Why Normal Returns Do Not Solve Mean-Variance Estimation Error
Summary
The document asks whether normally distributed asset returns would let a Markowitz mean-variance portfolio recover strong out-of-sample performance. Its answer is no: normality and standardized moments do not remove the difficulty of estimating expected returns. The mean estimate remains noisy, with standard error proportional to return volatility and inversely proportional to the square root of the sample length.
As evidence, it cites a simulation study in which mean-variance performance approaches the true Sharpe ratio only after a very long sample of monthly observations. This illustrates how uncertain mean estimates can distort portfolio weights even when returns follow a normal distribution. The discussion is limited to the stated simulation and does not compare alternative portfolio methods or explain how covariance estimation affects results. Its central lesson is that distributional assumptions alone cannot guarantee reliable out-of-sample portfolio performance.
Key ideas
- Normally distributed returns do not guarantee that mean-variance portfolios perform well out of sample.
- Standardized means and moments do not eliminate uncertainty in estimated expected returns.
- The standard error of a sample mean declines with the square root of the observation count.
- The cited simulation requires an unusually long sample before mean-variance performance approaches the true Sharpe ratio.
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Full text
# Do normal returns make the mean-variance portfolio model perform properly?
# Do normal returns make the mean-variance portfolio model perform properly?
The Markowitz mean-variance model is known to suffer from estimation error due to financial returns not meeting the assumptions of a normal distribution, providing portfolio weights that underperform out-of-sample.
Does this mean that if asset returns that have a:
- Mean of 0,
- standard deviation of 1,
- skewness of 0 and
- excess kurtosis of 0
are fed into the model, this allows the model to perform best and fully (or partially) recover perfect out-of-sample performance/accuracy (given that the out-of-sample returns are also normally distributed)?
## Answer by phdstudent (score 4, accepted)
https://quant.stackexchange.com/a/55294
No, even if returns were perfectly normal (it really doesn't matter whether mean is zero and standard deviation is 1 - they can be anything), it wouldn't ensure that markowitz would perform well out of sample. The reason is because even if data is normally distributed it is hard to estimate means of returns.
The standard error for an estimate of a mean like a mean return - is:
$$SE(\bar{r}) = \frac{\sigma}{\sqrt{T}}$$
Now for the stock market, if $\sigma = 0.2$ and you have 100 years of data, then the confidence interval for the mean is fairly wide (approx +/- 2%).
Take a look at the example below from De Miguel et al:
The row you are interested in is the third row ($mv$). They simulate normally distributed data, and realize that only when you have 6000 months of data (i.e. 500 years), mean variance starts to be close to the true sharpe ratio (0.15 in their economy).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.