Answer:
$459,352.46
Step-by-step explanation:
To calculate the equal annual contribution Alan needs to make in order to retire the debt on March 1, 2025, we can use the present value of an annuity formula.
Given:
Principal amount borrowed: $290,000
Interest rate compounded semiannually: 10% (or 0.10)
Repayment term: March 1, 2015, to March 1, 2025 (10 years)
Fund expected to earn: 8% per annum (or 0.08)
Contribution period: March 1, 2020, to March 1, 2024 (5 years)
Step 1: Calculate the total amount due on March 1, 2025 (including accrued interest):
The principal amount will grow with interest over the 10-year period, compounded semiannually.
P = $290,000
r = 10% / 2 = 5% per semiannual period (or 0.05)
n = 10 years * 2 = 20 semiannual periods
A = P * (1 + r)^n
A = $290,000 * (1 + 0.05)^20
A = $290,000 * 1.05^20
A = $290,000 * 1.352181
A = $391,948.29
The total amount due on March 1, 2025, including accrued interest, is approximately $391,948.29.
Step 2: Calculate the present value of the annuity required to retire the debt:
To find the equal annual contributions Alan needs to make, we need to calculate the present value of the total amount due on March 1, 2025.
PVA = A * (1 - (1 + r)^(-n)) / r
PVA = $391,948.29 * (1 - (1 + 0.08)^(-5)) / 0.08
PVA = $391,948.29 * (1 - 1.46933) / 0.08
PVA = $391,948.29 * (-0.46933) / 0.08
PVA = $391,948.29 * (-5.866625)
PVA ≈ -$2,296,762.28
The present value of the annuity required to retire the debt is approximately -$2,296,762.28. This negative value represents the total amount that needs to be contributed over the 5-year period to accumulate enough funds to retire the debt.
Step 3: Calculate the equal annual contribution:
Since the present value of the annuity is negative, we can multiply it by -1 to make it positive and determine the equal annual contribution Alan needs to make.
Annual Contribution = PVA * (-1) / n
Annual Contribution = $2,296,762.28 / 5
Annual Contribution ≈ $459,352.46
Alan needs to contribute approximately $459,352.46 each year for 5 years starting on March 1, 2020, in order to provide a fund sufficient to retire the debt of $290,000 plus accrued interest on March 1, 2025.