Monthly Payment loans

DonaldH_

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Dec 11, 2012
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Mike plans to purchase a hybrid car. He has applied for a loan of $32150. Loans are available for 3 years and 5 years at 8.3% per year, compounded monthly.
Determine Mike's monthly payment for the 3-year loan.

I've tried: 32150/1000 x ? But I don't what to do next?
 
Mike plans to purchase a hybrid car. He has applied for a loan of $32150. Loans are available for 3 years and 5 years at 8.3% per year, compounded monthly.
Determine Mike's monthly payment for the 3-year loan.

I've tried: 32150/1000 x ? But I don't what to do next?
What is x? Why divide by 1000?

What does your book say about net present value?
 
If you have not been given a formula for the monthly payment amount, this can be derived as follows:

Let P = monthly payment, A = amount borrowed, i = monthly interest rate, and n = the number of payments.

Also, let \(\displaystyle D_n\) be the debt amount after payment n.

Consider the recursion:

(1) \(\displaystyle D_{n}=(1+i)D_{n-1}-P\)

(2) \(\displaystyle D_{n+1}=(1+i)D_{n}-P\)

Subtracting (1) from (2) (symbolic differencing) yields the homogeneous recursion:

\(\displaystyle D_{n+1}=(2+i)D_{n}-(1+i)D_{n-1}\)

whose associated auxiliary equation is:

\(\displaystyle r^2-(2+i)r+(1+i)=0\)

\(\displaystyle (r-(1+i))(r-1)=0\)

Thus, the closed-form for our recursion is:

\(\displaystyle D_n=k_1(1+i)^n+k_2\)

Using initial values, we may determine the coefficients \(\displaystyle k_i\):

\(\displaystyle D_0=k_1+k_2=A\)

\(\displaystyle D_1=k_1(1+i)+k_2=(1+i)A-P\)

Solving this system, we find:

\(\displaystyle k_1=\dfrac{Ai-P}{i},\,k_2=\dfrac{P}{i}\) and so we have:

\(\displaystyle D_n=\left(\dfrac{Ai-P}{i} \right)(1+i)^n+\dfrac{P}{i}=\dfrac{(Ai-P)(1+i)^n+P}{i}\)

Now, equating this to zero (when the loan is paid off), we can solve for P:

\(\displaystyle \dfrac{(Ai-P)(1+i)^n+P}{i}=0\)

\(\displaystyle (Ai-P)(1+i)^n+P=0\)

\(\displaystyle (P-Ai)(1+i)^n=P\)

\(\displaystyle P((1+i)^n-1)=Ai(1+i)^n\)

\(\displaystyle P=\dfrac{Ai(1+i)^n}{(1+i)^n-1}\)

\(\displaystyle P=\dfrac{Ai}{1-(1+i)^{-n}}\)
 
Hello, DonaldH!

Mike plans to purchase a hybrid car; he has applied for a loan of $32,150.
Loans are available for 3 years and 5 years at 8.3% per year, compounded monthly.
Determine Mike's monthly payment for the 3-year loan.

This is an Amortization problem.

The Amortization Formula: .\(\displaystyle A \;=\;P\dfrac{i(1+i)^n}{(1+i)^n-1}\)

. . where: .\(\displaystyle \begin{Bmatrix}A &=& \text{periodic payment} \\ P &=& \text{amount of loan} \\ i &=& \text{periodic interest rate} \\ n &=& \text{number of periods} \end{Bmatrix}\)


Your problem has: .\(\displaystyle \begin{Bmatrix}P \:=\:\$32,150 \\ i \:=\:\frac{0.083}{12} \\ n \:=\:36 \end{Bmatrix}\)
Go for it!
 
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