How much should you deposit at the end of each month in an IRA that pays 8% compounded monthly to earn $120,000 per year from interest alone, while leaving theprincipal untouched, to be withdrawn at the end of each year after you retire in 50 years? Determine the monthly deposit in dollars. Round up to the nearest dollar in the final answer.

To determine the monthly deposit needed in an IRA account that pays 8% compounded monthly, we must follow these steps:

### Step 1: Understanding the Requirements

1. **Annual Interest Income Requirement**: You need to earn $120,000 per year from interest alone during retirement.
2. **Interest Rate**: The IRA pays 8% compounded monthly.
3. **Retirement Timeline**: You will start withdrawing at the end of 50 years, and you need the principal untouched.
4. **Principal Preservation**: The principal amount must remain intact, only earning interest to be withdrawn each year.

### Step 2: Calculate the Principal Needed at Retirement

To earn $120,000 annually from interest alone, we first need to calculate the principal amount required at the time of retirement.

The principal amount $ P $ required can be found using the annual interest formula:

\[
\text{Annual Interest} = P \times r
\]

where:
- $ P $ is the principal required at retirement.
- $ r $ is the annual interest rate.

Given:
- Annual Interest = $120,000
- Annual Interest Rate $ r $ = 8% or 0.08.

So, the principal amount $ P $ is:

\[
P = \frac{120,000}{0.08} = 1,500,000
\]

### Step 3: Calculate the Monthly Deposit Required

To accumulate $1,500,000 over 50 years with monthly deposits into an account that compounds monthly, we use the future value of an annuity formula:

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

where:
- $ FV $ = future value of the annuity ($1,500,000).
- $ PMT $ = monthly deposit (what we are solving for).
- $ i $ = monthly interest rate = $\frac{0.08}{12} = 0.006667$.
- $ n $ = number of compounding periods per year (12).
- $ t $ = number of years (50).

Substituting the values:

\[
1,500,000 = PMT \times \frac{(1 + 0.006667)^{600} - 1}{0.006667}
\]

Let's solve for $ PMT $.The monthly deposit required to accumulate $1,500,000 over 50 years at an 8% interest rate compounded monthly is approximately **$189**.

### Additional Questions:
1. How would the required monthly deposit change if the interest rate were higher, say 10%?
2. What would be the effect on the required deposit if the investment period were 40 years instead of 50 years?
3. How does compounding frequency (e.g., quarterly vs. monthly) affect the final required deposit amount?
4. What would happen if the desired annual withdrawal amount were increased to $150,000?
5. How would inflation impact the real value of the withdrawals over time?

**Tip:** Start saving early and contribute consistently to take full advantage of compound interest over time!

How much should you deposit at the end of each month in an IRA that pays 8% compounded monthly to earn $120,000 per year from interest alone, while leaving the principal untouched, to be withdrawn at the end of each year after you retire in 50 years? Determine the monthly deposit in dollars. Round up to the nearest dollar in the final answer.

Calculate the monthly deposit required for an IRA that pays 8% compounded monthly to accumulate enough principal to earn $120,000 annually from interest alone after 50 years. This problem uses the future value of annuity formula and compound interest principles. Suitable for advanced students in Grade 12 and college level.

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