Why Expected Returns Are Harder to Estimate Than Volatility

# Are asset return means difficult to predict because they have no lower bound?


# Are asset return means difficult to predict because they have no lower bound?












In finance, it is widely known that the volatility of asset returns ($\sigma$) are easier to predict than the expected value of asset returns ($\mu$) , otherwise known as the average return or mean.

Is this partly due to the fact that asset volatility is restricted to be a positive value ($\sigma \in (0,+\infty)$), whereas asset returns and mean can take on negative percentage values ($\mu \in (-\infty,+\infty)$)? If so, why would positive boundedness of a variable makes its estimation more reliable and lower estimation error?

## Answer by RRL (score 8, accepted)

https://quant.stackexchange.com/a/54501

To answer, the assertion that volatility is easier to predict than expected return requires clarification. The phrase "easier to predict" is particularly ambiguous.

To me this means that the estimation of volatility from a sample of returns is more robust than the estimation of expected return in the context of relative sampling error.

Suppose over a time period $T$ we observe asset prices $S_0,S_1, \ldots, S_N$ at uniformly spaced time intervals of length $\delta t$ where $T = N \delta t$. Assume that the log-return (over an interval of length $\delta t$) has a stable distribution and returns over non-overlapping intervals are independent. Let $\mu$ and $\sigma$ denote the annualized mean return and volatility, respectively.

The $\delta t$-period log-return has expected value $\mu \delta t$ and variance $\sigma^2 \delta t$, where the $\delta t$ scaling of the variance is a consequence of the independence. We now have an iid sample $X_1,X_2,\ldots, X_N$ where

$$X_j = \log \frac{S_j}{S_{j-1}}$$

and the estimators for expected retrun and volatility are

$$\hat{\mu}\delta t = \frac{1}{N}\sum_{j=1}^N X_j, \quad \hat{\sigma}^2\delta t = \frac{1}{N-1}\sum_{j=1}^N (X_j - \hat{\mu}\delta t)^2$$

Asymptotically, the sampling distributions for the estimators are

$$\hat{\mu}\delta t \sim \mathcal{N}(\mu \delta t, \sigma^2 \delta t/N),\quad \frac{(N-1) \hat{\sigma}^2 \delta t}{\sigma^2 \delta t} \sim \chi^2(N-1),$$ that is, normal and chi-square with $N-1$ degrees of freedom, respectively. The standard errors for the estimates of expected return and volatility are, respectively, $\sigma\sqrt{\frac{\delta t}{N}}$ and $\frac{\sqrt{2} \sigma^2 \delta t}{\sqrt{N-1}}$.

As expected, the absolute sampling error (given by standard error) for the both the expected return and the volatility diminish as $1/\sqrt{N}$ as the number of samples $N$ increases.

However, the relative errors tell a different story. The relative sampling error for the volatility is

$$\frac{\frac{\sqrt{2} \sigma^2 \delta t}{\sqrt{N-1}}}{\sigma^2 \delta t} = \sqrt{\frac{2}{N-1}}$$

This shows that the relative error improves simply by increasing the number of samples. Given a fixed time period $T$, we only need to sample returns at a higher frequency to improve the estimate of volatility. Sampling daily is more accurate than sampling monthly, sampling monthly is more accurate than sampling quarterly, etc.

On the other hand, the relative sampling error for the expected return is

$$\frac{\sigma \sqrt{\frac{\delta t}{N}}}{\mu \delta t} = \frac{\sigma}{\mu \sqrt{N \delta t}}= \frac{\sigma}{\mu \sqrt{T}}$$

The only way to get a better estimate for expected return is to increase the length of the period $T$ over which the samples are observed. For a fixed period $T$, say 3 years, the relative error cannot be improved by increasing the sampling frequency, regardless of how many additional samples are taken. In other words, in order to improve the accuracy of the estimated return by a factor of 5, we must increase the sampling period by a factor of 25 to 75 years -- clearly problematic.

The root cause of this phenomenon would seem to be the fact that return scales like $\delta t$ and volatility, with independent returns, scales like $\sqrt{\delta t}$ with respect to the measurement period $\delta t$.

## Answer by Brian B (score 2)

https://quant.stackexchange.com/a/54518

The essential difference arises not from the lower bound on volatility, but rather the fact that volatility is mean-reverting and asset values are not.

To make this clearer, note that a period-$T$ return prediction $\hat{r}=\hat{r}_T^{(0)}$ at time $t=0$ for an asset with price $P_0$ is equivalent to a price prediction of $P_T=P_0 e^{\hat{r} T}$. And, of course prices are bounded below by zero just as volatility is. And yet, they are harder to predict than volatility.

The real difference is that any sane stochastic model for volatility has mean reverting terms, for example

$$ d \sigma = \kappa (\sigma_0 - \sigma) dt + \eta \sigma^p dW $$

which for reasonable values of $\kappa, \eta, p$ cannot go below zero. A long-term average of $\sigma$ is then a good estimate of $\sigma_0$ and hence of long-term future volatility.

In contrast, reasonable stochastic models for $P$ have no such mean reversion, and the simplest ones like Black-Scholes can be proven to wander infinitely far from their initial values. Thus the returns themselves can be infinitely far from zero as well, making them much harder to predict than mean-reverting quantities.