investment strategy case analysis (mgt 3050)

DECISION SCIENCE(MGT 3050)

GROUP ASSIGNMENT

CASE STUDY: INVESTMENT STRATEGY

NAME MATRIC NO.NUR HASEENA BT HANAFE 1322762

AFIFAH NABILAH BT MOHAMAD SAFEI

1321976

KHAN AKMAM BINTE ARIF 1124392SITI HAJAR BINTI MUSTAFFA 1328680

Lecturer’s Name : Dr. Rafikul IslamSection : 7Session : Semester 2, 2014/2015

TABLE OF CONTENTS

CONTENTS PAGES

TABLE OF CONTENTS 2

SUMMARY OF THE CASE PROBLEM 3

ANSWER

CONCLUSION 12

APPENDIX 13 - 21

REFERENCES 22

SUMMARY

The case problem is about J.D. Williams which specializes in investment advisory. It

basically suggests its clients on how much of their capital to invest in each of the three

different type of funds which are the growth fund, money fund and the income fund. It also

opts for a diversified portfolio for each client by implementing certain requirements for each

type of fund. For instance growth fund must be more than 20% but less than 40% and at least

30% should be allocated to the money fund.

The case revolves around one client with a fund of $800000. Apart from the

requirements of the portion of each type of fund on the client’s portfolio, another constraint of

risk index has been placed. The clients overall risk index should not be over 0.05. Also,

individual risk index for each type of fund has been given. The company predicted the annual

yield of growth, income and money funds to be 18%, 12.5% and 7.5%.

We are required to develop a model where the yield from the client’s portfolio is

maximized given the constraints.

Let growth fund be X1, income fund X2 and money fund X3.

The formulation of the above problem is:

Objective Function:

Maximize Z= 0.18 X1 + 0.125 X2 + 0.075 X3 (maximizing yield)

Subject to:

1 X1 + 1 X2 + 1 X3 ≤ 800000 (available fund)

0.80 X1 -0.20 X2 -0.20 X3 ≥ 0 (requirement of X1 to be more than 20%)

0.60 X1 -0.40 X2 -0.40 X3 ≤ 0 (requirement of X1 to be less than 40%)

-0.20 X1 + 0.80 X2 -0.20 X3 ≥ 0 (requirement of X2 to be greater than 20%)

-0.50 X1 + 0.50 X2 -0.50 X3 ≤ 0 (requirement of X2 to be less than 50%)

-0.30 X1 -0.30 X2 + 0.70 X3 ≥ 0 (requirement of X3 to be at least 30%)

0.05 X1 + 0.02 X2 -0.04 X3 ≤ 0 (requirement of overall risk index to be not more than 0.05)

X1, X2, X3 ≥ 0

The first question of the case deals with solving the stated problem. The second

question asks for the suggestion when the overall risk index is increased by 0.005. The third

question asks whether there will be any change in the previous optimal solution if the yield

for the growth fund is decreased by 2% first and then a further 2%. The fourth question deals

with a constraint where the growth fund cannot be more than the income fund. The last

question asks to what extent we feel this asset allocation model is useful.

Apart from the last question, we will be solving each of them by using

PHPSimplexfrom www.phpsimplex.com/en/. The solution and comments of each question

will be discussed in the subsequent sections.

http://www.phpsimplex.com/en/

QUESTION 1

Model formulation:

Maximize Z = 0.18 X1 + 0.125 X2 + 0.075 X3

subject to 1X1 + 1 X2 + 1 X3 ≤ 800000

0.8 X1 - 0.2 X2 - 0.2 X3 ≥ 0

0.6 X1 - 0.4 X2 - 0.4 X3 ≤ 0

-0.2 X1 + 0.8 X2 - 0.2 X3 ≥ 0

-0.5 X1 + 0.5 X2 - 0.5 X3 ≤ 0

-0.3 X1 - 0.3 X2 + 0.7 X3 ≥ 0

0.05 X1 + 0.02 X2 - 0.04 X3 ≤ 0

X1, X2, X3 ≥ 0

The problem is converted to canonical form by adding slack, surplus and artificial variables as

appropriate.

As the constraint 1 is of type '≤' we should add the slack variable X4.

As the constraint 2 is of type '≥' we should add the surplus variable X5 and the

artificial variable X11.








Maximize: 0.18 X1 + 0.125 X2 + 0.075 X3 + 0 X4 + 0 X5 + 0 X6 + 0 X7 + 0 X8 + 0 X9 + 0 X10 +

0 X11 + 0 X12 + 0 X13

1 X1 + 1 X2 + 1 X3 + 1 X4 = 800000

0.8 X1 - 0.2 X2 - 0.2 X3 -1 X5 + 1 X11 = 0

0.6 X1 - 0.4 X2 - 0.4 X3 + 1 X6 = 0

-0.2 X1 + 0.8 X2 - 0.2 X3 -1 X7 + 1 X12 = 0

-0.5 X1 + 0.5 X2 - 0.5 X3 + 1 X8 = 0

-0.3 X1 - 0.3 X2 + 0.7 X3 - 1 X9 + 1 X13 = 0

0.05 X1 + 0.02 X2 - 0.04 X3 + 1 X10 = 0

X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13 ≥ 0

There is infinitely many values of X1, X2, X3 for the optimal value Z = 0, which are contained

in the part of the plane 0 X1 + 0 X2 + 0 X3 = 0 that satisfies the constraints of the problem.

One is:

X1 = 248888.88888889

X2 = 160000

X3 = 391111.11111111

Thus, the allocation of the funds is as follows:

Growth Fund, X1 = $ 248,889

Income Fund, X2 = $160,000

Money Market Fund, X3 = $391,111

The allocation of fund will provide maximum annual yield at $94,133

QUESTION 2

If the client’s risk index is increased by one half of a percentage point, from 0.05 to

0.005, then the annual yield of the investment would also be increased by $4,667. This means

that the total annual yield will increases from $94,133 to a new projection of $98,800.

The investment recommendation would change as the following:

Fund Allocation Projected Annual Yield

Growth fund, X1 = $293,333 $293,333 x 0.18 = $52,800

Income Fund, X2 = $160,000 $160,000 x 0.125 = $20,000

Money Market Fund, X3 = $346,667 $346,667 x 0.075 = $26,000

Total = $800,000 Total = $ 98,800

Thus, the new total annual yield will be $98,800 compared to before which was $94,133

QUESTION 3

When annual yield of the growth fund would decrease from 18% to 16%, keeping everything

else constant, the only change in our formulation would be the coefficient of X1 in the

objective function. The new formulation would be:

Maximize: 0.16 X1 + 0.125 X2 + 0.075 X3

Subject to:

1 X1 + 1 X2 + 1 X3 ≤ 800000

0.80 X1 -0.20 X2 -0.20 X3 ≥ 0

0.60 X1 -0.40 X2 -0.40 X3 ≤ 0

-0.20 X1 + 0.80 X2 -0.20 X3 ≥ 0

-0.50 X1 + 0.50 X2 -0.50 X3 ≤ 0

-0.30 X1 -0.30 X2 + 0.70 X3 ≥ 0

0.05 X1 + 0.02 X2 -0.04 X3 ≤ 0

X1, X2, X3 ≥ 0

Tableau Form:

Maximize: 0.16 X1 + 0.125 X2 + 0.075 X3 + 0 X4 + 0 X5 + 0 X6 + 0 X7 + 0 X8 + 0 X9 + 0 X10

+ 0 X11 + 0 X12 + 0 X13

Subject to:

1 X1 + 1 X2 + 1 X3 + 1 X4 = 800000

0.8 X1 -0.2 X2 -0.2 X3 -1 X5 + 1 X11 = 0

0.6 X1 -0.4 X2 -0.4 X3 + 1 X6 = 0

-0.2 X1 + 0.8 X2 -0.2 X3 -1 X7 + 1 X12 = 0

-0.5 X1 + 0.5 X2 -0.5 X3 + 1 X8 = 0

-0.3 X1 -0.3 X2 + 0.7 X3 -1 X9 + 1 X13 = 0

0.05 X1 + 0.02 X2 -0.04 X3 + 1 X10 = 0

X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13 ≥ 0

Solving this by the simplex method gives the optimal solution:

Z = 89155.555555556

X1 (growth fund) = 248888.88888889

X2 (income fund) = 160000

X3 (money fund) = 391111.11111111

As we can see, the optimal solution did not change from the original recommendation

even though the annual yield decreased to 16%. However, if it is reduced to 14% the

recommendation will change to optimize Z. Here the coefficient of X1 in the objective

function will change again giving us:

Maximize: 0.14 X1 + 0.125 X2 + 0.075 X3

Subject to:

1 X1 + 1 X2 + 1 X3 ≤ 800000

0.80 X1 -0.20 X2 -0.20 X3 ≥ 0

0.60 X1 -0.40 X2 -0.40 X3 ≤ 0

-0.20 X1 + 0.80 X2 -0.20 X3 ≥ 0

-0.50 X1 + 0.50 X2 -0.50 X3 ≤ 0

-0.30 X1 -0.30 X2 + 0.70 X3 ≥ 0

0.05 X1 + 0.02 X2 -0.04 X3 ≤ 0

X1, X2, X3 ≥ 0

Tableau Form:


+ 0 X11 + 0 X12 + 0 X13

Subject to:

1 X1 + 1 X2 + 1 X3 + 1 X4 = 800000

0.8 X1 -0.2 X2 -0.2 X3 -1 X5 + 1 X11= 0

0.6 X1 -0.4 X2 -0.4 X3 + 1 X6= 0

-0.2 X1 + 0.8 X2 -0.2 X3 -1 X7 + 1 X12 = 0

-0.5 X1 + 0.5 X2 -0.5 X3 + 1 X8 = 0

-0.3 X1 -0.3 X2 + 0.7 X3 -1 X9 + 1 X13 = 0

0.05 X1 + 0.02 X2 -0.04 X3 + 1 X10 = 0

X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13 ≥ 0

Solving this gives us the optimal solution:

Z = 85066.666666667

X1 (growth fund) = 160000

X2 (income fund) = 293333.33333333

X3 (money fund) = 346666.66666667

As we can see, the optimal solution is quite different from the previous cases. If annual

yield of growth fund decreases to14%, investment in growth and money fund will decrease,

first drastically and the latter slightly, and investment in income fund will increase by a

significant amount in order to maximise the client’s return.

QUESTION 4

The current optimal solution shows the amount of money invested in growth fund is

more than in the income fund which are $248889 and $160000 respectively. However, in

question 4, the amount invested in growth fund is not allowed to exceed the amount invested

in income fund. Therefore, to add this new requirement, we will add a new constraint to the

equation:

Maximize: 0.18 X1 + 0.125 X2 + 0.075 X3

Subject to constraints:

1 X1 + 1 X2 + 1 X3 ≤ 800000

0.8 X1 - 0.2 X2 - 0.2 X3 ≥ 0

0.6 X1 - 0.4 X2 - 0.4 X3 ≤ 0

-0.2 X1 + 0.8 X2 - 0.2 X3 ≥ 0

-0.5 X1 + 0.5 X2 - 0.5 X3 ≤ 0

-0.3 X1 - 0.3 X2 + 0.7 X3 ≥ 0

0.05 X1 + 0.02 X2 - 0.04 X3 ≤ 0

1 X1 - 1 X2 + 0 X3 ≤ 0 New Constraint

X1, X2, X3 ≥ 0

Tableau Form


+ 0 X11 + 0 X12 + 0 X13 + 0 X14

Subject to:

1 X1 + 1 X2 + 1 X3 + 1 X4 = 800000

0.8 X1 - 0.2 X2 - 0.2 X3 - 1 X5 + 1 X12 = 0

0.6 X1 - 0.4 X2 - 0.4 X3 + 1 X6 = 0

-0.2 X1 + 0.8 X2 - 0.2 X3 - 1 X7 + 1 X13 = 0

-0.5 X1 + 0.5 X2 - 0.5 X3 + 1 X8 = 0

-0.3 X1 - 0.3 X2 + 0.7 X3 - 1 X9 + 1 X14 = 0

0.05 X1 + 0.02 X2 - 0.04 X3 + 1 X10 = 0

1 X1 - 1 X2 + 1 X11 = 0 New Constraint

X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14 ≥ 0

Solving this using the Simplex Method in Tableau format, we got:

The optimal solution value is Z = 93066.666666667

X1 (Growth Fund) = 213333.33333333

X2 (Income Fund) = 213333.33333333

X3 (Money Fund) = 373333.33333333

When the new constraint is added to the equation, we found out that the amount to be

invested in both growth fund and income fund will be the same at the risk of 0.18 and 0.125

respectively. And both amounts are decreased from $248889 and $160000 to

$213333.33333333 respectively. The projected annual yield when this constraint is applied is

also much lower than the original of $94,133, which is $93067. This is only a decrease of

0.2% in yield . Since the yield decrease is so small, we may prefer this portfolio.

QUESTION 5

We would recommend the use of this model only when the potential new clients meet

the requirement and outlined criteria of similar objectives and constraints. The company’s

mission, however is to provide the professional, financial advice that best meets the individual

investors’ needs. We would therefore, not recommend the use of this asset allocation model as

a general guide to financial investment.

REFERENCES

Anderson, Sweeney, Williams, Martin; An Introduction to Management Science:

Quantitative Approaches to Decision Making (International Edition); 13th Edition;

South-Western Cengage Learning.

PHP Simplex; http://www.phpsimplex.com/simplex/simplex.htm?l=en

investment strategy case analysis (mgt 3050)

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