question 11\nyou want to be able to withdraw $50,000 from your account each year for 15 years after you…

question 11\nyou want to be able to withdraw $50,000 from your account each year for 15 years after you retire.\nyou expect to retire in 25 years.\nif your account earns 4% interest, how much will you need to deposit each year until retirement to achieve your retirement goals?\nhint: you must first find the \required ending balance\ and then use that ending balance to calculate the \regular deposit amount\.\n$

question 11\nyou want to be able to withdraw $50,000 from your account each year for 15 years after you retire.\nyou expect to retire in 25 years.\nif your account earns 4% interest, how much will you need to deposit each year until retirement to achieve your retirement goals?\nhint: you must first find the \required ending balance\ and then use that ending balance to calculate the \regular deposit amount\.\n$

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 $n = 15$. $$PV=50000\times\frac{1-(1 + 0.04)^{-15}}{0.04}$$ First, calculate $(1 + 0.04)^{-15}\approx0.555264$. Then $1-(1 + 0.04)^{-15}=1 - 0.555264 = 0.444736$. And $\frac{1-(1 + 0.04)^{-15}}{0.04}=\frac{0.444736}{0.04}=11.1184$. So, $PV = 50000\times11.1184=$555920$ (this is the required ending balance).

Step2: Calculate the annual deposit amount

We now use the future - value of an ordinary annuity formula $FVA = P\times\frac{(1 + r)^{n}-1}{r}$, where $FVA=$555920$, $r = 0.04$, and $n = 25$. We need to solve for $P$. Rearranging the formula for $P$ gives $P=\frac{FVA\times r}{(1 + r)^{n}-1}$. $(1 + 0.04)^{25}\approx2.665837$. Then $(1 + 0.04)^{25}-1=2.665837 - 1 = 1.665837$. And $\frac{(1 + 0.04)^{25}-1}{0.04}=\frac{1.665837}{0.04}=41.645925$. So, $P=\frac{555920\times0.04}{(1 + 0.04)^{25}-1}=\frac{22236.8}{1.665837}\approx$13349.87$

Answer:

$13349.87$