- #1
Eclair_de_XII
- 1,083
- 91
- Homework Statement
- Suppose the present-value of some annuity is some positive number ##A##. Suppose that its periodic payments are some positive number ##P##. Find the interest rate ##i##, given the number of periods.
- Relevant Equations
- ##A=P\,a_{n|i}##
A 30-year monthly-payment mortgage loan for 300,000 is offered at a nominal rate of 0.072 converted monthly. Thus the monthly effective interest rate is 0.006 and the calculated monthly payment is 2036.36. (Calculate the payment (PMT) on your calculator and leave it there for the moment.)
When the loan closes, the lender applies a fee of 3 'points' for which no service is performed. It is taking 0.03 of the loan amount (9000) as a fee that raises the lender's yield. In effect, the borrower is receiving a loan of only 291,000. This increases the borrower's interest rate. To see this, modify the loan in your calculator by setting PV = 291,000 and CPT I/Y. The result is a monthly rate of 0.006257. Multiply this by 12 to find the borrowers actual actual nominal annual rate: 0.075083.
##\textbf{Actual ``Work''}##
##\mathrm{PV}=A\,a_{n|i}##
##\frac{\mathrm{PV}}{A}i=1-(1+i)^{-n}##
I'm not sure if I could solve for ##i## algebraically. Is there any way of doing this without using a calculator?
When the loan closes, the lender applies a fee of 3 'points' for which no service is performed. It is taking 0.03 of the loan amount (9000) as a fee that raises the lender's yield. In effect, the borrower is receiving a loan of only 291,000. This increases the borrower's interest rate. To see this, modify the loan in your calculator by setting PV = 291,000 and CPT I/Y. The result is a monthly rate of 0.006257. Multiply this by 12 to find the borrowers actual actual nominal annual rate: 0.075083.
##\textbf{Actual ``Work''}##
##\mathrm{PV}=A\,a_{n|i}##
##\frac{\mathrm{PV}}{A}i=1-(1+i)^{-n}##
I'm not sure if I could solve for ##i## algebraically. Is there any way of doing this without using a calculator?