# Interpretation of portfolio standard deviation

I have computed an efficient frontier using quadratic optimization algorithm for some stock data and then plotted it.

However, I have troubles understanding how to interpret standard deviation of portfolio returns. For example, if you look at the plot, you can see that for daily return of 0.0006, the standard deviation is roughly 0.017.

What does that mean? Does it mean that on average daily return of my portfolio is going to be $0.0006\pm0.017$?

In other words, should I expect daily return to be in the interval from $-0.0164$ to $0.0176$?

P.S. Also, how do I go from daily values to yearly in terms of returns and standard deviation, knowing that there are 250 trading days? Is the following way correct?

$r_{yearly} = (1 + r_{daily})^{250} - 1$

$SD_{yearly} = SD_{daily} \times \sqrt{250}$

It depends on the distribution of the returns. If you assume that it's roughly normally distributed, then you have a ~68% chance for a return in the range of 1 standard deviation, ~95% chance for 2 standard deviations, and so on.

@Richard I assume your $V$ stands for variance so that your formula is correct. The question was about standard deviation, though, and there the square root needs to be taken.

• Yes,but this is trivial and your answer is a comment. – Ric Jan 14 '16 at 15:32
• @Richard Absolutely! Unfortunately I wasn't able to write a comment at that point in time (missing the necessary reputation which I now have). – Dr_Be Jan 14 '16 at 16:01

The first part of the question is correct.

The second is wrong:

If you model daily log returns: $$r_t = \log(P_t)-\log(P_{t-1}$$ then your yearly return $r_y$ is just $$\sum_{t=0}^{250} r_t,$$ assuming $250$ days. Then $$E[r_y] = 250 E[r_t],$$ and $$V[r_y] = 250 V[r_t]$$ if we assume that returns are uncorrelated.

In the case of arithmetic returns $$r_t = P_t/P_{t-1}-1.$$ The exact expressions for passing from daily mean to yearly are not that easy. The same holds for variance.