TAGPUNO, RIKI JONAS A.
LINEAR PROGRAMMING ASSIGNMENT
The ABC Company manages approximately ₱15 million for clients. For each client, ABC
chooses a mix of three investment vehicles: a growth stock fund, an income fund, and a
money market fund. Each client has different investment objectives and different
tolerances for risk. To accommodate these differences, ABC places limits on the
percentage of each portfolio that may be invested in the three funds and assigns a
portfolio risk index to each client. Here’s how the system works for Juan dela Cruz, one
of ABC’s clients. Based on an evaluation of Juan’s risk tolerance, ABC has assigned Juan’s
portfolio a risk index of 0.05. Furthermore, to maintain diversity, the fraction of Juan’s
portfolio invested in the growth and income funds must be at least 10% for each, and at
least 20% must be in the money market fund. The risk ratings for the growth, income,
and money market funds are 0.10, 0.05, and 0.01, respectively. A portfolio risk index is
computed as a weighted average of the risk ratings for the three funds, where the weights
are the fraction of the portfolio invested in each of the funds. Juan has given ABC
₱300,000 to manage. ABC is currently forecasting a yield of 20% on the growth fund,
10% on the income fund, and 6% on the money market fund.
a. Develop a linear programming model to select the best mix of
investments for Juan’s portfolio.
G = Growth Stock Fund
I = Income Stock Fund
M = Money Market Fund
Objective = Maximize = 0.20G + 0.10I + 0.06M
Constraints:
0.10G + 0.05I + 0.01M ≤ 15,000 Max Risk for Juan
G ≥ 30,000 Minimum for Growth Stock Fund
I ≥ 30,000 Minimum for Income Stock Fund
M ≥ 60,000 Minimum for Money Market Fund
G + I + M ≤ 300,000 Funds Available
G, I, M ≥ 0
b. Solve the model you developed in part (a).
Growth Stock = ₱120,000; Income Stock = ₱30,000; Money Market = ₱150,000
c. How much may the yields on the three funds vary before it will be
necessary for ABC to modify Juan’s portfolio?
Growth Stock = 0.15 ≤ G ≤ 0.6
Income Stock = -Infinity ≤ I ≤ 0.12
Money Market = 0.02 ≤ M ≤ 0.2
d. If Juan were more risk tolerant, how much of a yield increase could he
expect? For instance, what if his portfolio risk index is increased to 0.06?
0.10G + 0.05I + 0.01M ≤ (0.06)(300,000)
0.10G + 0.05I + 0.01M ≤ 18,000
18,000 – 15,000 = 3,000 Increase
3,000 x 1.56 = 4,680
0.20G + 0.10I + 0.06M = 0.20(120,000) + 0.10(30,000) + 0.06(150,000)
0.20G + 0.10I + 0.06M = 24,000 + 3,000 + 9,000
0.20G + 0.10I + 0.06M = 36,000
Yield if Risk Index = 0.05
Yield = 36,000/300,000
Yield = 0.12
Yield if Risk Index = 0.06
Yield = (36,000 + 4,680)/300,000
Yield = 0.1356
Yield Increase = Y06 – Y05 = 0.1356 – 0.12
Yield Increase = 0.0156
e. If ABC revised the yield estimate for the growth fund downward to 0.10,
how would you recommend modifying Juan’s portfolio?
Growth Stock Fund = 0.20 – 0.10
Growth Stock Fund = 0.10
Objective = Maximize = 0.10G + 0.10I + 0.06M
Constraints:
0.10G + 0.05I + 0.01M ≤ 15,000 Max Risk for Juan
G ≥ 30,000 Minimum for Growth Stock Fund
I ≥ 30,000 Minimum for Income Stock Fund
M ≥ 60,000 Minimum for Money Market Fund
G + I + M ≤ 300,000 Funds Available
G, I, M ≥ 0
Growth Stock = ₱48,000; Income Stock = ₱192,000; Money Market = ₱60,000
f. What information must ABC maintain on each client in order to use this
system to manage client portfolios?
Maintain the client’s Risk Index and the Amount of Funds Available for each
client in order to use this system to manage client portfolios.
g. On a weekly basis ABC revises the yield estimates for the three funds.
Suppose ABC has 50 clients. Describe how you would envision ABC
making weekly modifications in each client’s portfolio and allocating the
total funds managed among the three investment funds.
He would have to solve a new linear program with the new yield estimates to find
the optimal portfolio mix for each client. The total amount that should be placed
in three investment options would then be determined by summing across all 50
clients. Then he'd have to adjust the amount invested in each fund to match the
sums he calculated earlier. Individual clients would not see any actual switching of
funds.