I have set aside $50,000 to invest on securities. I am considering
buying an index mutual fund which promises an annual rate of return of
8%. Currently, Treasury Bills promise to yield around 3% per year. If
I wish to make a portfolio rate of return of at least 6%, by investing
in both the index fund and the T-Bills, how much money should I invest
in each security?

Hello holla:

You should invest at least $30,000 in the index fund and at most
$20,000 in treasury bills to obtain a return of at least 6%.



Here is the mathematical solution:

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Total funds: $50,000
IntRate for Index Fund: 8% or 0.08
IntRate for TBills: 3% or 0.03



Let x represent the fraction of monies that are invested in the
IndexFund
Let y represent the fraction of monies that are invested in the TBills

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Eqn1: 50000(0.08)x + 50000(0.03)y = 50000(0.06)
Eqn1: 4000x + 1500y = 3000

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Eqn2: x + y = 1
Eqn2: 1500x + 1500y = 1500

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Subtract Eqn2 from Eqn1
4000x + 1500y - (1500x + 1500y) = 3000 - 1500
4000x + 1500y - 1500x - 1500y = 1500
2500x = 1500
x = 1500/2500
x = 15/25
x = 3/5
x = 0.6 (let this be Eqn3)

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Substitute Eqn3 into Eqn2
x + y = 1
(0.6) + y = 1
y = 1 - 0.6
y = 0.4


Therefore, the fraction of monies that you should invest in the
IndexFund is 0.6 and the fraction of monies that you should invest in
the TBills is 0.4.

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To calculate the actual amount of money:
IndexFund = (total funds)(x)
IndexFund = ($50,000)(0.6)
IndexFund = $30,000

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TBills = (total funds)(y)
TBills = ($50,000)(0.4)
TBills = $20,000

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Therefore, for a return of 6%, $30,000 should be invested in the index
mutual fund and $20,000 should be invested in the treasury bills.



Note that these amounts are for an exact return of 6%. If you want a
greater return, then you should invest more in the index fund and
invest the remaining monies in treasury bills.

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I hope this is an adequate mathematic solution.

If you are investing for more than a year, the answer given does not
account for the compounding effect.  The question does not say how
many years you are invested for which is a necessary piece of
information.  For a complete answer you need to state an assumption as
to the number of years invested.  Lets assume 20 years. $30,000 at 8%
per year grows to $30,000 x (1.08 to the power of 20) which is
$139,828.71. $20,000 at 3% per year grows to $20,000 x (1.03 to the
power of 20) which is $36,122.22.  The sum of $139,828.71 and
$36,122.22 is $175,950.94. The growth rate which goes from $50,000 to
$175,950.94 in 20 years is 6.4930043 per cent per year which you
should be interested to see is more than 6%.  Check: $50,000 x
(1.064930043 to the power of 20) equals $175,950.94.