Question 1209539: Gabriella and Mario plan to send their son to university. To pay for this they will contribute 9 equal yearly payments to an account bearing interest at the APR of 3.4%, compounded annually. Five years after their last contribution, they will begin the first of five, yearly, withdrawals of $35,600 to pay the university's bills. How large must their yearly contributions be?
Found 2 solutions by ElectricPavlov, ikleyn:
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**1. Calculate the present value of the withdrawals:**
We need to find the present value of the series of withdrawals at the *start* of the withdrawal period. This is because the withdrawals are made over time, and we need to discount them back to a single point in time to determine how much money is needed at the start of the withdrawals.
We can treat the withdrawals as an ordinary annuity. The formula for the present value of an ordinary annuity is:
PV = PMT * [1 - (1 + r)^-n] / r
Where:
* PV = Present Value
* PMT = Periodic Payment (withdrawal amount) = $35,600
* r = Interest rate per period = 3.4% = 0.034
* n = Number of periods (withdrawals) = 5
PV = $35,600 * [1 - (1 + 0.034)^-5] / 0.034
PV = $35,600 * [1 - 0.8562] / 0.034
PV ≈ $35,600 * 4.2221
PV ≈ $150,200.32
**2. Discount the present value back to the start of the contributions:**
Gabriella and Mario make their last contribution 5 years *before* the withdrawals begin. We need to discount the present value of the withdrawals back another 5 years to the end of the contributions.
PV_start = PV / (1 + r)^t
Where:
* PV_start = The present value at the start of the contributions
* PV = Present value of withdrawals = $150,200.32
* r = Interest rate = 0.034
* t = Number of years = 5
PV_start = $150,200.32 / (1 + 0.034)^5
PV_start ≈ $150,200.32 / 1.1803
PV_start ≈ $127,248.04
**3. Calculate the yearly contribution:**
Now we know how much money they need to have saved at the *end* of their 9 years of contributions. Their contributions are also an ordinary annuity so we use the future value of an ordinary annuity formula to find the yearly contribution.
FV = PMT * [(1 + r)^n - 1] / r
Where:
* FV = Future Value = $127,248.04
* PMT = Periodic Payment (yearly contribution) - what we are solving for
* r = Interest rate = 0.034
* n = Number of periods (contributions) = 9
Rearranging the formula to solve for PMT:
PMT = FV * r / [(1 + r)^n - 1]
PMT = $127,248.04 * 0.034 / [(1 + 0.034)^9 - 1]
PMT ≈ $4,326.43 / [1.3669 -1]
PMT ≈ $4,326.43/0.3669
PMT ≈ $11,791.80
**Answer:**
Gabriella and Mario must make yearly contributions of approximately $11,791.80.
Answer by ikleyn(52864)  (Show Source):
You can put this solution on YOUR website! .
In the post by @ElectricPavlov, the logic of the solution is correct.
The formulas that he uses for calculations are correct, too.
But all his output numbers, that are the results of his calculations, are WRONG.
His final answer is wrong, too.
It is because they systematically use inadequate calculator or inadequate calculation procedure
for their calculations, which does not provide the necessary precision.
So, if you are looking to get a precise answer for your problems in Finance,
which would be correct at the reference level of accuracy for your answer book,
especially for annuity problems, then @ElectricPavlov is a bad source for such a goal.
As I look at their performance, I clearly see that they do not understand, at all,
what approximate calculations are and which requirements such calculations must satisfy.
So, their level of understanding approximate calculations corresponds to a 5th grade school student,
who makes such calculations for the first time in his life and who never took/got lessons
from an experienced mentor on the subject.
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