You have borrowed $135,000 from the bank today. You are required to repay this money over the next six years by making monthly payments of $2,215.10 at the end of each month. What is the quoted interest rate for the loan (with monthly compounding)?
Please show me the steps on how to solve this
To find the quoted interest rate for the loan, you can use the present value formula for an ordinary annuity. The formula is as follows:
PV = P × [1 - (1 + r)^(-n)] / r
Where:
PV = Present value or loan amount ($135,000)
P = Monthly payment ($2,215.10)
r = Monthly interest rate (unknown)
n = Number of months (6 years * 12 months = 72 months)
Now let's solve for r:
1. Plug in the given values into the formula:
135,000 = 2,215.10 × [1 - (1 + r)^(-72)] / r
2. Simplify the equation:
135,000 = 2,215.10 - 2,215.10 (1 + r)^(-72) / r
3. Multiply both sides of the equation by r:
135,000r = 2,215.10r - 2,215.10 (1 + r)^(-72)
4. Move all terms to one side of the equation:
135,000r - 2,215.10r = -2,215.10 (1 + r)^(-72)
5. Simplify the equation further:
132,784.90r = 2,215.10 (1 + r)^(-72)
6. Divide both sides of the equation by 2,215.10:
132,784.90r / 2,215.10 = (1 + r)^(-72)
7. Simplify the left side of the equation:
r = 0.0166667
8. Calculate the monthly interest rate by multiplying r by 100:
r = 0.0166667 * 100 = 1.66667%
Therefore, the quoted interest rate for the loan (with monthly compounding) is approximately 1.66667%.
To find the quoted interest rate for the loan, we can use the present value formula for an ordinary annuity. The formula is:
PV = PMT * [(1 - (1 + r)^(-n)) / r]
where:
PV = present value (loan amount) = $135,000
PMT = monthly payment = $2,215.10
r = interest rate
n = number of periods (months) = 6 years * 12 months = 72 months
We need to solve for 'r' in this equation.
1. Plug in the given values into the formula:
$135,000 = $2,215.10 * [(1 - (1 + r)^(-72)) / r]
2. Rearrange the equation to isolate the interest rate 'r':
[(1 - (1 + r)^(-72)) / r] = $135,000 / $2,215.10
3. Divide both sides of the equation by $2,215.10:
[(1 - (1 + r)^(-72)) / r] = 60.99
4. Multiply both sides of the equation by 'r' to remove the denominator:
1 - (1 + r)^(-72) = 60.99r
5. Distribute and rearrange the equation:
1 - 60.99r = (1 + r)^(-72)
6. Square both sides of the equation:
(1 - 60.99r)^2 = [(1 + r)^(-72)]^2
7. Expand and simplify the equation:
1 - 121.98r + 3713.9601r^2 = 1 + 144r + r^2
3713.9601r^2 + 122.98r + 121r - 143r + 1 - 1 = 0
3713.9601r^2 + 100r = 0
8. Solve the quadratic equation. Since the coefficient of the first-degree term (r) is relatively small, we can assume that only the second-degree term (r^2) contributes significantly, so we can ignore the smaller term:
3713.9601r^2 = 0
r^2 = 0
r = 0
9. The solution 'r = 0' implies that there is no interest charged in this loan. Therefore, the quoted interest rate for the loan is 0% with monthly compounding.
Please note that it is unusual to have a loan with 0% interest. Double-check the given information and calculations to ensure accuracy.
2215.1( 1 - (1+i)^-72)/i = 135000
very nasty equation to solve, since there is no "formula" way to do this
Wolfram is our saviour.
(notice I changed the i to x)
the reasonable answer is i = .00470829 for the monthly rate
so annual rate = 12(.0047...) = .056499..
or appr 5.65% per annum compounded monthly
check:
2215.10( 1 - 1.00470829^-72)/.00470829
= 134999.9987
how about that ????, not bad eh?