Question

attempt 2: 1 out of 2 parts have been answered correctly. finance and growth: the parents of a disabled child have set up a payout annuity to supplement social security and to care for their child after they are gone. if the account starts with $50,000.00 and the child is expected to live for 30 years, and will earn 4.8% compounded monthly. what is the amount of the monthly distribution available to the child? how much interest is earned during the distribution period? round to the nearest cent (or 2 decimal places). do not include dollar sign or commas.

Explanation:

Step1: Identify the annuity - formula variables

We use the present - value of an ordinary annuity formula $PV = PMT\times\frac{1-(1 + r)^{-n}}{r}$, where $PV$ is the present value of the annuity, $PMT$ is the monthly payment, $r$ is the monthly interest rate, and $n$ is the total number of periods. The initial amount $PV=50000$, the annual interest rate $i = 4.8\%=0.048$, so the monthly interest rate $r=\frac{0.048}{12}=0.004$, and the number of months $n = 30\times12=360$.

Step2: Solve for the monthly payment $PMT$

From $PV = PMT\times\frac{1-(1 + r)^{-n}}{r}$, we can re - arrange it to $PMT=\frac{PV\times r}{1-(1 + r)^{-n}}$. Substitute $PV = 50000$, $r = 0.004$, and $n = 360$ into the formula:
\[

$$\begin{align*} PMT&=\frac{50000\times0.004}{1-(1 + 0.004)^{-360}}\\ &=\frac{200}{1-(1.004)^{-360}}\\ \end{align*}$$

\]
Using a calculator, $(1.004)^{-360}\approx0.2477$, so $PMT=\frac{200}{1 - 0.2477}=\frac{200}{0.7523}\approx262.33$.

Step3: Calculate the total amount paid out

The total amount paid out over 30 years (360 months) is $A_{total}=PMT\times n=262.33\times360 = 94438.8$.

Step4: Calculate the interest earned

The interest earned $I=A_{total}-PV$. Substitute $A_{total}=94438.8$ and $PV = 50000$ into the formula, $I=94438.8 - 50000=44438.8\approx44439.76$ (there may be a small difference due to rounding in the previous steps).

Answer:

Monthly distribution: 262.33
Interest earned: 44439.76