  What return should you expect when you take on a given amount of risk? How
should that return depend upon other people's behavior? What principles can you
use to answer these questions? In this paper, we approach these topics by
exploring the consequences of two simple hypotheses about risk.
  The first is a common-sense invariance principle: assets with the same
perceived risk must have the same expected return. The second hypothesis
concerns the perception of time. We conjecture that in times of speculative
excitement, short-term investors may instinctively imagine stock prices to be
evolving in a time measure different from that of calendar time. They may
instead perceive and experience the risk and return of a stock in intrinsic
time, a dimensionless time scale that counts the number of trading
opportunities that occur.
  The most noteworthy result is that, in the short-term, a stock's trading
frequency affects its expected return. We show that short-term stock
speculators will expect returns proportional to the temperature of a stock,
where temperature is defined as the product of the stock's traditional
volatility and the square root of its trading frequency. We hope that this
model will have some relevance to the behavior of investors expecting
inordinate returns in highly speculative markets.
