What is the present value of an annuity consisting of monthly payments of an amount C. What is the present value of an annuity consisting of monthly payments of an amount C continuing for n years? Express the answer in terms of the effective rate re.
I know that the effective rate $re = (1 + \frac {r}{m}) ^ m $. Where m is the amount of payments per year. But apart from that I don't understand how to solve this question. 
 A: The future value of an monthly payed annuity C after $t$ months is
$$FV=C+C\cdot q_m+C\cdot q_m^2+C\cdot q_m^3+\ldots +C\cdot q_m^{t-1}$$
$$=C\cdot\left( 1+q_m+q_m^2+q_m^3+\ldots +q_m^{t-1}\right)$$
with $q_m=\left(1+\frac{i}{12} \right)\Rightarrow q_m^{12}=\left(1+\frac{i}{12} \right)^{12}=r_e\qquad (*)$
$i$ is the yearly interest rate. $q_m$ is the monthly interest factor. The term in the brackets is a geometric series. We can use the closed form of it.
$$FV=C\cdot \frac{q_m^t-1 }{q_m-1}$$
In $n$ years we have $12\cdot n$ months. Thus $t=12\cdot n$ 
$$FV=C\cdot \frac{q_m^{12\cdot n}-1 }{q_m-1}=C\cdot \frac{\left(q_m^{12}\right)^n-1 }{q_m-1}$$
To get the present value $FV$ has to be discouted $12\cdot n$ times.
$$PV=C\cdot \frac{\left(q_m^{12}\right)^n-1 }{q_m-1}\cdot \frac1{\left(q_m^{12}\right)^ n}$$
And $q_m=\sqrt[12]{q_m^{12}}$
$$PV=C\cdot \frac{\left(q_m^{12}\right)^n-1 }{\sqrt[12]{q_m^{12}}-1}\cdot \frac1{\left(q_m^{12}\right)^ n}$$
Using the relation (*) we get 
$$PV=C\cdot \frac{r_e^n-1 }{\sqrt[12]{r_e}-1}\cdot \frac1{r_e^n}$$
