Interpreting Portfolio Return Standard Deviation and Annualization
Summary
The document discusses how to interpret a portfolio’s daily return standard deviation on an efficient-frontier plot. Under an approximate normality assumption, the accepted answer relates one standard deviation around the mean to roughly a 68% probability range, and two standard deviations to roughly 95%. This explains why mean plus or minus one standard deviation is a range with a probability interpretation, rather than a guarantee about typical returns in every setting.
It also examines converting daily returns and variability to annual figures. For daily log returns, the response says returns add across days and that their mean and variance scale with the number of days when returns are uncorrelated; standard deviation is the square root of variance. The proposed compounding expression is raised for arithmetic returns, but the response cautions that exact annual mean and variance conversions are more involved. These statements rely on assumptions about distribution and dependence, and the excerpt does not assess skew, fat tails, or changing volatility.
Key ideas
- Standard deviation measures dispersion around the mean and is not a guaranteed daily return range.
- Under an approximate normal model, one and two standard deviations correspond to roughly 68% and 95% ranges.
- Daily log returns add across periods.
- For uncorrelated log returns, annual mean and variance scale with the number of trading days.
- Arithmetic returns do not generally have equally simple exact annual mean and variance conversions.
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Full text
# Interpretation of portfolio standard deviation
# 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?
$r_{yearly} = (1 + r_{daily})^{250} - 1$
$SD_{yearly} = SD_{daily} \times \sqrt{250}$
## Answer by amsh (score 5, accepted)
https://quant.stackexchange.com/a/22737
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.
## Answer by Dr_Be (score 1)
https://quant.stackexchange.com/a/22745
@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.
## Answer by Richi Wa (score 0)
https://quant.stackexchange.com/a/22744
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.