Your daughter, Ramona, has just celebrated her 7th birthday. (Happy
Birthday Ramona!!)
In just 11 years she will be ready to begin her college education. The cost of a
4-year college education at a leading university is now $35,000/year, and is projected to
increase indefinitely at the annual rate of 6%. You receive dividends in the amount
of $500.00 (net) from a trust at the end of each quarter. You can invest these
funds in an account which pays 7% per annum, net, compounded monthly.
What additional funds should you invest in this account at the beginning of
each month in order to accumulate the funds necessary to fund Ramona's education when
she reaches 18?
You will need to determine the future financial equivalent of today's
$140,000 ($35,000 x 4) when Ramona
begins college in 11 years.
If education costs escalate 6% per year for 11 years, you will need $265,761.80
in nominal dollars at that time. This figure is the Future Value of $140,000,
inflated at the rate of 6% p.a. for 11 years.
Your trust fund furnishes $500.00 at the end of each quarter. These funds can
be re-invested in an account earning
7% per year, compounded monthly.
But there is a mismatch between the timing of the trust PMT and the compounding period of the account into which these funds will be placed..
Therefore you will need to determine the equivalent quarterly rate of a PMT which is compounded monthly.
Here's how:
First, solve for the Future Value of $1.00 (PV), compounded monthly at the
rate of 7% p.a. for 1 year (12 periods).
This result is 1.07229.
Now, with 1.07229 in FV, convert to a quarterly determination of
i by changing n = 4.
Solve for i, the equivalent quarterly rate.
The result, 1.76023..., is the quarterly rate which will result in
the same annual effective rate of 7.229%.
(Prove this to yourself by solving for (pressing) FV. Result should be 1.07229...)
Use this quarterly rate (1.076023...), to determine the Future Value of the $500 quarterly payments (reset the calculator to EOP mode ), over 44 quarters. Set PV = 0
The result is $32,806.54..., the amount to which the quarterly EOP PMTs will grow over 11 years.
Therefore the remaining amount to be accumulated is $232,955.26 ($265,761.80 - 32,806.56) by means of an additional beginning-of-the-month PMT.
Set calculator to BOP and
n = 11*12 = 132;
i = 7/12 = 0.5833... ;
PV = 0;
FV = $232,955.26
Solve for PMT.
Answer = $1,169.78
Therefore, by investing the proceeds of the trust ($500 per quarter) in an account paying 7% p.a., compounded monthly, and to this account adding an amount equal to $1169,78 at the beginning of each month, the client will accrue $265,761.80, the sum needed to fund a four-year education account in 11 years.
This is not a situation calling for the use of the Inflation Adjusted Rate (IAR).
The IAR applied to the determination of the future cost of tuition will render a FV which
is expressed in constant (today's) dollars.
But the school bill must be paid in nominal dollars at that time.
It is this future amount which must be accumulated by monthly (BOP) investment sums
earning a 7% yield, compounded monthly.
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