Final answer:
To find the annual effective interest rate i, we must calculate the present value of a perpetuity payment of 1500 starting in year 31 at 1% interest, then solve for i to equal this value with the series of 1000 deposits made annually for 30 years.
Step-by-step explanation:
The student is asking to find the annual effective interest rate i for a fund where deposits of 1000 are made at the beginning of each year for 30 years, and from the 31st year onwards, annual payments of 1500 are made indefinitely. The interest rate changes to 1% after the 30th year.
To solve this, we have to find the present value of the perpetuity of 1500 starting from year 31 and discount it back to year 0 at the effective interest rate of 1%. Then we can use this as an equivalent lump sum to find the rate i where the present value of the annual deposits of 1000 matches this amount.
First, let's calculate the present value of the perpetuity using the formula PV = P / r, where P = 1500 (annual payment) and r = 0.01 (interest rate from the 31st year). This gives us PV = 1500 / 0.01 = 150,000. We discount this amount back to the start by dividing by (1 + 0.01)30. The present value of the perpetuity at year 0 is 150,000 / 1.0130.
Now, let's find the annual effective interest rate i that equates this present value to the series of annual deposits. We can set up the equation:
1000 + 1000(1+i) + 1000(1+i)2 + … + 1000(1+i)29 = 150,000 / 1.0130.
This is an annuity immediate equation which we would solve to get the value of i. Due to the complexity of this equation, it would typically be solved using financial calculator or numerical methods such as iteration or software that can handle time value of money calculations.