1. $115,779 earned, $95,779 interest.
2. Deposit $60,589, interest reduces goal by $954,428.
1. Annuity and Interest for $2000 quarterly deposit:
- Formula: A = p * ((1 + r/n)^(n*t) - 1) / (1 + r/n)
- A: Future value
- p: Periodic deposit ($2000)
- r: Annual interest rate (5.75% / 4 = 1.4375% per quarter)
- n: Number of compounding periods per year (4 quarters)
- t: Time in years (8 years)
- Calculation:
- A = $2000 * ((1 + 0.014375)^(4*8) - 1) / (1 + 0.014375) ≈ $115,779.04
Therefore, the annuity is $115,779.04, and the total interest earned is $95,779.04 (A - p*n*t).
2. Periodic Deposit and Financial Goal Contribution:
- Goal: $15,000
- Time:4 years
- Interest rate:5.5% compounded quarterly (n = 4)
- Formula:** A = p * ((1 + r/n)^(n*t) - 1) / (1 + r/n)
- Solve for p (periodic deposit)
- Calculation:
- p = A / ((1 + 0.01375)^(4*4) - 1) / (1 + 0.01375) ≈ $60,589.31
Therefore, the required periodic deposit is $60,589.31.
- Deposits contribution: p * n * t = $60,589.31 * 4 * 4 = $969,428.92
- Interest contribution: A - p * n * t = $15,000 - $969,428.92 = -$954,428.92 (negative value indicates interest reduces the goal)
So, $969,428.92 comes from deposits, and the interest actually reduces the goal by $954,428.92.