Question

a student has a total of 2000 in student loan that will be paid with 48 month installment loan with monthly payments of 49.30 determine the apr of the loan to the nearest one half of a percent

Answer 1

Annuities

The initial value of a loan (P) that will be paid with n installments of R dollars each is calculated as:

$P=R\cdot\frac{1-(1+i)^{-n}}{i}$

Where i is the adjusted interest rate:

$i=\frac{APR}{m}$

And m is the number of payments per year.

We are given the data:

P = 2000

m = 121

n = 48

R = 49.30

Substituting:

$2000=49.30\cdot\frac{1-(1+i)^{-48}}{i}$

This equation cannot be solved with a fixed formula or by isolating the unknown variable i.

We need to use successive approximations until we find a reasonable precision for the equation above.

Starting with the value:

i = 0.01, we get the equation: 2000 = 1872

For:

i = 0.005, we get the equation 2000 = 2099

i = 0.007, we get 2000 = 2004

i = 0.0071, we get 2000 = 1999

This value is close enough to produce an accurate answer, so:

i = 0.0071

Now calculate the APR:

$APR=i\times m=0.0071\times12=0.0852$

Converting to % and rounding as required: APR = 8.5%