ricky is 35 years old. he plans to retire whe...
ricky is 35 years old. he plans to retire when he is 63. he has opened a traditional retirement account that pays 1% interest compounded monthly. if he makes monthly deposits of $400, how much will he have in the account by the time he retires?
Answer
# Explanation:
## Step1: Identify the time - period and interest rate parameters
The number of years from 35 to 63 is $n_y=63 - 35=28$ years. Since the interest is compounded monthly and deposits are made monthly, the number of periods $n = 28\times12=336$ months. The monthly interest rate $r=\frac{0.01}{12}$. The monthly deposit $P = 400$.
## Step2: Use the future - value of an ordinary annuity formula
The formula for the future - value of an ordinary annuity is $FVA=P\times\frac{(1 + r)^n-1}{r}$.
Substitute $P = 400$, $r=\frac{0.01}{12}$, and $n = 336$ into the formula:
\[
\begin{align*}
FVA&=400\times\frac{(1+\frac{0.01}{12})^{336}-1}{\frac{0.01}{12}}\\
\end{align*}
\]
First, calculate $(1+\frac{0.01}{12})^{336}$. Let $x=\frac{0.01}{12}\approx0.000833$. Then $(1 + x)^{336}=(1+0.000833)^{336}$. Using the formula $a^b = e^{b\ln(a)}$, we have $(1 + 0.000833)^{336}=e^{336\ln(1.000833)}$. $\ln(1.000833)\approx0.0008326$ and $336\times0.0008326 = 0.28075$. So, $(1 + 0.000833)^{336}=e^{0.28075}\approx1.3247$.
\[
\begin{align*}
FVA&=400\times\frac{1.3247 - 1}{\frac{0.01}{12}}\\
&=400\times\frac{0.3247}{\frac{0.01}{12}}\\
&=400\times\frac{0.3247\times12}{0.01}\\
&=400\times\frac{3.8964}{0.01}\\
&=400\times389.64\\
&=155856
\end{align*}
\]
# Answer:
$155856$