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Math  /  Algebra

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An account paying 4.6%4.6 \% interest compounded quarterly has a balance of $506,732.32\$ 506,732.32. Determine the amount that can be withdrawn quarterly from the account for 20 years, assuming ordinary annuity. a. $9,722.36\$ 9,722.36 b. $6,334.15\$ 6,334.15 c. $23,965.92\$ 23,965.92 d. $7,366.99\$ 7,366.99
Please select the best answer from the choices provided

Studdy Solution

STEP 1

1. The interest rate is 4.6% 4.6\% compounded quarterly.
2. The account balance is $506,732.32 \$506,732.32 .
3. Withdrawals are made quarterly for 20 years.
4. The problem assumes an ordinary annuity.

STEP 2

1. Determine the quarterly interest rate.
2. Calculate the total number of withdrawal periods.
3. Use the formula for the present value of an ordinary annuity to find the quarterly withdrawal amount.
4. Compare the calculated withdrawal amount with the given options.

STEP 3

The annual interest rate is 4.6% 4.6\% . Since the interest is compounded quarterly, divide the annual rate by 4 to find the quarterly interest rate:
Quarterly interest rate=4.6%4=1.15%\text{Quarterly interest rate} = \frac{4.6\%}{4} = 1.15\%
Convert the percentage to a decimal for calculations:
Quarterly interest rate as a decimal=1.15100=0.0115\text{Quarterly interest rate as a decimal} = \frac{1.15}{100} = 0.0115

STEP 4

Calculate the total number of withdrawal periods over 20 years, with quarterly withdrawals:
Total periods=20×4=80\text{Total periods} = 20 \times 4 = 80

STEP 5

Use the present value of an ordinary annuity formula:
PV=W×(1−(1+r)−nr)PV = W \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
Where: - PV PV is the present value (\$506,732.32), - \( W \) is the quarterly withdrawal amount, - \( r \) is the quarterly interest rate (0.0115), - \( n \) is the total number of periods (80).
Rearrange the formula to solve for W W :
W=PV×r1−(1+r)−nW = \frac{PV \times r}{1 - (1 + r)^{-n}}
Substitute the known values:
W=506,732.32×0.01151−(1+0.0115)−80W = \frac{506,732.32 \times 0.0115}{1 - (1 + 0.0115)^{-80}}
Calculate W W :
W≈5,827.421−(1.0115)−80W \approx \frac{5,827.42}{1 - (1.0115)^{-80}}
Calculate (1.0115)−80 (1.0115)^{-80} and complete the calculation:
(1.0115)−80≈0.3640(1.0115)^{-80} \approx 0.3640
W≈5,827.421−0.3640≈5,827.420.6360≈9,165.02W \approx \frac{5,827.42}{1 - 0.3640} \approx \frac{5,827.42}{0.6360} \approx 9,165.02

STEP 6

Compare the calculated withdrawal amount 9,165.02 9,165.02 with the given options:
a. $9,722.36 \$ 9,722.36 b. $6,334.15 \$ 6,334.15 c. $23,965.92 \$ 23,965.92 d. $7,366.99 \$ 7,366.99
The closest option to our calculated amount is:
a. $9,722.36 \$ 9,722.36
The best answer is:
$9,722.36 \boxed{\$ 9,722.36}

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