Question

You want to borrow $27,500 and can afford monthly payments of $650 for 48 months, but no more. Assume monthly compounding. What is the highest APR rate you can afford

Answer 1

Answer:

Max APR=6.34%

Explanation:

Present Value of Payments

If someone borrows an amount PV and will make regular payments of R dollars, then the relation between them is

$PV=F_a\cdot R$

Where

$\displaystyle F_a=\frac{1-(1+i)^{-n}}{i}$

We know the maximum value for R is $650, thus we can know the minimum value for Fa with:

$\displaystyle F_a=\frac{PV}{R}=\frac{27,500}{650}=42.31$

It means that we need to find the value of i such that (for n=48):

$\displaystyle \frac{1-(1+i)^{-48}}{i}=42.308$

This equation cannot be solved in terms of natural or algebraic functions. We need to find the best possible value of i by any numerical approximate method. Let's start off by setting i=0.01

$\displaystyle \frac{1-(1+0.01)^{-48}}{0.01}=37.97$

It's too far away from the required value. Now we adjust to i=0.005

$\displaystyle \frac{1-(1+0.005)^{-48}}{0.005}=42.58$

This is a much better value. A final iteration for i=0.00528 gives

$\displaystyle \frac{1-(1+0.00528)^{-48}}{0.00528}=42.302$

This value is close enough to the required answer. Converting it tho APR:

$i=0.00528\cdot 12\cdot 100=6.34\%$