You plan to save for your child's education by depositing $699 at the end of each month into an account that earns an annual interest rate of 7%, compounded monthly. If you plan to invest for 10 years, calculate the value at the end of this period. Please write the answer in two decimal places.

Question

BlackTom AI · Accepted Answer

Question restatement: You are depositing 699 at the end of each month into an account that earns 7% annual interest, compounded monthly, for 10 years. Find the value at the end of the period, to two decimal places.

Step 1: Identify the cash flow type and formula. The deposits are made at the end of each month, so this is an ordinary annuity. The future value of an ordinary annuity with monthly deposits is given by FV = P × [((1 + i)^n − 1) / i], where P is the monthly payment, i is the monthly interest rate, and n is the total number of payments.

Step 2: Compute the monthly interest rate and number of payments. The annual nominal rate is 7%, so the monthly rate i is 0.07 / 12 ≈ 0.0058333333. Over 10 years with monthly deposits, the number of payments n = 10 × 12 = 120.

Step 3: Evaluate the growth factor: (1 + i)^n = (1.0058333333)^{120}. This exponentiation yields a growth factor slightly over 2, reflecting long-term compounding.

Step 4: Compute the numerator ( (1 + i)^n − 1 ). Subtract 1 from the growth factor to account for the cumulative effect of all deposits being compounding over time.

Step 5: Divide by the monthly rate i to convert the accumulated growth into a multiplier for the monthly deposit amount.

Step 6: Multiply by the monthly payment P = 699 to obtain the future value after 120 deposits.

Step 7: Numerical result and rounding. Substituting i = 0.0058333333 and n = 120 into FV = 699 × [((1 + 0.0058333333)^{120} − 1) / 0.0058333333] gives a value around 120,807.20 when rounded to two decimal places.

Note on precision: Depending on the calculator or rounding during intermediate steps, you may see a very close value (e.g., within a few dollars) to 120,807.20. The method and formula are correct for determining the end-of-period value of an ordinary annuity with monthly compounding.

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