EFF/KRF: The tools we develop in "Disagreement, Tastes, and Asset Pricing," published in the Journal of Financial Economics 83 (March 2007), 667-689, are helpful here. To keep the analysis simple, let's assume that (i) there is only one stock, (ii) I have $100,000 to invest, and (iii) for some reason, I want to own as much of the stock as possible. Compare two scenarios. In the first I cannot borrow so I just buy $100,000 of equity. In the second scenario, you are willing to lend me money to buy more stock. You are not crazy, however, so you limit my leverage to four to one. With $100,000 to invest I can borrow $400,000 and buy $500,000 of stock.

What happens if the stock price falls by 10%? In the first scenario, not much. The value of my portfolio falls to $90,000. The second scenario is more interesting. My stock falls to $450,000. Since I still owe you $400,000, the 10% drop in price reduces my wealth from $100,000 to $50,000. My new 8 to 1 leverage ratio is well above your 4 to 1 limit. As a result, you recall half my loan, forcing me to sell $200,000 of stock. I now have $250,000 of stock, financed by a $200,000 loan and $50,000 of my own money.

So how does leveraging and deleveraging affect prices? My initial position is $100,000 of stock in the first scenario and $500,000 in the second. The ability to lever increases the amount of equity I can buy so, everything else the same, the price is initially a bit higher and the expected return is a bit lower. The 10% drop in price forces me to sell $200,000 in the second scenario. Because prices have to fall further to induce other investors to absorb my stock, deleveraging does amplify the 10% drop. Notice, however, that I still own more stock in the second scenario than in the first. Thus, the total impact of leverage on the stock price is positive.

The key here is that leverage allows me to own more stock than I could without leverage. In essence, leverage expands the economy's risk bearing capacity. Perhaps, for example, I want to hold as much stock as possible because I am almost risk neutral. Increasing the amount of equity I can buy reduces the weighted average of all investors' risk aversion. When my levered portfolio pays off and I can buy more, the weighted average risk aversion falls further, and the expected return on equity falls with it. When I have to delever because my portfolio has done poorly, both the weighted average risk aversion and the expected equity return rise. Thus, if the desire for leverage arises because some investors are less risk averse than others, leveraging and deleveraging lowers and raises the weighted average level of risk aversion and amplifies swings in stock prices.

The story is not much different if the desire for leverage is caused by heterogeneous expectations. We can think of the price as a weighted average of all investors' expectations. The weights in this average are a function of each investor's wealth, risk aversion, and available leverage. Suppose I want to own as much equity as possible because I think it is severely undervalued. When my levered portfolio does well, my relative wealth increases and my optimistic vote gets more weight in the price. This reinforces the price increase. Similarly, when my levered portfolio does poorly, my relative wealth declines and my vote gets less weight, which reinforces the price decline.

So conceptually, we expect leveraging and deleveraging to have an impact on volatility. Without a lot more information, however, it is hard to assess the magnitude of this effect.