**Table: Financial Data**

The risk measure indicates the relative
uncertainty associated with the stock in terms of it realising the projected
annual return: higher values indicate greater risk.

National’s top management has
stipulated the following investment guidelines

1.
The annual rate of return for the portfolio must be 9%

2.
No one stock can account for more than 50% of the total sterling
investment

They request you to find the investment decisions.

In this problem we need to find the
number of stocks A, B, C and D need to be bought with the provided
guidelines and with minimum risk.

Now this problem needs to be presented in
the mathematical form. This will involve three steps

(1)
Formulate an LP that minimises risk

(2)
Identifying the Decision Variables

The
decision that National faces is to decide how much of each stock to buy.

Let x_{1
}be the number of shares of stock A bought

x_{2
}be the number of shares of stock B bought

x_{3
}be the number of shares of stock C bought

x_{4
}be the number of shares of stock D bought

(3)
Determine the values of these four variables in order to minimise

National’s risk

In our example we wish to minimise risk.
We risk £0.10 on each pound invested in stock A, similarly for stock B the
risk is 0.07 per pound, for stock C it is 0.05, and for stock D the
corresponding risk is 0.08.

Thus if we buy x_{1}
shares of stock A, we have a risk exposure of
0.10*100*x_{1}
since each share costs £100. Similarly, if we buy
x_{2}
shares of stock B we risk 0.07*50*x_{2},
while for stocks C and D the risk measures are
0.05*80*x_{2}
and 0.10*40*x_{2}.
Therefore this leads to the following quantity that we wish to minimise

Risk =0.10*100 x_{1}
+ 0.07*50 x_{2}
+ 0.05*80 x_{1}
+ 0.10*40 x_{2}

The first constraint concerns the
budget. That is we can’t invest more than the money we have available. This
leads to the following constraint

100* x_{1}
+ 50* x_{2}
+ 80* x_{3}
+ 40* x_{4}
200000

The second constraint concerns the rate
of return of the portfolio and is as follows

100*0.12* x_{1}
+ 50*0.08* x_{2}
+ 80*0.06* x_{3}
+ 40*0.10* x_{4}
200000*.09

Finally, the cash investment in any one
stock cannot exceed 50% of the total investment

100*x_{1}
100000

50*x_{2}
100000

80*x_{3}
100000

40*x_{4}
100000 and** **x_{1}
0, x_{2}
0, x_{3}
0 , x_{4}
0

AMPL is mainly an algebraic language.
That means it follows the algebraic syntax used in the mathematical
representation of the problems. AMPL’s main keyword declarations are **set,
param**, **var** and **maximize**/**minimize**

Since AMPL deal with plain text files the
above problem can be rewrite as the following AMPL model as follows. Where x_{1},
x_{2}, x_{3 }and x_{4 }are replaced with the most
suitable variable names StockA, StockB, StockC and StockD.

Minimize

Risk
= 10*StockA + 3.5*StockB + 4*StockC + 4*StockD

Variables

StockA 1000

StockB 2000

StockC 1250

StockD 2500

Subject
to

100*StockA
+ 50*StockB + 80*StockC + 40*StockD
200000

12*StockA + 4*StockB + 4.8*StockC + 4*StockD
18000

In order to create a new workspace for
NIA’s problem create a new workspace with the following steps.

**
**

**Step 1**:
Choose **New Workspace** from the **File** menu.

**
File**

**Step 2**:
Write workspace name as NIA and choose your appropriate folder (C:\) by
clicking the ellipsis (…) button where you want to create your workspace.

Click OK to create the workspace at
your chosen folder.

###

**
Step 1**:
Having created a workspace, we now define a new project by selecting **
Insert New Project** from the **Project** menu.

**
Step 2**:
Enter the Project name as “**StepByStep1**”
and choose your preferred folder by clicking the ellipsis (…) button.

Check the
R
**Add templates** checkbox and write the **Model name** as “**StepByStep1.mod**”
and the **Data instance** as “**StepByStep1.dat**”.

Clicking OK will create a new
project with the model and data template files within the created workspace.

###

###

###

**
**

Double clicking on the model file
will open the model template file. The lines with # at the beginning are
comment lines. The AMPL key words will be in blue and the numbers in red.

**Step 1**:
Now write your AMPL model in this window.

**Step 2**: To
check the syntax of your model choose Build Model menu from Build menu.

**Build Menu**

If any syntax errors occurred then the appropriate error
messages will be displayed in the Console Window. In the above model
displays the following syntax error.

** **

**Step 3**:
Double click on the error line (line 30) will display the following screen.

**Step
4: **The line has two errors.

1.
Risk = should be replaced by Risk :

2.
Semicolon is missing at the end of line.

**Step
5:** Make these corrections and compiling it
again will show the following.

**Step 1**: In
order to solve the model you need to select the solver. By default you will
receive FortMP solver with your AMPL studio distribution. FortMP is a
powerful solver and capable to handle this simple problem.

**Solver**

** **

**
Step 2**: Now you can
run this problem by choosing the **Solve Problem** menu from the **Build**
Menu.

Immediately the problem will be
solved and the results will be displayed in the **Editing Area**.

X

##

The following
is the investment problem exploiting structure.