suppose you want to have $800,000 for retirement in 30 years. your account earns 8% interest. a) how much…

Explanation:

Step1: Identify the relevant formula for the future - value of an ordinary annuity

The formula for the future - value of an ordinary annuity is $F = A\times\frac{(1 + r)^{n}-1}{r}$, where $F$ is the future value of the annuity, $A$ is the amount of each payment, $r$ is the interest rate per period, and $n$ is the number of periods. The annual interest rate $i = 8%=0.08$. The number of years $t = 30$ years. Since the deposits are made monthly, the number of periods $n=30\times12 = 360$ months, and the monthly interest rate $r=\frac{0.08}{12}$. The future value $F = 800000$. We need to solve the formula for $A$: [A=\frac{F\times r}{(1 + r)^{n}-1}]

Step2: Calculate the monthly deposit amount $A$

Substitute $F = 800000$, $r=\frac{0.08}{12}$, and $n = 360$ into the formula: [r=\frac{0.08}{12}\approx0.00667] [(1 + r)^{n}=(1+\frac{0.08}{12})^{360}\approx10.9357] [A=\frac{800000\times\frac{0.08}{12}}{(1+\frac{0.08}{12})^{360}-1}=\frac{800000\times0.00667}{10.9357 - 1}=\frac{5336}{9.9357}\approx537.05]

Step3: Calculate the total amount of deposits

The total amount of deposits is the monthly deposit amount times the number of months. So the total amount of deposits $D=A\times n=537.05\times360 = 193338$.

Step4: Calculate the interest earned

The interest earned $I$ is the future value minus the total amount of deposits. So $I=F - D=800000-193338 = 606662$.

Answer:

a) $$537.05$ b) $$606662$