Question

future value of annuity. kirk can take his $980 income tax refund and invest it in a 36 - month certificate of deposit at 7% compounded monthly, or he can take the money to purchase a home entertainment system and put $35 a month in a bank savings account that will pay him 8% annual interest, compounded monthly. which choice will give him more money at the end of three years? after depositing his income tax refund, at the end of three years kirk will have $ (use your financial calculator. round your final answer to the nearest cent.)

Explanation:

Step1: Identify the annuity - formula

The future - value of an ordinary annuity formula is $FVA = A\times\frac{(1 + r)^{n}-1}{r}$, where $A$ is the annuity payment, $r$ is the interest rate per period, and $n$ is the number of periods.
The monthly deposit $A=\$35$, the annual interest rate for the savings account is $i = 8\%=0.08$, so the monthly interest rate $r=\frac{0.08}{12}$, and the number of months $n = 36$.

Step2: Calculate the future - value of the annuity

Substitute the values into the formula:
\[

$$\begin{align*} FVA&=35\times\frac{(1+\frac{0.08}{12})^{36}-1}{\frac{0.08}{12}}\\ \end{align*}$$

\]
First, calculate $(1+\frac{0.08}{12})^{36}$. Let $x=\frac{0.08}{12}\approx0.00667$, then $(1 + x)^{36}=(1 + 0.00667)^{36}$. Using the formula $(a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}$, or simply using a calculator, $(1+0.00667)^{36}\approx1.270237$.
Then $(1+\frac{0.08}{12})^{36}-1\approx1.270237 - 1=0.270237$.
And $\frac{(1+\frac{0.08}{12})^{36}-1}{\frac{0.08}{12}}=\frac{0.270237}{\frac{0.08}{12}}=\frac{0.270237\times12}{0.08}=40.53555$.
So $FVA = 35\times40.53555=\$1418.74425$.

Answer:

$\$1418.74$