Questions: Question 10 Multi-Step Applications Solve the following problems. Round your results to the nearest cent as needed. You want to be able to withdraw 40,000 from your account each year for 20 years after you retire from an account earning 8% interest compounded annually. How much will need in your retirement account at the beginning of your retirement? You want to open account to achieve this goal. You will make annual payments and earn 8% interest. If you expect to retire in 30 years, how much should your annual payments be to reach this goal?

Solution Steps

To determine how much you need in your retirement account at the beginning of your retirement, we calculate the present value of an annuity using the formula:

\[ PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]

where:

• \( PV \) is the present value,
• \( PMT = 40000 \) is the annual withdrawal,
• \( r = 0.08 \) is the interest rate,
• \( n = 20 \) is the number of years of withdrawals.

Substituting the values, we find:

\[ PV = 40000 \times \left( \frac{1 - (1 + 0.08)^{-20}}{0.08} \right) \approx 392725.90 \]

Next, we need to find the annual payments required to reach the retirement goal. We use the future value of an annuity formula:

\[ FV = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right) \]

Rearranging this formula to solve for \( PMT \):

\[ PMT = \frac{FV \times r}{(1 + r)^n - 1} \]

where:

• \( FV = 392725.90 \) is the amount needed at retirement,
• \( r = 0.08 \) is the interest rate,
• \( n = 30 \) is the number of years until retirement.

Substituting the values, we find:

\[ PMT = \frac{392725.90 \times 0.08}{(1 + 0.08)^{30} - 1} \approx 3466.76 \]

Final Answer