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.
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
$$PV=C\cdot \frac{r_e^n-1 }{\sqrt[12]{r_e}-1}\cdot \frac1{r_e^n}$$