Suppose you want to have $$ \$ 800,000 $$ for retirement in 20 years. Your account earns $$ 8 \% $$ interest. Round your answers to the nearest cent. a) How much would you need to deposit in the account each month? b) How much interest will you earn?

Question

UpStudy AI · Accepted Answer

We want to accumulate a future value of  
$$
FV = \$800{,}000
$$
in $$20$$ years by making equal monthly deposits into an account that earns a nominal annual interest rate of $$8\%$$ compounded monthly.

Let  
$$
i = \frac{0.08}{12} \quad \text{and} \quad n = 20 \times 12 = 240.
$$

The future value of an annuity (a series of equal deposits) is given by  
$$
FV = P \times \frac{(1+i)^n - 1}{i},
$$
where $$P$$ is the monthly deposit and $$i$$ is the monthly interest rate.

### a) Finding the Monthly Deposit

Solve for $$P$$:
$$
P = FV \times \frac{i}{(1+i)^n - 1}.
$$

Substitute the given values:
$$
P = 800{,}000 \times \frac{\frac{0.08}{12}}{(1+\frac{0.08}{12})^{240}-1}.
$$

Here,  
$$
i = \frac{0.08}{12} \approx 0.00666667.
$$

Now, calculate $$(1+i)^{240}$$:
$$
(1.00666667)^{240}.
$$
Using exponentiation (by taking natural logs or using a calculator) we find:
$$
(1.00666667)^{240} \approx 4.898.
$$

Now the expression in the denominator is:
$$
(1+i)^n - 1 \approx 4.898 - 1 = 3.898.
$$

Thus,
$$
P \approx 800{,}000 \times \frac{0.00666667}{3.898}.
$$

Calculate the numerator:
$$
800{,}000 \times 0.00666667 \approx 5333.336.
$$

Finally, divide:
$$
P \approx \frac{5333.336}{3.898} \approx 1367.03.
$$

So, the monthly deposit required is approximately  
$$
\boxed{\$1367.03.}
$$

### b) Calculating the Total Interest Earned

The total amount contributed by making monthly deposits for 240 months is:
$$
\text{Total Contributions} = P \times n \approx 1367.03 \times 240 \approx 328087.20.
$$

The interest earned is the difference between the future value and the total contributions:
$$
\text{Interest Earned} = FV - (\text{Total Contributions}) \approx 800{,}000 - 328087.20 \approx 471912.80.
$$

Thus, the interest earned is approximately  
$$
\boxed{\$471912.80.}
$$
a) You need to deposit approximately \$1,367.03 each month. b) You will earn approximately \$471,912.80 in interest.

Suppose you want to have \( \$ 800,000 \) for retirement in 20 years. Your account earns \( 8 \% \) interest. Round your answers to the nearest cent. a) How much would you need to deposit in the account each month? b) How much interest will you earn?

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Economics Questions