Hi there,
I am having some trouble solving this problem, can you give some guidance as to the solution.
What is the amount of 10 equal annual deposits that can provide five annual withdrawals, when a first withdrawal of $1000 is made at the end of year 11, and subsequent withdrawals increase at the rate of 6% per year over the previous year's, if
(a)The interest rate is 8%, compounded annually?
(b)The interest rate is 6%, compounded annually?
An excel spreadsheet is very helpful for these types of problems.
Let's first assume that both deposits and withdrawls are made at the end of the year. (Deposits at the beginning of the year slightly change the answer, but not the methodology).
So B1 (balance at the end of year 1) = 1000.
B2 = 1000 + B1*(1.08) = B1*(1+1.08)
B3 = B1*(1+1.08+1.08^)
so
B10 = B1*(sum(1.08)^(n-1) n goes from 1 to 10
find B10. Now for the withdrawals.
B11 = B10*(1.08) - W
B12 = B11*(1.08) - W*1.06
= B10*(1.08)^2 - W*(1.08) - W*(1.06)
B13 = B10*(1.08)^3 - W*(1.08)^2 - W*(1.06)*(1.08) - W*(1.06)
so
B15 = B10*(1.08)^5 - W*[(1.08)^4 + (1.06)*(1.08)^3 + (1.06)^2*(1.08)^2 + (1.06)^3*(1.08) + 1.06^4]
solve for W such that B15=0.
You can do this with algebra or you could use Excel and plug in values and iterate until you find the right solution.
Good luck.
Of course! I'd be happy to help you solve this problem.
To find the amount of 10 equal annual deposits that can provide five annual withdrawals, we can break the problem into a few steps. Let's begin by understanding the problem and organizing the information given.
We have two scenarios to consider: one with an interest rate of 8% and another with an interest rate of 6%. We'll solve them one by one, starting with scenario (a) where the interest rate is 8%, compounded annually.
(a) With an interest rate of 8%, compounded annually:
Step 1: Determine the withdrawal amounts
We are given that the first withdrawal amount is $1000 at the end of year 11. We also know that subsequent withdrawals increase by 6% per year over the previous year's amount.
Step 2: Find the future value of the withdrawals
Using the compound interest formula, we can find the future value of the withdrawals by applying the interest rate and compounding period. In this case, we are compounding annually.
Step 3: Find the amount of annual deposits
To find the amount of annual deposits, we need to calculate how much we need to accumulate in order to make the withdrawals. We will use the present value annuity formula, taking into account the interest rate and the number of deposits.
Let's calculate the amount of annual deposits using these steps:
1. Determine the withdrawal amounts:
- Year 11: $1000
- Year 12: $1000 + (6% increase)
- Year 13: $1000 + (6% increase) + (6% increase on previous year's amount)
...
- Year 15: $1000 + (6% increase) + (6% increase on the previous year's amount) + (6% increase on the previous year's amount)
2. Find the future value of the withdrawals:
Using the compound interest formula (FV = PV * (1 + r)^n), where FV is the future value, PV is the present value, r is the interest rate, and n is the number of periods, calculate the future value of the withdrawals from years 11-15. The present value (PV) in this case is $1000, and we will use an interest rate of 8%. The number of periods (n) is equal to the number of years from 11-15.
3. Find the amount of annual deposits:
Using the present value annuity formula (PV = PMT * [1 - (1 + r)^-n] / r), where PV is the present value, PMT is the annual deposit, r is the interest rate, and n is the number of deposits. We want to solve for PMT in this case.
Now, let's move on to scenario (b) where the interest rate is 6%, compounded annually.
(b) With an interest rate of 6%, compounded annually:
Follow the same steps as in scenario (a), but this time, use an interest rate of 6% instead of 8% to find the amount of annual deposits.
By following these steps and performing the calculations, you should be able to find the amount of annual deposits that can provide the given withdrawals for each scenario. If you need further assistance or have any additional questions, feel free to ask!