We are tasked with calculating the future value of a series of monthly deposits into an account with a 2.7% APR compounded monthly. We are also asked to assess the impact of adding an extra $25 to each deposit.

Step 1: Given Information

• Regular deposit amount (K): $175
• Extra deposit (second scenario): $25, so total monthly deposit = $175 + $25 = $200
• Interest rate (APR): 2.7% per year, compounded monthly.
• Monthly interest rate (r): 2.7% ÷ 12 = 0.225% = 0.00225 (as a decimal).
• Time period for first scenario (age 25 to age 65): 40 years = 40 × 12 = 480 months.
• Time period for second scenario (age 25 to age 70): 45 years = 45 × 12 = 540 months.

Step 2: Future Value Formula

The future value of an annuity (series of monthly deposits) with compounding interest can be calculated using the formula:

$FV = K \times \left(\frac{(1 + r)^n - 1}{r}\right)$

Where:

• $K$ = monthly deposit amount,
• $r$ = monthly interest rate,
• $n$ = total number of payments (months),
• $FV$ = future value of the account at the end of the period.

Scenario 1: Regular Monthly Deposits of $175

For age 65 (40 years, 480 months):

We substitute the values:

• $K = 175$
• $r = 0.00225$
• $n = 480$

$FV_{65} = 175 \times \left(\frac{(1 + 0.00225)^{480} - 1}{0.00225}\right)$

Scenario 2: Extra Monthly Deposits of $200

For age 65 (40 years, 480 months):

We substitute the values:

• $K = 200$
• $r = 0.00225$
• $n = 480$

$FV_{65} = 200 \times \left(\frac{(1 + 0.00225)^{480} - 1}{0.00225}\right)$

For age 70 (45 years, 540 months):

We substitute the values:

• $K = 200$
• $r = 0.00225$
• $n = 540$

$FV_{70} = 200 \times \left(\frac{(1 + 0.00225)^{540} - 1}{0.00225}\right)$

Let me now calculate the results.Here are the results based on the calculations:

1. At age 65 with regular monthly deposits of $175, you will have approximately $150,975.
2. At age 65 with extra monthly deposits of $200, you will have approximately $172,543.
3. At age 70 with extra monthly deposits of $200, you will have approximately $210,284.

Would you like more details or have any questions?

Related Questions:

1. What would the value be if you increased the APR to 3%?
2. How much would you accumulate if you started at age 30 instead of 25?
3. What would happen if the interest was compounded weekly instead of monthly?
4. What is the total interest earned in both scenarios by age 65?
5. How does increasing the monthly deposit to $300 change the results?

Tip:

Small increases in monthly contributions early on can significantly boost your savings due to the effects of compound interest over time.