An investor has up to $450,000 to invest in three types of investments. Type A pays 6% annually and has a
risk factor of 0. Type B pays 10% annually and has a risk factor of 0.06. Type C pays 12% annually and has a
risk factor of 0.08. To have a well‐balanced portfolio, the investor imposes the following conditions. The
average risk factor should be no greater than 0.05. Moreover, at least one‐half of the total portfolio is to be
allocated to type A investments and at least one‐fourth of the portfolio is to be allocated to type B
investments. How much should be allocated to each type of investment to obtain a
maximum return and what is the maximum return?
Why did the investor go to the circus? Because they heard there were high-risk factors and big returns! But let's crunch the numbers anyway.
Let's say the amount allocated to Type A investments is x, to Type B investments is y, and to Type C investments is z.
The conditions given are:
1. Average risk factor ≤ 0.05, so we have the equation: (0*x + 0.06*y + 0.08*z) / (x + y + z) ≤ 0.05.
2. At least one-half should be allocated to Type A, so we have the inequality: x ≥ (1/2) * (x + y + z).
3. At least one-fourth should be allocated to Type B, so we have the inequality: y ≥ (1/4) * (x + y + z).
4. The total amount to invest is $450,000, so we have the equation: x + y + z = 450,000.
Now, let's maximize the return.
The return function can be written as: R = 0.06x + 0.1y + 0.12z.
Now, let's solve these equations using linear programming or optimization techniques to find the values of x, y, and z that maximize the return R.
To solve this problem, we can use a technique called linear programming. Linear programming is a mathematical optimization method used to find the best outcome in a mathematical model with multiple constraints.
Let's define the variables:
- Let X be the amount allocated to type A investments
- Let Y be the amount allocated to type B investments
- Let Z be the amount allocated to type C investments
Now, let's set up the constraints based on the given conditions:
1) The average risk factor should be no greater than 0.05:
- (0 * X + 0.06 * Y + 0.08 * Z) / (X + Y + Z) <= 0.05
2) At least one-half of the total portfolio is to be allocated to type A investments:
- X >= (1/2) * (X + Y + Z)
3) At least one-fourth of the total portfolio is to be allocated to type B investments:
- Y >= (1/4) * (X + Y + Z)
4) The total investment amount should not exceed $450,000:
- X + Y + Z <= 450,000
Now, let's define the objective function, which is the maximum return:
- Objective function: Maximize 0.06 * X + 0.10 * Y + 0.12 * Z
We have set up the linear programming problem with the constraints and the objective function. Now, we can solve it using a linear programming solver or by graphically analyzing the feasible region.
After solving the problem, we will find the optimal allocation and the maximum return.