question archive Innis Investments manages funds for a number of companies and wealthy clients
Innis Investments manages funds for a number of companies and wealthy clients
Subject:ManagementPrice:3.87 Bought12
Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client's needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%. The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innis' risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis's client also specified that at least $300,000 be invested in the money market fund.
Let S = units purchased in the stock fund
M = units purchased in the money market fund
Min 8S + 3M
st
50S + 100M <= 1,200,000
5S + 4M >= 60,000
M >= 3,000
S, M >= 0
- What is the optimal solution, and what is the minimum total risk?
- Specify the objective coefficient ranges.
- How much annual income will be earned by the portfolio?
The computer solution was taken out of Lindo.
- What is the rate of return for the portfolio?
- What is the dual value for the funds available constraint?
- What is the marginal rate of return on extra funds added to the portfolio?
- Suppose the risk index for the stock fund (the value of ) increases from its current value of 8 to 12. How does the optimal solution change, if at all?
- Suppose the risk index for the money market fund (the value of ) increases from its current value of 3 to 3.5. How does the optimal solution change, if at all?
- Suppose increases to 12 and increases to 3.5. How does the optimal solution change, if at all?
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Solution
Given
?Min: 8S+3M?
subject to
- Investment : ? 50S+100M≤1200000?
- Annual Income: ? 5S+4M≥60000?
- Money Market: ? M≥3000?
- ?S, M≥0?
Part 1
A)
The optimal solution is
?S=$4,000?
?M=$10,000?
The minimum risk is
?8⋅4000+3⋅10000=$62,000?
B)
The lower and upper limits for the object coefficients can be computed using the following formulas
?Lower limit=Object coefficient−Allowable decrease?
?Upper limit=Object coefficient+Allowable increase?
For variable S:
?Lower limit=8−4.25=3.75?
?Upper limit=∞?
Specified range:
?(3.75,∞)?
For variable M:
?Lower limit=−∞?
?Upper limit=3+3.4=6.4?
Specified range:
?(−∞,6.4)?
C)
The annual income earned is
?Annual Income=5⋅4000+4⋅10000=$60,000?
Part 2
A)
?Rate of return=Available FundsAnnual Income?=120000060000?=0.05=5%?
B)
The dual value is ?−0.057? units. See the column under shadow value of the sensitivity analysis.
C)
?Marginal rate of return=Amount of risk×100=−0.057⋅100=−5.7%?
D)
According to the specified range on S, there is no upper limit as to how much the value of S can increase. Increasing the value of S from 8 to 12 has no effect on the optimal solution.
E)
Since the increase of M from 3 to 3.5 is less the the upper limit of 6.4, the optimal solution will remain optimal.
F)
Since there is no report for simultaneous change in the coefficient of the objective function, we have to check following ratio for optimal result. The solution is still optimal if:
?∑Allawable ChangeProposed Change?<1?
But
?4.2512−8?+3.43.5−3?≈1.09>1?
The solution will no longer be optimal.
Please see the attached file for the complete solution
