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 behaviour? What principles can you use to answer these questions? In this paper, I 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. It leads directly to the well known Sharpe ratio and the classic risk-return relationships of arbitrage pricing theory and the capital asset pricing model. The second hypothesis concerns the perception of time. I 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 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, but pays no attention to the calendar time that passes between them. Applying the first hypothesis in the intrinsic time measure suggested by the second, I derive an alternative set of relationships between risk and return. Its most noteworthy feature is that, in the short-term, a stock's trading frequency affects its expected return. I 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. Furthermore, I derive a modified version of the capital asset pricing model in which a stock's excess return relative to the market is proportional to its traditional beta multiplied by the square root of its trading frequency. I also present a model for the joint interaction of long-term calendar-time investors and short-term intrinsic-time speculators that leads to market bubbles characterized by stock prices that grow super-exponentially with time. Finally, I show that the same short-term approach to options speculation can lead to an implied volatility skew. I hope that this model will have some relevance to the behaviour of investors expecting inordinate returns in highly speculative markets.