the monthly discounting factor. Equate 20000 -= M + Mx + Mx^2 + ... +M^71 and solve for M.

Usually, if you take out a loan today you do not have to make the first payment today (which is what you are assuming when you have $M/(1+i)^n$ with $n = 0,1, \ldots, 71$.) Instead, your first payment should be in one month's time, so you should have $n = 1, 2, \ldots, 72.$