Why Expected Asset Returns Are Hard to Estimate Precisely
Summary
The document discusses how to assess estimates of the mean return, emphasizing that sampling uncertainty remains substantial even with long histories. It gives the standard error of a sample mean as return volatility divided by the square root of the observation count, and illustrates that high volatility can leave a wide confidence interval after a century of data. This makes a single realized sample mean an uncertain benchmark for judging forecasts.
It also summarizes an asymptotic result attributed to Merton: under the stated estimator setup, the variance estimate becomes precise as the number of observations grows, while precision of the mean estimate depends on elapsed time. The material does not provide a complete procedure for comparing competing forecasting models, nor does it specify how to handle changing return distributions, dependence, or forecast horizons. Its central lesson is that the mean is much harder to estimate precisely than variance, so forecast evaluation should account for estimation error rather than treating an observed historical mean as unquestioned truth.
Key ideas
- The standard error of a sample mean decreases with the square root of the sample size.
- High return volatility can leave considerable uncertainty about the mean even across long samples.
- The cited asymptotic results distinguish the precision of mean and variance estimates.
- Mean estimation depends on elapsed time, while variance estimation improves with additional observations under the stated setup.
- The document does not specify a full method for comparing competing forecasts or addressing changing return behavior.
Tags
Full text
# How to evaluate prediction(s) made of the asset return mean?
# How to evaluate prediction(s) made of the asset return mean?
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?
## Answer by phdstudent (score 2, accepted)
https://quant.stackexchange.com/a/55418
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:
- 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).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.