Question

a portfolio manager wants to distribute funds between two investments. the risk function is ( r(x) = 0.0263x^2 - 0.0162x + 0.0081 ). if ( x ) is the portion invested in the first option, how should the manager distribute the funds to minimize the risk? round to the nearest thousandth. first option: 0.081 second option: 0.919 first option: 0.692 second option: 0.308 first option: 0.919 second option: 0.081 first option: 0.308 second option: 0.692

Explanation:

Step1: Identify min of quadratic function

The risk function $R(x)=0.0263x^2 - 0.0162x + 0.0081$ is a quadratic function in the form $ax^2+bx+c$ with $a>0$, so its minimum occurs at $x=-\frac{b}{2a}$.

Step2: Calculate x value

Substitute $a=0.0263$, $b=-0.0162$:

$$ x = -\frac{-0.0162}{2\times0.0263} = \frac{0.0162}{0.0526} \approx 0.308 $$

Step3: Find second portion

The total portion is 1, so the portion for the second option is $1 - x$.

$$ 1 - 0.308 = 0.692 $$

Answer:

First option: 0.308
Second option: 0.692