You wish to retire in 12 years and currently have $50,000 in a savings account yielding 5 percent annually and $100,000 in quality "blue chip" stocks yielding 10 percent. If you expect to add $30,000 at the end of each year to your stock portfolios, how much will you have in your retirement fund when you retire? What rate of return must you earn on your retirement funds if you want to withdraw $102,000 per year for the next 15 years after retiring?) .
first let the 50,000 and the 100,000 ride for the 12 years.
Amount = 50000(1.05)^12 + 100000(1.10)^12
= 403,635.65
PLus our annuity of 30,000 for 12 years at 10%
= 30000(1.1^12 - 1)/.1
=641,528.51
for a total of $1,045,164.17
Last part:
let the rate be i
1,045,164.17 = 102,000 (1 - (1+i)^-15)/i
10.24670752 i = 1 - (1+i)^-15
that is going to be hard to solve, and will need something like Newton's Method,
I will "cheat" and use Wolfram
http://www.wolframalpha.com/input/?i=10.24670752x+%3D+1+-+%281%2Bx%29%5E-15
(I had to switch to x, since Wolfram reads i as √-1)
x = .0519134
So you will need a return of 5.19% per annum
check:
102000(1 - 1.0519134^-15)/.0519134
= 1,045,164.20
(how about that? Out by 3 cents in over 1 Million dollars !!!!)
To calculate the retirement fund at the end of 12 years, we need to consider the contributions to savings account, contributions to the stock portfolio, and the growth of both.
1. Contributions to the savings account:
You currently have $50,000 in the savings account, which yields a 5% annual return. After 12 years, with no additional contributions, the amount in the savings account would be:
$50,000 * (1 + 0.05)^12 = $82,032.42
2. Contributions to the stock portfolio:
You currently have $100,000 in the stock portfolio, which yields a 10% annual return. Additionally, you plan to contribute $30,000 at the end of each year. We can use the future value of an annuity formula to calculate the amount in the stock portfolio after 12 years.
FV = PMT * ((1 + r)^n -1) / r
Where:
FV = Future Value
PMT = Payment (annual contribution)
r = Interest rate
n = Number of periods (years)
FV = $30,000 * ((1 + 0.10)^12 - 1) / 0.10 = $659,813.88
3. Total retirement fund:
Adding the amounts from both the savings account and the stock portfolio, we get:
Retirement Fund = $82,032.42 + $659,813.88 = $741,846.30
So, you would have approximately $741,846.30 in your retirement fund when you retire.
Now, let's calculate the rate of return you need to earn on your retirement funds if you want to withdraw $102,000 per year for the next 15 years after retiring.
Using the present value of an annuity formula, we can find the rate of return.
PV = PMT * ((1 - (1 + r)^-n) / r)
Where:
PV = Present Value
PMT = Payment (withdrawal per year)
r = Interest rate
n = Number of periods (years)
PV = $102,000 * ((1 - (1 + r)^-15) / r)
Since we want to find the interest rate, we need to solve for r. However, this equation is not easy to solve algebraically. We can use iterative techniques or financial calculators/tools to find the rate.
Approximately, the rate of return you need to earn on your retirement funds to withdraw $102,000 per year for the next 15 years would be around 4.02%.
Note: The calculations provided are based on the information given and assumptions made. It is always recommended to consult with a financial advisor for personalized advice.
To determine how much you will have in your retirement fund after 12 years, we need to calculate the future value of your savings account and stocks.
1. Future Value of Savings Account:
In the savings account, you currently have $50,000. This will earn a 5% annual return for 12 years.
FV = PV * (1 + r)^n
FV = $50,000 * (1 + 0.05)^12
FV = $50,000 * 1.795853
FV = $89,792.65
2. Future Value of Stocks:
In the stocks, you currently have $100,000. You will add $30,000 at the end of each year for 12 years. The stocks yield a 10% annual return.
To calculate the future value, we can use the formula for the future value of an annuity:
FV = PMT * ((1 + r)^n - 1) / r
FV = $30,000 * ((1 + 0.10)^12 - 1) / 0.10
FV = $30,000 * (3.138428 - 1) / 0.10
FV = $30,000 * 2.138428 / 0.10
FV = $641,528.40
3. Total Retirement Fund:
The total retirement fund will be the sum of the future values of the savings account and stocks:
Total Retirement Fund = FV of Savings Account + FV of Stocks
Total Retirement Fund = $89,792.65 + $641,528.40
Total Retirement Fund = $731,321.05
To determine the rate of return on your retirement funds needed to withdraw $102,000 per year for the next 15 years after retiring, we can use the formula for the future value of an ordinary annuity:
FV = PMT * ((1 + r)^n - 1) / r
We need to solve for r, the rate of return. The future value (FV) will be $102,000, the annual payment (PMT) you wish to withdraw, n will be 15 years, and we'll rearrange the formula to solve for r:
r = (FV / PMT - 1)^(1/n) - 1
r = ($102,000 / $102,000 - 1)^(1/15) - 1
r = 1^(1/15) - 1
r = 1 - 1
r = 0
According to the calculations, you will have a total retirement fund of $731,321.05. To withdraw $102,000 per year for the next 15 years, you will not need to earn any rate of return on your retirement funds since you have sufficient funds to cover your withdrawals without any extra growth.