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:

1. Mean of 0,
2. standard deviation of 1,
3. skewness of 0 and
4. 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)?

• If asset returns have a mean of zero, then all portfolios on that universe have expected return of zero, whether you are using Markowitz, Talmudic, Risk-Parity, or whatever. Jul 16, 2020 at 19:34
• There is an expected upper bound on the achieved SNR of a portfolio based on the Cramer-Rao bound, proved here. The Markowitz portfolio, which is built on Maximum Likelihood Estimators, will asymptotically achieve that upper bound. The real culprit is very low signal-noise ratios in the real world, long bias in commonly used test data, and a lot of rhetorical noise and cargo-cult thinking in this space. Jul 16, 2020 at 19:38

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).

• can it be said then that downsampling data (decreasing observations), regardless of distribution, ruins out of sample performance, in other words, the attainment of perfect prediction? Jun 28, 2020 at 16:34
• The less data you have the worse it will be, yes. Jun 28, 2020 at 16:36
• Can it be said that the mean of normal returns are easier to estimate than the mean of non-normal returns? or is it just as difficult Jun 28, 2020 at 16:50
• I would say just as difficult. Imagine that returns follow a binomial distribution, i.e. they can only take two values. This is an example of a non-normal distribution with a mean that is easier to estimate. Jun 28, 2020 at 16:55
• I've come across discussions saying that Markowitz never assumed normality in asset returns in the first place for his mean-variance model. and that only knowledge of their mean and variance (first two moments) are assumed. would this explain why out-of-sample performance is unaffected by either normal or non-normal inputs, that there was no normality assumption to begin with? Aug 16, 2020 at 7:54