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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)?

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migrated from May 1 '12 at 19:55

This question came from our site for professional and academic economists and analysts.

Does anything strike you as odd about the resulting value of sigma squared, when you follow your intuition? Could there be a mistake in the transcription of the question? (I could be wrong, I don't do MPT, it just strikes me as a bit peculiar, that's all) – EnergyNumbers Jan 7 '12 at 7:08

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?

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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).

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