The question requires us to find the APR on the loan. APR is the Annual percentage rate. It is basically the amount charged for borrowing (a loan) or amount made by investing (an investment).
This question is an example of an annuity. Annuities are defined as a series of fixed payments made by you or made to you over a fixed period of time. The common payment periods are yearly, semi-annually, quarterly or monthly.
This question is a typical case of present value annuity. Because we are taking the loan now, we are determining the future payments of the loan based on today's value of the loan.
The present value formula is as follows:
`PV = x ((1 - (1+i)^(-n))/i)`
PV = Present Value
We are told out loan value is 75% of the mortgage value:
`PV = 0.75 * $4 200 000 = $3 150 000`
x= monthly payments
`x =$18 300 `
n = number of payments
We are making payments monthly for 25 years:
`n = 25 * 12 = 300 `
i = interest rate . We are required to find this to answer the question.
Now we can substitute into our equation:
`3 150 000 = 18300 ((1-(1+i)^(-300))/i)`
`(3150000/18300) = ((1-(1+i)^(-300))/i)`
This question requires us to iterate using software such as wolfram alpha or excel. Otherwise, trial and error can be used but will very time consuming.
`i = 0.00041135`
Now we need to determine the annual percentage rate:
`APR = ((1+i)^n -1 )*100`
`APR = (((1+0.00041135)^300)-1)*100`
`APR = 13.13%`
Answer: APR = 13.13%