The Perception of Time, Risk and Return During Periods of Speculation

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.