I am doing some revision questions on my Portfolio Theory module, and have come across the following question:
Consider an investor who has constructed a risky portfolio from N securities. The investment opportunity set is described by the equation:
$$\sigma^2 = 10 - 5{\times}E(r) + 0.5\times(E(r))^2$$
Find the minimum variance portfolio.
I can't find any info in my notes, but my intuition says differentiate, set to zero and rearrange for E(r)?
On
We want to minimize $\sigma^2$ as a function of $E(r)$. We also likely want $E(r) \ge 0$, but this won't be necessary.
$$0 =\frac{d \sigma^2}{dE(r)}=-5+E(r).$$
Now, looking at the second derivative
$$\frac{d^2 \sigma^2}{dE(r)^2} = 1>0$$
It is convex at every point, so we only need to check minima. And $E(r)=5$ is the minimum variance portfolio.
However, the variance at that point, $\sigma^2=-2.5$, so perhaps there is a mistake in the question. Can you clarify?
If you are trying to minimize sigma-squared, then the points where the derivative is zero will be at least local minima or maxima. If this is not intuitive, imagine a parabola and calculate the derivative at various points.
Another step would be to prove that the function is globally concave so that the local minima/maxima are in fact global, but your prof probably won't require that. In comparison with the parabola example, finding where the dy/dx is zero in y = x ^ 3 won't find the global.
I'm not sure what you mean by rearrange for E(r).