  We discuss - in what is intended to be a pedagogical fashion - a criterion,
which is a lower bound on a certain ratio, for when a stock (or a similar
instrument) is not a good investment in the long term, which can happen even if
the expected return is positive. The root cause is that prices are positive and
have skewed, long-tailed distributions, which coupled with volatility results
in a long-run asymmetry. This relates to bubbles in stock prices, which we
discuss using a simple binomial tree model, without resorting to the stochastic
calculus machinery. We illustrate empirical properties of the aforesaid ratio.
Log of market cap and sectors appear to be relevant explanatory variables for
this ratio, while price-to-book ratio (or its log) is not. We also discuss a
short-term effect of volatility, to wit, the analog of Heisenberg's uncertainty
principle in finance and a simple derivation thereof using a binary tree.
