1. Tyler Wallace and Byron Price
Professor Derek Bruff
Math 194
8 Dec 2009
Application Project
The Stock Market

2. In the 1950’s, Henry Markowitz introduced linear programming to Wall Street as
a scientific method to optimize returns on a stock portfolio (Fernando). Today, many
researchers, known as quantitative analysts, use similar methods to develop stock
portfolios while considering various constraints including the investor’s budget and the
risk involved in purchasing a corporation’s stock. Occasionally, investors also consider
CEO compensation and environmental factors in order to make a socially responsible
investment (one that does not contribute to inhumane treatment of employees or
degradation of the global environment). With these factors in mind, our goal is to
develop a five-stock portfolio that will maximize returns over the next 52-week period.
The main limitation of our analysis as an investment tool is a direct result of the
uncertainty of the stock market: information analyzed over the past year’s trends will not
necessarily indicate similar results in the following year. A more precise analysis would
certainly need to more closely follow trends and make an attempt to predict the future.
Therefore, our application will be running under the assumption that general market
trends will hold into the next year. However, were we to perform a more intensive,
“future predicting” analysis, we could include a number of other constraints. One could
establish a “power of the executive board constraint” by researching past decisions made
by the company and investigating the backgrounds of various officers. One could also
create a constraint that involves a company’s potential for growth: is the market saturated
in their industry or is there room for improvement? Each of these constraints develops a
more accurate and powerful investment analysis, however they also require many hours
of research. For this reason, we have chosen to limit ourselves to six basic constraints.

3. In order to satisfy the need for diversity in our stock portfolio, we are beginning
our analysis with five corporations from five of the major investment sectors: energy,
pharmaceuticals, restaurants, finance and technology. We have chosen Exxon Mobil
(XOM), Abbott Laboratories (ABT), Starbucks (SBUX), JPMorgan Chase (JPM) and
Microsoft (MSFT). These companies will be used in our first trial. However, we will
tease out underperforming companies by replacing them with a different company from
the same sector in hopes of concluding with the most diverse, profitable and socially
responsible portfolio. In order to establish constraint equations, we must specify methods
with which to measure risk, CEO compensation and environmental factors. Each of these
constraints depends on our objective function, which we have decided to depict as
follows:
Total Profit = x(52-week return of company x, pct.) + y(52-week return of company y,
pct.) + z(52-week return of company z) + … w … u
where x, y, z, w, and u represent the total amounts of money invested in companies x
(MSFT), y (JPM), z (XOM), w (SBUX) and u (ABT). With the values of x, y, z, w and u
defined as total amounts of money invested, we can set a budget on our investment at
10,000 USD with the following constraint: x + y + z + w + u = 10,000. In order to
maintain diversity within our portfolio, we have chosen to limit the percentage of the
initial 10,000 USD we invest in a particular company to 33.33%. This limit creates the
following diversity constraints: (x/10,000) ≤ 0.333, (y/10,000) ≤ 0.333, (z/10,000) ≤
0.333, (w/10,000) ≤ 0.333, and u ≤ 0.333. We also have to consider that none of our
values can be negative, since it is impossible to invest negative amounts of money. This
yields the following constraints: x ≥ 0, y ≥ 0, z ≥ 0, w ≥ 0 and u ≥ 0.

4. The other three constraints on our objective function (risk, CEO compensation
and environmental factors) are much more complicated and therefore require further
discussion. The risk involved with making investments is very difficult to quantify,
especially when considering corporations as well established as the five above. In the
financial world, researchers and investors usually look to a value called beta to quantify
“risk.” Essentially, beta describes the relationship between the returns of an individual
company, or an entire index, with respect to that of the entire financial market (“Beta
financial”). Therefore, beta is normalized to 1.0 for the returns of the stock market as a
whole, and then each company is given a value of beta above or below 1.0 to indicate,
basically, the fluctuation/volatility of their returns. A value of beta less than 1.0 indicates
that a company’s returns are fluctuating less than the stock market as a whole, which
means less risk is involved. A value of beta greater than 1.0 indicates that a company’s
returns are fluctuating more than the stock market as a whole, therefore more risk is
involved. This information allows us to create the following constraint:
1 ≥ (x/10,000)*(beta of company x) + (y/10,000)*(beta of company y) + … z … w … u
We have chosen to divide each of the variables by 10,000 USD (the total amount
invested) in order to establish percentages. Since we are multiplying the percentage
invested in a particular company by the beta of each company, the sum on the right side
is an average value of beta for our portfolio. We have chosen to set beta = 1.0 as our
upper limit to reduce the risk of our investment to the intrinsic risk involved with
gambling in the stock market.
For the CEO compensation constraint, we used a method similar to that used
while creating the beta constraint: we set an upper limit on the average CEO

5. compensation of our portfolio. This effectively weights companies with lower CEO
compensation over companies with higher CEO compensation (we are allowed to invest
more money into a company that does not overpay its chief executive officer).
Considering the nature of our five companies (all Fortune 500), the CEO compensations
are absurdly high, averaging 21,464,065 USD (“CEO …”). Therefore, if we were really
adamant about CEO overcompensation, we would probably choose to invest 0 dollars in
our five companies. However, we chose to establish a more reasonable constraint by
looking at the average CEO compensation of companies in our five sectors that belong to
the Dow Jones Industrial Average. This average was calculated to be a meager
20,260,090 USD (“CEO …”). With this information, our CEO compensation constraint
is defined as follows:
20,260,090 ≥ (x/10,000)*(CEO compensation of company x) + (y/10,000)*(CEO
compensation of company y) + z … w … u
For our final constraint, the environmental one, we chose to search the Internet for
a standardized rating system of various corporations. We found a particularly useful
rating system created by Green America Today, which established for each corporation a
grade from A to F based on their environmental impact and overall efforts to support a
healthy global ecosystem (“Responsible …”). With this rating system, we can establish
our environmental constraint as follows:
1.7 ≤ (x/10,000)*(Rating of company x) + (y/10,000)*(rating of company y) + z…w …u
Where we have chosen to institute a lower limit of 1.7 on the “environmental GPA” of
our portfolio. A rating of 1.7 corresponds to a C- grade using Vanderbilt’s system from

6. 0.0 to 4.0. Similar to the high CEO compensation constraint, a C- average does not
sound very promising for the environment. However, looking at the average rating for
our five companies of 1.34, setting the lower limit at 1.7 truly inhibits the investment we
can make in the more environmentally unfriendly corporations.
In order to execute our linear programming model with the previously described
objective function and constraints, we need data:
Company
Avg.
52-­‐
week
return
Beta
Environmental
Impact
Rating
(w/
assigned
GPA
numerical
value)
CEO
Compensation
X
Microsoft
(MSFT)
50.88% 0.96 D
=
1
$1,350,834
Y
JPMorgan Chase &
Co. (JPM)
25.16% 1.18 C
=
2
$35,764,557
Z
Exxon Mobil Corp
(XOM)
-3.07% 0.35 F
=
0
$32,211,079
W
Starbucks Corp.
(SBUX)
136.84% 1.36 C-­‐
=
1.7
$9,740,471
U
Abbott Laboratories
(ABT)
2.44% 0.27 C
=
2
$28,253,387
(Yahoo!)
This information yields the following input into Mathematica’s “Maximize” function:
In[67]:= Maximize [{.5088 x + .2516 y - .0307 z + 1.3684 w
+ .0244 u,
x + y + z + w + u == 10000 &&
.96 (x/10000) + 1.18 (y/10000) + .35 (z/10000) +
1.36 (w/10000) + .27 (u/10000) <= 1.0 &&
0 <= (x/10000) <= 1/3 && 0 <= (y/10000) <= 1/3 &&
0 <= (z/10000) <= 1/3 && 0 <= (w/10000) <= 1/3 &&
0 <= (u/10000) <= 1/3 &&
1 (x/10000) + 2 (y/10000) + 0 (z/10000) + 1.7 (w/10000)
+ 2 (u/10000) >= 1.7 &&
1350834(x/10000) + 35764557(y/10000) + 32211079(z/10000)
+ 9740471 (w/10000) + 28253387 (u/10000) <= 20260090.2},
{x, y, z, w, u}]
Out[67]= {6263.71, {x -> 2000., y -> 2512.82, z -> 0., w ->
3333.33, u -> 2153.85}}

7. Our results showed the highest performing portfolio consisting of investments of
2000 USD in MSFT, 2512.82 in JPM, no money in Exxon Mobil, 3333.33 in SBUX, and
2153.85 in ABT. Given these investments, our model predicts that we will profit 6263.71
over the next 52-week period.
Due to Exxon’s low performance in stock market returns and a failing grade in
environmental impact, our constraints kept Exxon out of our portfolio. In order to add
more diversity to our portfolio, and more importantly to improve our returns, we decided
to search for a different company from the energy sector that held more rigid
environmental standards and shows more promising returns. We looked again to Green
America website to find another oil company with a better environmental impact rating.
Of Sunoco and Hess, the two such companies with better environmental standards, the
Hess Corporation had far higher stock returns over the past year. Therefore, we revised
our portfolio to include the Hess Corporation as our investment in the energy sector.
The data for our second trial is as follows:
Variable
Company
Avg.
52-­‐
week
return
Beta
Environmental
Impact
Rating
(w/
assigned
GPA
numerical
value)
CEO
Compensation
X
Microsoft
(MSFT)
50.88% 0.96 D
=
1
$1,350,834
Y
JPMorgan Chase & Co.
(JPM)
25.16% 1.18 C
=
2
$35,764,557
Z
Hess Corporation
(HES)
26.11% 0.95 D-­‐
=
0.7
$26,334,067
W
Starbucks Corp.
(SBUX)
136.84% 1.36 C-­‐
=
1.7
$9,740,471
U
Abbott Laboratories
(ABT)
2.44% 0.27 C
=
2
$28,253,387
(Yahoo!)

8. Our objective function and constraints remain the same, with one minor
exception. To ensure that our portfolio includes an investment in the oil sector, we were
forced to reduce the required environmental impact rating of our portfolio from 1.7 to
1.5. The environmental impact constraint is now:
1 (x/10000) + 2 (y/10000) + 0.7 (z/10000) + 1.7 (w/10000) + 2 (u/10000) ≥ 1.5
Using the simplex method in Mathematica to maximize our objective function, our
results are as follows:
In[73]:= Maximize [{.5088 x + .2516 y + .2611 z + 1.3684 w
+ .0244 u,
x + y + z + w + u == 10000 &&
.96 (x/10000) + 1.18 (y/10000) + .95 (z/10000) +
1.36 (w/10000) + .27 (u/10000) <= 1.0 &&
0 <= (x/10000) < 1/3 && 0 <= (y/10000) <= 1/3 &&
0 <= (z/10000) <= 1/3 && 0 <= (w/10000) <= 1/3 &&
0 <= (u/10000) <= 1/3 &&
1 (x/10000) + 2 (y/10000) + 0.7 (z/10000) + 1.7
(w/10000) + 2 (u/10000) >= 1.5 &&
1350834(x/10000) + 35764557(y/10000) + 26334067(z/10000)
+ 9740471 (w/10000) + 28253387 (u/10000) <= 20260090.2},
{x, y, z, w, u}]
Out[73]= {6714.2, {x -> 3333.33, y -> 1118.62, z ->
512.821, w -> 3333.33, u -> 1701.89}}
The results from our second trial show increased performance from our portfolio.
After the change in oil companies and a decreased promise to uphold environmental
standards, our returns have increased 450.49 USD from our first trial to 6714.20. We are
also pleased to see that our portfolio is more diverse, as it contains investments in all five
of our chosen sectors. This current portfolio consists of investments of 3333.33 USD in
MSFT, 1118.62 in JPM, 512.82 in HES, 3333.33 in SBUX, and 1701.89 in ABT.
However, we decided that our portfolio was underachieving and could perform
better with the removal of the Hess Corporation. It appears that our environmental

9. standards are still restraining our investments in the energy sector, so we will choose
another, more environmentally friendly sector in its place. For this change, we decided to
make an educated guess as to which company would most positively affect our profits,
with the given constraints. We found that the retail sector shows promising figures in its
environmental ratings and its 52-week returns. Therefore, for our final trial, we replaced
the Hess Corporation with The Home Depot (HD).
Our data is now as follows:
Variable
Company
Avg.
52-­‐
week
return
Beta
Environmental
Impact
Rating
(w/
assigned
GPA
numerical
value)
CEO
Compensation
X
Microsoft
(MSFT)
50.88% 0.96 D
=
1
$1,350,834
Y
JPMorgan Chase & Co.
(JPM)
25.16% 1.18 C
=
2
$35,764,557
Z
Home Depot, Inc. (HD) 19.27% 0.63 B
=
3
$9,244,533
W
Starbucks Corp.
(SBUX)
136.84% 1.36 C-­‐
=
1.7
$9,740,471
U
Abbott Laboratories
(ABT)
2.44% 0.27 C
=
2
$28,253,387
(Yahoo!)
Our objective function and constraints remain the same for this trial.
Using Mathematica to solve:
In[109]:= Maximize [{.5088 x + .2516 y + .1927 z + 1.3684 w
+ .0244 u,
x + y + z + w + u == 10000 &&
.96 (x/10000) + 1.18 (y/10000) + .63 (z/10000) +
1.36 (w/10000) + .27 (u/10000) <= 1.0 &&
0 <= (x/10000) <= 1/3 && 0 <= (y/10000) <= 1/3 &&
0 <= (z/10000) <= 1/3 && 0 <= (w/10000) <= 1/3 &&
0 <= (u/10000) <= 1/3 &&
1 (x/10000) + 2 (y/10000) + 3 (z/10000) + 1.7 (w/10000)
+ 2 (u/10000) >= 1.5 &&
1350834(x/10000) + 35764557(y/10000) + 9244533(z/10000)
+ 9740471 (w/10000) + 28253387 (u/10000) <= 20260090.2}
, {x, y, z, w, u}]

10. Out[109]= {6917.52, {x -> 3333.33, y -> 303.03, z ->
3030.3, w -> 3333.33, u -> 0.}}
The results from our final trial improved our returns from the previous trial by
203.32 USD and improved upon our initial trial by 653.81. Although our portfolio is less
diverse than we desired, we are pleased by its predicted returns of 6917.52 USD or 69.2%
of our original investment. Our final portfolio calls for investments of 3333.33 USD in
MSFT, 303.03 in JPM, 3030.3 in HD, 3333.33 in SBUX, and no investment in ABT.
One secondary question that we had at the beginning of our analysis was: Will
one of our chosen constraint functions dominate the results? Comparing the input data to
our results, we see that the optimization of our portfolio is highly sensitive to the betas of
individual stocks. This indicates that we could require stricter environmental impact and
CEO compensation constraints in future analysis. The high sensitivity of our objective
function to the risk constraint also indicates a practical limitation of our method of
analysis. We performed a simple measure of this sensitivity by altering only one number
in the risk constraint from our final trial. By changing the beta for Starbucks from 1.36 to
1.46, a 3.7% change, our results surprisingly changed to the following: 0 USD in JPM
and ABT, and a full 3333.33 in MSFT, SBUX and HD. This at first seems very
surprising because our final trial already recommended a maximum investment in SBUX.
However, since SBUX yields such high returns, and therefore greatly influences our
objective function, it makes sense to maintain a maximum investment in SBUX. With
the increased beta, the simplex method must hold the average beta for the portfolio under
one, and we must invest more money into a company with a lower beta. For this reason,
money shifts from JPM (beta 1.18) to HD (beta 0.63). This method of measuring the

11. sensitivity of our objective function provides wonderful insight into the simplex method,
and especially highlights the complexity required to find a viable solution with so many
variables.
At the conclusion of this project, we are very much content with the results that
our methods have returned us. If we were confident enough to invest $10,000 in the stock
market today, knowing for a fact that we would gain $6,917.52 in profits over the coming
year, we would be thrilled. However, we realize that our methods made predictions based
upon very shallow observations, and that 2010 will likely not hold as much fortune for
companies like Starbucks as the past 52 weeks have. Even though our application may
not provide an accurate dollar amount for the returns that our portfolio will earn over the
next 52 weeks, we do believe that our constraints enabled us to make more educated
decisions in our investments. Working on this project has demonstrated that using linear
programming can be a very helpful tool in choosing investments, especially when there
are many variables to choose between and constraints that need to be satisfied. Had we
had access to more thoughtful and detailed data about companies, we would be
comfortable using linear programming to help us invest our money.

12. Works Cited
"Beta financial definition of Beta. Beta finance term by the Free Online
Dictionary." Financial Dictionary. Web. 09 Dec. 2009. <http://financial-
dictionary.thefreedictionary.com/Beta>.
"CEO Pay Database." AFL-CIO - America's Union Movement. Web. 09 Dec. 2009.
<http://www.aflcio.org/corporatewatch/paywatch/ceou/>.
Fernando, K V. Practical Portfolio Optimization. NAG Ltd. Web. 29 Nov. 2009.
<http://www.nag.co.uk/doc/TechRep/Pdf/tr2_00.pdf>.
"Responsible Shopper: Profiles of Major Corporations on Human Rights, Social Justice,
Environmental Sustainability and more." Green America: Economic Action for a
Just Planet. Web. 09 Dec. 2009.
<http://www.greenamericatoday.org/programs/responsibleshopper/learn_hub.cfm
>.
Yahoo! Finance - Business Finance, Stock Market, Quotes, News. Web. 09 Dec. 2009.
<http://finance.yahoo.com/>.