In finance, it is well-known that the expected value of asset returns, $\mu$, otherwise known as the average return or mean or first statistical moment, is difficult to predict. I think it was Mandelbrot or Merton who first showed proof of this.

Could someone summarise how, and what is the procedure, for evaluating the accuracy and precision of predictions made of a time series' first statistical moment (which is a scalar value), based on historical returns data? Is it simply the prediction compared to the actual mean when the new data arrives?

And if there are multiple models individually giving a unique prediction of the asset mean, how can these different predictions be compared against one another? Would the comparison be really against one another, or each against some sort of truth benchmark like the true mean, if obtainable?

  • 1
    $\begingroup$ Please don't give to Mandlebrot what belongs to R. C. Merton ;) $\endgroup$
    – nbbo2
    Commented Jul 3, 2020 at 15:58

1 Answer 1


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 σ=0.2 and you have 100 years of data, then the confidence interval for the mean is fairly wide (approx +/- 2%).

To expand on @noob2 comment above, yes it was indeed Merton. A summary of Merton's insight below:

  1. Log prices follow: $dp_t=\mu dt+\sigma dW_t$

  2. Then: $r_{t+h,h}=p_{r+h,h}-p_t ~ N(\mu h, \sigma^2 h)$

  3. standard ML estimators:

  • $\hat{\mu}=\frac{1}{nh}\sum_{k=1} r_{kh,h}$
  • $\hat{\sigma^2}=\frac{1}{nh}\sum_{k=1} (r_{kh,h}-\hat{\mu}h)^2$

Assymptotic distribution of estimators:

  • $\sqrt T(\hat{\mu}-\mu) \rightarrow N(0,\sigma^2)$
  • $\sqrt n (\hat{\sigma^2}-\sigma^2)\rightarrow N(0,\sigma^4)$

So when $n$ tends to infinity we get precise estimator of $\sigma^2$ , and when $T$ tends to infinity we get it for $\mu$.

This was first noted by Merton (1980).

  • $\begingroup$ if a daily return series is originally $T=1000$ observations long, but then we downsample to a lower frequency so that $T=750$ observations instead, such that the observations become larger due to the new time aggregation, is it a given that predictions of the mean for the new series will become less precise because $T$ has now decreased away from infinity? $\endgroup$
    – develarist
    Commented Jul 3, 2020 at 22:45

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