Question

Beginning three months from now, you want to be able to withdraw $3,000 each quarter from your bank account to cover college expenses over the next four years. If the account pays .57 percent interest per quarter, how much do you need to have in your bank account today to meet your expense needs over the next four years? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Present value_________ .

Answer 1

To cover $3,000 in expenses each quarter for four years with a 0.57% quarterly interest rate, you would need approximately $42,947.47 in your bank account today. This is calculated using the present value of an annuity formula.

Step-by-step explanation:

To determine how much you need to have in your bank account today to meet your expense needs over the next four years with withdrawals of $3,000 each quarter and a quarterly interest rate of 0.57%, we can use the formula for the present value of an annuity. Each withdrawal is a fixed payment made at regular intervals, which is precisely what an annuity represents. To calculate this, we will use the present value of an annuity formula:

PV = P * [(1 - (1 + r)^(-n)) / r]

Where:

• PV is the present value or initial amount required
• P is the payment each period ($3,000)
• r is the interest rate per period (0.57% or 0.0057)
• n is the total number of payments (4 years * 4 quarters/year = 16 payments)

Let's calculate it:

PV = 3000 * [(1 - (1 + 0.0057)^(-16)) / 0.0057]

We can use a calculator to find that PV equals approximately $42,947.47 when calculated with intermediate steps not rounded. Therefore, you would need to have $42,947.47 in your account today to withdraw $3,000 each quarter for four years at a 0.57% quarterly interest rate.

Answer 2

You need to have $45,751.83 in your bank account today to meet your expense needs over the next four years.

How to solve

To calculate the present value of the future cash flows without using code, you can follow these steps:

Determine the future value of each quarterly withdrawal.

For each quarterly withdrawal, calculate its future value at the end of the four years. Use the formula for compound interest:

Future Value = Present Value * (1 + Interest Rate)^Number of Periods

Since the interest rate is given as a quarterly rate, you'll need to adjust the number of periods accordingly. In this case, the total number of periods is 16 quarters (4 years * 4 quarters/year).

For the first quarterly withdrawal, the future value would be:

Future Value = $3,000 * (1 + 0.0057)^16 = $4,575.18

Repeat this calculation for each of the 16 quarterly withdrawals to determine their respective future values.

Calculate the total future value of all quarterly withdrawals.

Sum the future values of all 16 quarterly withdrawals to find the total future value needed at the end of the four years.

Total Future Value = $4,575.18 (for the first withdrawal) + $4,669.93 (for the second withdrawal) + ... + $5,813.54 (for the sixteenth withdrawal)

Discount the total future value to its present value.

Use the formula for present value to discount the total future value back to its present value (the amount you need in your bank account today).

Present Value = Total Future Value / (1 + Interest Rate)^Number of Periods

Using the given interest rate and the total number of periods of 16 quarters, you'll get:

Present Value = $76,098.05 / (1 + 0.0057)^16 = $45,751.83

Therefore, you need to have $45,751.83 in your bank account today to meet your expense needs over the next four years.