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An investor has up to $250,000 to invest in three types of in-vestments. Type A pays 8% annually and has a risk factor of0. Type B pays 10% annually and has a risk factor of 0.06.Type C pays 14% annually and has a risk factor of 0.10. Tohave a well-balanced portfolio, the investor imposes the fol-lowing conditions. The average risk factor should be nogreater than 0.05. Moreover, at least one-fourth of the totalportfolio is to be allocated to Type A investments and at leastone-fourth of the portfolio is to be allocated to Type B invest-ments. How much should be allocated to each type of invest-ment to obtain a maximum return?

Sagot :

Answer:

Answer is explained below in the explanation section.

Explanation:

Solution:

An investor has up to $250,000 to invest in three types of investment.

Type A pays 8% annually and has risk factor of 0.

Type B pays 10% annually and has risk factor of 0.06.

Type C pays 14% annually and has risk factor of 0.10.

So,

Decision Variables are:

[tex]X_{1}[/tex] = Total Amount invested in Type A.

[tex]X_{2}[/tex] = Total Amount invested in Type B.

[tex]X_{3}[/tex] =  Total Amount invested in Type C.

So, the Objective Function will be:

Objective function:

Max Z = 0.08[tex]X_{1}[/tex] + 0.10[tex]X_{2}[/tex]  + 0.14[tex]X_{3}[/tex]

And the Constraints will be:

1. Total Amount Variable:

[tex]X_{1}[/tex] + [tex]X_{2}[/tex]  + [tex]X_{3}[/tex]  [tex]\leq[/tex] 250000

2. Total Risk is no greater than 0.05:

0[tex]X_{1}[/tex]  + 0.06[tex]X_{2}[/tex]  + 0.10[tex]X_{3}[/tex] [tex]\leq[/tex] 0.05

3. At least one fourth of the total amount invested to be allocated to Type A investment.

[tex]X_{1}[/tex] [tex]\geq[/tex] 0.25 ( [tex]X_{1}[/tex] + [tex]X_{2}[/tex]  + [tex]X_{3}[/tex]  )

0.75[tex]X_{1}[/tex]  - 0.25[tex]X_{2}[/tex] - 0.25[tex]X_{3}[/tex] [tex]\geq[/tex] 0

4. At least one fourth of the total amount to be allocated to Type B investment.

[tex]X_{2}[/tex]  [tex]\geq[/tex] 0.25 ( [tex]X_{1}[/tex] + [tex]X_{2}[/tex]  + [tex]X_{3}[/tex]  )

-0.25[tex]X_{1}[/tex]  + 0.75[tex]X_{2}[/tex]  - 0.25[tex]X_{3}[/tex] [tex]\geq[/tex] 0

5. And the non- negativity constraints are:

[tex]X_{1}[/tex],[tex]X_{2}[/tex], and [tex]X_{3}[/tex]  [tex]\geq[/tex] 0

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