To pay for your child's education, you wish to have accumulated $58705 at the end of 12years. To do this, you plan on depositing an equal amount into the bank at the beginning of each month. a) If the bank pays 5.5% p.a. compounded monthly, how much must you deposit monthly to reach your goal (round to 2 decimal places)?

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BlackTom AI · Accepted Answer

We begin by identifying the type of annuity and the interest rate conversion that matches the problem setup. The deposits are made at the beginning of each month (an annuity due), and interest is compounded monthly.

Step 1: Convert the annual nominal rate to a monthly rate. The monthly rate i is 5.5% per year compounded monthly, so i = 0.055 / 12.

Step 2: Determine the total number of deposits. Since deposits are made monthly for 12 years, the number of periods n is 12 × 12 = 144 months.

Step 3: Recall the formula for the future value of an annuity due. The future value after n periods of an annuity due with payment P per period at monthly rate i is:
FV = P × [((1 + i)^n − 1) / i] × (1 + i).
Here, FV is the target amount, which is 58,705.

Step 4: Solve for the monthly deposit P. Rearranging the formula gives:
P = FV / { [((1 + i)^n − 1) / i] × (1 + i) }.

Step 5: Plug in the known values. i = 0.055 / 12, n = 144, and FV = 58,705. You would compute (1 + i)^n first, then compute the bracketed term ((1 + i)^n − 1) / i, then multiply by (1 + i), and finally divide 58,705 by that product to obtain P.

Step 6: Numerical evaluation notes. The calculation involves handling a fairly large exponent ((1 + i)^n with n = 144) and precise division. Depending on rounding at intermediate steps, you may obtain results that differ slightly when carried to two decimal places. In practice, you would use a financial calculator or software (or careful spreadsheet functions) to carry out:
- i = 0.055 / 12
- n = 144
- factor = ((1 + i)^n − 1) / i
- P = 58,705 / (factor × (1 + i))

Step 7: Interpret the result. The computed P is the monthly amount to deposit at the beginning of each month to reach 58,705 after 12 years, given the stated monthly compounding rate. If you compare this P to nearby values, you would observe how changes in i or n affect the required periodic payment: increasing i or increasing n generally lowers the required P, while decreasing i or decreasing n raises P.

Step 8: Edge considerations. If you accidentally treated the payments as end-of-month (ordinary annuity) instead of at the beginning (annuity due), you would omit the extra multiplication by (1 + i) and obtain a slightly larger or smaller P depending on the direction of the misclassification. Always verify whether the problem specifies deposits at the beginning or end of each period.

Step 9: Summary. The key workflow is to convert the rate to a monthly rate, compute the future-value annuity factor for n = 144, adjust for payments at the beginning of each month by multiplying by (1 + i), and then solve for P by dividing the target amount by this total factor. The exact two-decimal answer should be obtained from a precise calculation using a calculator or software.}

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