Question

You plan to work for 40 years and then retire using a 25-year annuity. You want to arrange a retirement income of $4500 per month. You have access to an account that pays an APR of 7.2% compounded monthly. What size nest egg do you need to achieve the desired monthly yield? (Round your answer to the nearest cent.)

Answer 1

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;

`\begin{gathered} PV=P*(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}`
`\begin{gathered} P=\text{ \$4500} \\ r=(7.2)/(12)\text{ \% =}(0.072)/(12) \\ r=0.006 \\ n=25*12=300 \end{gathered}`

Substituting these values in the formula above,

`\begin{gathered} PV=4500*(1-(1+0.006)^(-25*12))/(0.006) \\ PV=4500*(1-(1.006)^(-300))/(0.006) \\ PV=4500*(1-0.166)/(0.006) \\ PV=4500*(0.834)/(0.006) \\ PV=4500*139 \\ PV=625500 \end{gathered}`

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

`\begin{gathered} FV=P*((1+r)^n-1)/(r) \\ FV=\text{ \$625,500} \\ P=\text{?} \\ r=0.006 \\ n=40*12=480\text{ months} \end{gathered}`

Substituting these values to get P,

`\begin{gathered} 625500=P*((1+0.006)^(480)-1)/(0.006) \\ 625500=P*(1.006^(480)-1)/(0.006) \\ 625500=P*(17.6616-1)/(0.006) \\ 625500=P*(16.6616)/(0.006) \\ 625500*0.006=16.6616P \\ 3753=16.6616P \\ \text{Divide both sides by 16.6616;} \\ P=(3753)/(16.6616) \\ P=\text{ \$225.25} \end{gathered}`

Therefore, the desired monthly yield is $225.25