To solve this, we use the formula for the future value of an ordinary annuity:

$FV = P \cdot \frac{(1 + r)^n - 1}{r}$

Where:

• $FV$ is the future value ($650,000)
• $P$ is the periodic payment (quarterly deposit, the value we want to find)
• $r$ is the interest rate per period ($2.4\% \text{ annual rate} \div 4$ quarters = $0.024 \div 4 = 0.006$)
• $n$ is the total number of compounding periods ($40 \text{ years} \cdot 4 = 160$ quarters)

Rearranging for $P$:

$P = \frac{FV \cdot r}{(1 + r)^n - 1}$

Substituting the values:

$P = \frac{650,000 \cdot 0.006}{(1 + 0.006)^{160} - 1}$

Let me calculate this.# Given values FV = 650000 # Future Value r = 0.006 # Quarterly interest rate n = 40 * 4 # Total number of periods (quarters)

Calculate periodic payment P

P = (FV * r) / ((1 + r)**n - 1) P2431.0948635731072431.094863573107successThe quarterly deposit you need to make is approximately $2,431.09. The correct answer is:

4. 2,431.09

Would you like a detailed breakdown of the calculation or assistance with related concepts?

Relative Questions:

1. How does compounding frequency affect the required deposit amount?
2. What would be the impact of a higher interest rate on the quarterly deposit?
3. How would the required deposit change if the retirement goal were increased to $800,000?
4. What is the difference between an ordinary annuity and an annuity due in terms of deposits?
5. How would inflation affect the real value of the retirement amount over 40 years?

Tip:

Always verify whether interest is compounded annually, quarterly, or monthly, as it significantly impacts the results.