We have to find the present value of a annuity of $540 payable every end of the month at 7% compounded monthly for 4 years and 5 months.
We can express the present value PV as:
[tex]PV=M\cdot\frac{[1-(1+r\/m)^{-n\cdot m}]}{r\/m}[/tex]where M: monthly payment (M = 540), r: annual nominal rate (r = 0.07), m: number of subperiods of compounding per year (m = 12) and n: number of years (n = 4+5/12).
We can replace the variables with its value and calculate PV as:
[tex]\begin{gathered} PV=540\cdot\frac{[1-(1+\frac{0.07}{12})^{-53}]}{\frac{0.07}{12}} \\ PV\approx540\cdot\frac{[1-(1.005833)^{-53}]}{0.005833} \\ PV\approx540\cdot\frac{1-0.7347}{0.005833} \\ PV\approx540\cdot\frac{0.2653}{0.005833} \\ PV\approx540\cdot45.4826 \\ PV\approx24560.60 \end{gathered}[/tex]Answer: The present value of teh annuity is P 24560.60.
