Suppose a stock's rate of return has annual mean and variance of $\bar{r}$ and $\sigma^{2}$. To estimate these quantities, we divide 1 year into $n$ equal periods and record the return for each period. Let $\bar{r}_{n}$ and $\sigma_{n}^{2}$ be the mean and the variance for the rate of return for each period. Specifically, assume that $\bar{r}_{n}=\bar{r} / n$ and $\sigma_{n}^{2}=\sigma^{2} / n$. If $\hat{\bar{r}}_{n}$ and $\hat{\sigma}_{n}^{2}$ are the estimates of these, then $\hat{\bar{r}}=n \hat{\bar{r}}_{n}$ and $\hat{\sigma}^{2}=n \hat{\sigma}_{n}^{2}$. Let $\sigma(\hat{\bar{r}})$ and $\sigma\left(\hat{\sigma}^{2}\right)$ be the standard deviations of these estimates.

(a) Show that $\sigma(\hat{\vec{r}})$ is independent of $n$.

(b) Show how $\sigma\left(\hat{\sigma}^{2}\right)$ depends on $n$. (Assume the returns are normal random variables.) Answer the question posed as the title to this exercise.