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Math  /  Algebra

QuestionIr. Alaa borrows $5,000\$ 5,000 from a bank at 8 percent annually compounded terest to be repaid in five annual installments. repare an amortization schedule

Studdy Solution

STEP 1

1. The loan amount is \$5,000.
2. The annual interest rate is 8%.
3. The loan is to be repaid in 5 equal annual installments.
4. The interest is compounded annually.

STEP 2

1. Calculate the annual installment amount.
2. Create the amortization schedule.
3. Detail each year's payment breakdown into interest and principal.

STEP 3

To calculate the annual installment amount, we use the formula for an annuity:
A=P⋅r⋅(1+r)n(1+r)n−1A = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
where: - A A is the annual installment, - P=5000 P = 5000 is the principal, - r=0.08 r = 0.08 is the annual interest rate, - n=5 n = 5 is the number of installments.
Substitute the values into the formula:
A=5000⋅0.08⋅(1+0.08)5(1+0.08)5−1A = \frac{5000 \cdot 0.08 \cdot (1 + 0.08)^5}{(1 + 0.08)^5 - 1}
Calculate (1+0.08)5 (1 + 0.08)^5 :
(1.08)5≈1.46933(1.08)^5 \approx 1.46933
Substitute back into the formula:
A=5000⋅0.08⋅1.469331.46933−1A = \frac{5000 \cdot 0.08 \cdot 1.46933}{1.46933 - 1}
A=5000⋅0.08⋅1.469330.46933A = \frac{5000 \cdot 0.08 \cdot 1.46933}{0.46933}
A≈587.7320.46933≈1252.28A \approx \frac{587.732}{0.46933} \approx 1252.28
The annual installment amount is approximately \$1,252.28.

STEP 4

Create the amortization schedule by calculating the interest and principal for each year.
Year 1: - Beginning Balance: \$5,000 - Interest = \( 5000 \times 0.08 = 400 \) - Principal = \( 1252.28 - 400 = 852.28 \) - Ending Balance = \( 5000 - 852.28 = 4147.72 \)
Year 2: - Beginning Balance: \$4,147.72 - Interest = \( 4147.72 \times 0.08 = 331.82 \) - Principal = \( 1252.28 - 331.82 = 920.46 \) - Ending Balance = \( 4147.72 - 920.46 = 3227.26 \)
Year 3: - Beginning Balance: \$3,227.26 - Interest = \( 3227.26 \times 0.08 = 258.18 \) - Principal = \( 1252.28 - 258.18 = 994.10 \) - Ending Balance = \( 3227.26 - 994.10 = 2233.16 \)
Year 4: - Beginning Balance: \$2,233.16 - Interest = \( 2233.16 \times 0.08 = 178.65 \) - Principal = \( 1252.28 - 178.65 = 1073.63 \) - Ending Balance = \( 2233.16 - 1073.63 = 1159.53 \)
Year 5: - Beginning Balance: \$1,159.53 - Interest = \( 1159.53 \times 0.08 = 92.76 \) - Principal = \( 1252.28 - 92.76 = 1159.52 \) - Ending Balance = \( 1159.53 - 1159.52 = 0.01 \) (rounding error adjustment)

STEP 5

Detail each year's payment breakdown into interest and principal as shown in the amortization schedule above.
The amortization schedule is complete, and the annual installment is approximately \$1,252.28.

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