If deposit made at beginning of each year, then $4,686.87 per year is required.
If deposit made at end of each year, then $4,921.21 per year is required.
Assuming you can continue to get the 5% interest when you start withdrawing money, then this problem consists of two parts.
Part 1. How much money must you have in the account when you start withdrawing?
Part 2. How much must you deposit in order to reach the figure for part 1?
Part 1 is fairly easily calculated, just figure it out in reverse. For example
At the beginning of year 15, you have to have $30,000 in the account. Since you're getting 5% interest, that means that after you've made the withdraw at the beginning of year 14, you need to have $30,000/1.05 = $28,571.43 in the account. And since you'll be withdrawing $30,000 at the beginning of year 14, you need to have $28,571.43+$30,000 = $58,571.43 in the account at the beginning of year 14. Continuing this process in reverse (add 30000 to sum, divide by 1.05, repeat), you'll determine that your retirement fund has to have $326,959.23 in it when you make your first withdrawal of $30,000.
Now you need to figure out how much you need to invest each year for 30 years at an interest rate of 5% to get the sum of $326,959.23
Assuming that you make your contribution at the end of each year, then the formula for the worth of your investment is
V=C(((1 + r)Y - 1) / r)
where
C = Contribution
r = interest rate
Y = number of years
V = Value
Solving for C, gives
V=C(((1 + r)Y - 1) / r)
V/(((1 + r)Y - 1) / r)=C
Substituting known values into formula:
326959.23/(((1 + 0.05)^30 - 1) / 0.05)=C
326959.23/((1.05^30 - 1) / 0.05)=C
326959.23/((4.321942375 - 1) / 0.05)=C
326959.23/(3.321942375 / 0.05)=C
326959.23/66.4388475=C
4921.21=C
So if you invest $4,921.21 for 30 years at the end of each year, you'll have the required amount for your retirement fund.
But if you can make your investment at the beginning of each year, you'll have an extra interest period and the formula for that case is
V=C(((1 + r)^(Y + 1) - (1 + r)) / r)
Solving for C, gives
V/(((1 + r)^(Y + 1) - (1 + r)) / r)=C
Substituting known values
326959.23/(((1 +0.05)^(30 + 1) - (1 + 0.05)) / 0.05)=C
326959.23/((1.05^31 - 1.05) / 0.05)=C
326959.23/((4.538039494 - 1.05) / 0.05)=C
326959.23/(3.488039494 / 0.05)=C
326959.23/69.76078988=C
4686.87=C
So if you can make your investment at the beginning of each year for 30 years, then $4,686.87 will be enough to reach your retirement goal.
