Final answer:
To find the initial payment X for the 20-year annuity-immediate with payments growing by 5% annually, equate the annuity's present value to the present value of the remaining perpetuity after the 10th payment of $100 a year at a 5% interest rate. The present value of the perpetuity is $2000, and by solving the present value of the annuity using the growing annuity formula, we can find X.
Step-by-step explanation:
The student is asking to calculate the initial payment X for a 20-year annuity-immediate where each payment grows by 5% annually. The present value of this increasing annuity must be equal to the value of the remaining perpetuity after the 10th payment, where a perpetuity-immediate pays 100 per year, and the annual effective rate of interest is 5%. The present value of a perpetuity-immediate is calculated using the formula: PV = Payment / Interest rate. Therefore, immediately after the 10th payment, the present value of the remaining perpetuity is 100 / 0.05 = 2000.
Now we calculate the present value of the annuity. Each payment is 5% greater than the previous, so the payments form a geometric series. This series is valued using the formula for the present value of a growing annuity:PV = P * [(1 - (1 + g)^n) / (r - g)], where P is the first payment, g is the growth rate, r is the interest rate, and n is the number of payments.By setting the present value of the annuity equal to the present value of the remaining perpetuity (2000), we can solve for X, the first payment.