## A Question I Have Never Asked Nick Barberis

In his 2000 *Journal of Finance* paper, "Investing for the Long Run when Returns Are Predictable" http://badger.som.yale.edu/faculty/ncb25/alloc_jnl.pdf, Nick Barberis wrote:

In the buy-and-hold case, we find that predictability in asset returns leads to strong horizon effects: an investor with a horizon of 10 years allocates significantly more to stocks than someone with a one-year horizon. The reason is that time-variation in expected returns such as that in equation (1) induces mean-reversion in returns, lowering the variance of cumulative returns over long hor izons. This makes stocks appear less risky to long-horizon investors and leads them to allocate more to equities than would investors with shor ter horizons.

We also find strong horizon effects when we solve the dynamic problem faced by an investor who rebalances optimally at regular intervals. However, the results here are of a different nature. Investors again hold substantially more in equities at longer horizons, but only when they are more risk-averse than log utility investors. The extra stock holdings of long-horizon investors are “hedging demands” in the sense of Merton (1973). Under the specification g iven in equation (1), the available investment opportunities change over time as the dividend yield changes: When the yield falls, expected returns fall. Merton shows that investors may want to hedge these changes in the opportunity set. In our data, we find that shocks to expected stock returns are negatively correlated with shocks to realized stock returns. Therefore, when investors choose to hedge, they do so by increasing their holdings of stocks...

I have never understood this distinction. When stock prices are mean reverting, a rebalancing investor will want to borrow and buy more after stock prices go down (expecting that they will then go up as they revert to the mean) and will want to sell and hold less stock after stock prices go up (expecting that they will then go down as they revert to the mean). The ultimate wealth of the rebalancing investor thus has a higher mean and a lower variance than the wealth of the buy-and-hold investor. The Merton hedge is not of an alternative nature to the mean-reversion-reduces-risk effect; it is an intensification of it.