A Teachers Union is to pay their retired teachers a retirement income of $60,000 each for a period of 30 years. The retired teachers are to receive a 20% increase every 10 years over the previous 10 years. If the teachers union is able to get 5% annual return on their retirement fund, compounded annually, how much does each teacher contribute in union fees annually to their retirement fund? Use 35 as an average age for all teachers who will contribute for 30 years into the fund.
Thanks for any help.
Thanks for the problem.
Let's retrieve information from the question.
n(total) = 60 years
The interest rate is already effective; i=5%
An increase g=20% from years 50-60
Annual total = ?
Let's set up our notation:
60,000(P/A, 5%, 19) = X (Years 30 through 60 is the retirement stage - I'm finding the present worth (at years 30) of the annual income right before the 20% increase)
60,000((1-(1.2/1.05)^10)/(0.05-0.2))(P/F, 5%, 20) = Y (calculating the present worth of the 20% increase 10 year span then converting that value to year 30)
X + Y = Z (adding both values at year 30 together)
Z(P/F, 5%, 30)(A/P, 5%, 30) (Now converting the new value at year 30 to year 0 then finding the annual worth over tge 30 year contribution period of the monetary amount)
Let's evaluate:
60,000(12.0853) = $725,118.00
60,000((1-(1.2/1.05)^10)/(0.05-0.2))(0.3769) = $422,307.26
$725,118.00 + $422,307.26 = $1,147,425.26
$1,147,425.26(0.2314)(0.06505) = $17,271.69
Thank you for your help. But the answer I have is $15,895 per teacher. Can you please re-check your calculations?
Calculate the payments that each teacher will receive for the 30-year period as follows:
For the first 10 years =$60,000
For the next 10 years=$60,000 x 1.20=$72,000
For the last 10 years=$72,000 x 1.20 =$86,400
Next, you have to find the PV of all these 3 payments and bring it forward to the beginning of their retirements. For that purpose, you have to use 2 different formulas for the 2nd and 3rd 10-year periods.
1- PV=P{[1 + R]^N - 1/[1 + R]^N} / R=PV OF $1 PER PERIOD.
2-PV=FV[1 + R]^-N=PV OF $1 IN THE FUTURE
Using formula (1) above, the PV of $60,000 payments=$463,304.10
Using formula (1) above, the PV of $72,000 =$555,964.91. But this amount is 10 years in the future. Using formula(2) above we find its PV to be=341,314.23.
Using formula (1) above, the PV of $86,400=$667,157.90. But this amount is 20 years in the future. Using formula (2) above, we find its PV to be=$251,444.80.
Now we add the 3 sums together=$463,304.10+$341,314.23+$251,444.80=$1,056,063.13
Now, this amount becomes the FV of each teacher's retirement fund. To find out what each teacher's annual contributions towards this amount will be, we have to use a third formula:
3- FV=P{[1 + R]^N - 1/ R}=FV OF $1 PER PERIOD.
Using this last formula and plugging in $1,056,063.13 as FV, and 5% for R and 30 years for N and solving for P, or annual payment required, we get =$15,895.27 - annual contributions per teacher.
Hmm how did you come to the conclusion that the income period is every 10 years? As well as the 20% increase will be for the previous 10 years and the next 10 years based off of the information: retirement income of $60,000 each for a period of 30 years. The retired teachers are to receive a 20% increase every 10 years over the previous 10 years.
I'm lead to believe the retirement income is an annual value where as you declared it a decade value. I don't see how you made this conclusion. Based on the second part of the information it seems as if the 20% increase was only for a preiod of 10 years which occured in the previous 10 years. Care to explain Guest?
This is very clear in the question! And is also very common in increasing/decreasing annuities. It is tantamount to compensating the teachers for the cost of living increase, or the inflation rate, which comes to about 1.2% per year over a 30-year period, which is in line with the current inflation rate in the U.S.
