Respuesta :
Solution:
The plan is to work for 40 years with a 25-year annuity.
To do this, we need to find the present value needed to achieve this in 25 years;
[tex]\begin{gathered} PV=P\times\frac{1-(1+r)^{-n}}{r} \\ \text{where;} \\ PV\text{ is the present value} \\ P\text{ is the value of each payment} \\ r\text{ is the interest rate per period} \\ n\text{ is the number of periods} \end{gathered}[/tex][tex]\begin{gathered} P=\text{ \$4500} \\ r=\frac{7.2}{12}\text{ \% =}\frac{0.072}{12} \\ r=0.006 \\ n=25\times12=300 \end{gathered}[/tex]Substituting these values in the formula above,
[tex]\begin{gathered} PV=4500\times\frac{1-(1+0.006)^{-25\times12}}{0.006} \\ PV=4500\times\frac{1-(1.006)^{-300}}{0.006} \\ PV=4500\times\frac{1-0.166}{0.006} \\ PV=4500\times\frac{0.834}{0.006} \\ PV=4500\times139 \\ PV=625500 \end{gathered}[/tex]The value that will be accumulated for 25 years annuity is $625,500.
To get the desired monthly yield, we use the future value formula for annuity.
The present value now represents the future value for the next 40years
[tex]\begin{gathered} FV=P\times\frac{(1+r)^n-1}{r} \\ FV=\text{ \$625,500} \\ P=\text{?} \\ r=0.006 \\ n=40\times12=480\text{ months} \end{gathered}[/tex]Substituting these values to get P,
[tex]\begin{gathered} 625500=P\times\frac{(1+0.006)^{480}-1}{0.006} \\ 625500=P\times\frac{1.006^{480}-1}{0.006} \\ 625500=P\times\frac{17.6616-1}{0.006} \\ 625500=P\times\frac{16.6616}{0.006} \\ 625500\times0.006=16.6616P \\ 3753=16.6616P \\ \text{Divide both sides by 16.6616;} \\ P=\frac{3753}{16.6616} \\ P=\text{ \$225.25} \end{gathered}[/tex]Therefore, the desired monthly yield is $225.25
