question 11\nyou want to be able to withdraw $50,000 from your account each year for 25 years after you…

question 11\nyou want to be able to withdraw $50,000 from your account each year for 25 years after you retire.\nyou expect to retire in 20 years.\nif your account earns 4% interest, how much will you need to deposit each year until retirement to achieve your retirement goals?\nhint: you must first find the \required ending balance\ and then use that ending balance to calculate the \regular deposit amount\.\n$

question 11\nyou want to be able to withdraw $50,000 from your account each year for 25 years after you retire.\nyou expect to retire in 20 years.\nif your account earns 4% interest, how much will you need to deposit each year until retirement to achieve your retirement goals?\nhint: you must first find the \required ending balance\ and then use that ending balance to calculate the \regular deposit amount\.\n$

Answer

Explanation:

Step1: Calculate the present - value of the retirement withdrawals

We use the present - value of an ordinary annuity formula $PV = A\times\frac{1-(1 + r)^{-n}}{r}$, where $A=$50000$, $r = 0.04$, and $n = 25$. $$PV=50000\times\frac{1-(1 + 0.04)^{-25}}{0.04}$$ First, calculate $(1 + 0.04)^{-25}=1\div(1.04)^{25}\approx0.375094$. Then $1-(1 + 0.04)^{-25}=1 - 0.375094 = 0.624906$. And $\frac{1-(1 + 0.04)^{-25}}{0.04}=\frac{0.624906}{0.04}=15.62265$. So, $PV = 50000\times15.62265=$781132.5$ (this is the required ending balance).

Step2: Calculate the annual deposit amount

We now use the future - value of an ordinary annuity formula $FVA = P\times\frac{(1 + r)^{n}-1}{r}$, and we want to solve for $P$. We know $FVA=$781132.5$, $r = 0.04$, and $n = 20$. Rearranging the formula for $P$ gives $P=\frac{FVA\times r}{(1 + r)^{n}-1}$. Calculate $(1 + 0.04)^{20}=1.04^{20}\approx2.20804$. Then $(1 + 0.04)^{20}-1=2.20804 - 1 = 1.20804$. And $\frac{781132.5\times0.04}{1.20804}=\frac{31245.3}{1.20804}\approx$25864.43$

Answer:

$25864.43$