Mr. Lothbrok Has $10,000 To Invest In Two Types Of Mutual Fu
Mr Lothbrok Has 10000 To Invest In Two Types Of Mutual Funds Co
Mr. Lothbrok has $10,000 to invest in two types of mutual funds: Coleman High-yield Fund, which offers an annual yield of 6%, and Coleman Equity Fund, which earns 10%. He desires to invest at least $5,000 in the High-yield fund and at least $2,000 in the Equity fund. The goal is to determine how much money he should invest in each fund to maximize his total annual yield, and to find the maximum yield achievable.
Paper For Above instruction
In seeking to maximize Mr. Lothbrok’s annual yield through strategic investment in two mutual funds, it is essential to formulate this problem as a linear optimization task. The problem involves decision variables, constraints, and an objective function, which are central to operations research and financial optimization domains.
Problem formulation
Let x be the amount invested in the High-yield fund, and y the amount invested in the Equity fund. The total investment constraint is:
x + y ≤ 10,000
Since Mr. Lothbrok must invest at least $5,000 in the High-yield fund and at least $2,000 in the Equity fund, the constraints are:
x ≥ 5,000
y ≥ 2,000
The objective is to maximize his annual yield:
Maximize Z = 0.06x + 0.10y
This is a linear programming problem with an objective function and linear constraints. The feasible region is bounded by the constraints, and the maximum occurs at one of the vertices of the feasible region, based on the fundamental theorem of linear programming.
Solution
The feasible set is defined by:
- x ≥ 5,000
- y ≥ 2,000
- x + y ≤ 10,000
To find the optimal point, check the vertices of the feasible region:
1. When x = 5,000 (minimum investment in high-yield fund)
- y can be at most 10,000 - 5,000 = 5,000
- Check at y = 5,000
- Yield: Z = 0.06(5,000) + 0.10(5,000) = 300 + 500 = $800
2. When y = 2,000 (minimum investment in equity fund)
- x can be at most 10,000 - 2,000 = 8,000
- Yield: Z = 0.06(8,000) + 0.10(2,000) = 480 + 200 = $680
3. When x + y = 10,000
- With constraints x ≥ 5,000 and y ≥ 2,000, choose the point to maximize yield:
- The function increases with y (since 0.10 > 0.06), so invest the maximum possible in y:
- y = 10,000 - x, with x as low as 5,000:
- y = 10,000 - 5,000 = 5,000
- Yield: Z = 0.06(5,000) + 0.10(5,000) = 300 + 500 = $800
Alternatively, trying to allocate more to y:
- Since increasing y increases yield, and x is at least 5,000, the optimal is at x = 5,000, y = 5,000.
Comparing the yields at these vertices, the maximum yield of $800 is achieved at x = 5,000, y = 5,000.
Conclusion
Mr. Lothbrok should invest $5,000 in the high-yield fund and $5,000 in the equity fund. The maximum annual yield will then be $800, which arises from maximizing the investment in the fund with higher yield per dollar substantially and investing the minimum in the other fund to meet constraints.
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