You can break what Banner calls the “curse of compounding” if you have more than one stock to work with. “What if you’re allowed to dynamically allocate capital between two or more assets?” Banner asks. The compound return of a portfolio, like that of a single asset, is the average of its returns minus one-half of its variance. Thanks to modern portfolio theory—which Nobel laureate Harry Markowitz set out in 1952—you can also describe the portfolio return as equal to the weighted average return of its constituent stocks. “Now, the plot thickens,” Banner says. The individual stocks tend to have higher volatilities than the portfolio as a whole: So the average stock variance tends to be larger than the portfolio variance.
If you rearrange the terms of a couple of equations, Banner says, you end up with an expression that shows exactly how much excess return you can get from rebalancing: It’s equal to one-half times the weighted average stock variance minus the portfolio variance. “That’s essentially the math formula that shows how the variances and covariances contribute to the compound return,” Banner says.
Rendered with a gamma and a couple of other Greek letters, it’s also the master formula of what Fernholz called stochastic portfolio theory. Fernholz originally got there by combining stochastic calculus—“that branch of math that studies changing quantities like calculus but with a probabilistic component: hence the fancy word stochastic,” Banner says—with modern portfolio theory.
That may seem daunting. For Banner, though, it’s the alternative that’s difficult: picking stocks that outperform. “We know that’s very hard to do,” he says.