Most games are defined by their probabilities. Take a coin toss, for example. If the coin is fair, players should expect an average of zero wins over the long-term given the game’s 50-50 probability. By the same token, an investor with no information, assuming normally distributed returns and an unlimited number of tries, should expect to make 0% returns in the long run by attempting to time the market at random times.

But what about investors * with* information?

If one of our investment goals is to prevent a lot of money from becoming a little, then the basis for our investment must be a well-researched forecast of return. Next, given that forecasting is subject to all kinds of errors, we need to determine our level of confidence in those forecasted returns and adjust them accordingly. Finally, we should only enter the market once the probability of risk-adjusted return is positive (i.e., once the odds are in our favor).

This probability is a function of our return forecast and the level of volatility in the market at that time. For a given volatility level of X, and a mean forecasted return of Y, the probability of a positive return is given by the cumulative normal distribution function[1] (again, assuming normally distributed returns). Conversely, given a required probability for positive returns and a volatility estimate for the market, we can use the inverse of the cumulative normal function to derive the implied return assumptions of investors.

The chart below shows the probability frontier for the STOXX USA 900 index at varying levels of predicted risk by our AXUS4-SH fundamental factor risk model. The left-most curve (blue) on the chart represents the range of necessary implied return expectations an investor would need at a fixed predicted risk level of 10% for each probability level. For example, the lower red dot on that curve at 3.9% represents the implied minimum return expectations an investor must have at 10% predicted risk, if they require the probability of positive excess return to be 65%[2]. The higher red dot on that curve at 6.7% represents the minimum expected return at 10% risk, if the investor requires a 75% probability of positive excess returns. Better odds require higher expected returns at a given level of risk.

Each curve on the chart represents the range of implied returns at a higher (fixed) predicted risk level. The first curve, at 10% of predicted risk, represents the risk-return environment investors faced in January of this year. At that time, an investor requiring a 65% probability of positive returns would have only needed a confidence-adjusted return forecast of 3.9% for the STOXX USA 900 index.

As the level of predicted risk increased over the next three months, from 10% to 16% to 24% and as high as 34% at the peak in April, the same 65% probability level would have required much higher return expectations for the same investor to enter the market. In February, at 16% risk (green curve), that investor would have needed a return forecast of at least 6.2%, up from 3.9% just a month earlier, to maintain the same probability (65%) of a positive return. In March, when predicted risk rose to 24% (yellow curve), a forecast of return of at least 9.2% would have been required. And at the peak of the volatility cycle in April, when predicted risk went as high as 34% (dark blue curve), the minimum required return expectation would have climbed to 13.1%.

So, in just three months, investors’ expected returns would have needed to increase by a whopping 235% to maintain a 65% probability of positive excess return by investing in a STOXX USA 900 ETF. If that same investor required a 75% probability of positive excess return, the return expectation would have needed to rise by 487% between January and April!

Predicted risk levels are now back around the 24% level, where we were in early March (yellow curve), and investors entering the market today would “*only”* have to have a 9.2% expected return, if they required a 65% probability of gain. If the downtrend in predicted risk in the chart below continues as it indicates, the burden of proof, so-to-speak, for entering the market at this time will continue to decline along with risk estimates.

Many pundits have called the recent market rally a bubble, reminiscent of the tech bubble of the late ‘90s. But a bubble is characterized by a cycle of ever-increasing and lofty return expectations. Declining volatility is making the current market bull run possible with declining return expectations. To quote Alanis Morissette, “Isn’t it ironic, don’t you think?”

[1] That is, for a volatility of 10% and an expected return of 3.9%, the probability of a positive return over the long-term is 65% (NORM.DIST(3.9,0,10,TRUE)=0.65173)

[2] NORM.INV(10, 0, 65)=3.9