Virtual Volatility, an Elementary New Concept with Surprising Stock Market Consequences

Textbook investors start by predicting the future price distribution, PDF, of a candidate stock (or portfolio) at horizon T, e.g. a year hence. A (log)normal PDF with center (=drift =expected return) $\mu $T and width (=volatility) $\sigma \surd $T is often assumed on Central Limit Theorem grounds, i.e. by a random walk of daily (log)price increments $\Delta $s. The standard deviation, stdev, of historical (\textit{ex post}) $\Delta $s `s is usually a fair predictor of the coming year's (\textit{ex ante}) stdev($\Delta $s) = $\sigma _{daily}$, but the historical mean E($\Delta $s)\textit{ at best }roughly limits the true, to be predicted, drift by $\mu _{true}$T$\sim \quad \mu _{hist}$T $\pm $ $\sigma _{hist}\surd $T. Textbooks take a PDF with $\sigma \quad \sim $ $\sigma _{daily }$ and $\mu $ as somehow known, as if accurate predictions of $\mu $ were possible. It is elementary and presumably new to argue that an average of PDF's over a range of $\mu $ values should be taken, e.g. an average over forecasts by different analysts. We estimate that this leads to a PDF with a `virtual' volatility $\sigma \quad \sim $ 1.3$\sigma _{daily.}$ It is indeed clear that uncertainty in the value of the expected gain parameter increases the risk of investment in that security by most measures, e. g. Sharpe's ratio $\mu $T/$\sigma \surd $T will be 30{\%} smaller because of this effect. It is significant and surprising that there are investments which\textit{ benefit }from this 30{\%} virtual increase in the volatility