Given price of an annuity and its level payments, find the interest rate

[Eclair_de_XII]

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?

 

[anuttarasammyak]

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.)

I have poor literacy on economy. So let me learn your way. I read without interest monthly equal payment is
\frac{300,000}{30*12}=833.33...
You say both 0.072 and 0.006 are rate of month but I assume 0.072 is annually and 0.006 is monthly because of factor 12, right?
Accumulated interest in 30 yrs is
0.072*30=2.16
times mortage. So the amount we should pay once 30 yrs later divided by 360 month is
833.33*(1+2.16)=2633.32
,right ?
I have no idea how number 2036.36 comes from these numbers. I should appreciate it if you would teach its calculation formula to me so that I might be able to have mortgage in future.