Question

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?

Answer 1

Answer is explained below in the explanation section.

Step-by-step 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:

`X_(1)` = Total Amount invested in Type A.

`X_(2)` = Total Amount invested in Type B.

`X_(3)` = Total Amount invested in Type C.

So, the Objective Function will be:

Objective function:

Max Z = 0.08
`X_(1)` + 0.10
`X_(2)` + 0.14
`X_(3)`

And the Constraints will be:

1. Total Amount Variable:

`X_(1)` +
`X_(2)` +
`X_(3)`
`\leq` 250000

2. Total Risk is no greater than 0.05:

0
`X_(1)` + 0.06
`X_(2)` + 0.10
`X_(3)`
`\leq` 0.05

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

`X_(1)`
`\geq` 0.25 (
`X_(1)` +
`X_(2)` +
`X_(3)` )

0.75
`X_(1)` - 0.25
`X_(2)` - 0.25
`X_(3)`
`\geq` 0

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

`X_(2)`
`\geq` 0.25 (
`X_(1)` +
`X_(2)` +
`X_(3)` )

-0.25
`X_(1)` + 0.75
`X_(2)` - 0.25
`X_(3)`
`\geq` 0

5. And the non- negativity constraints are:

`X_(1)`,
`X_(2)`, and
`X_(3)`
`\geq` 0