  We build a simple model of leveraged asset purchases with margin calls.
Investment funds use what is perhaps the most basic financial strategy, called
"value investing", i.e. systematically attempting to buy underpriced assets.
When funds do not borrow, the price fluctuations of the asset are normally
distributed and uncorrelated across time. All this changes when the funds are
allowed to leverage, i.e. borrow from a bank, to purchase more assets than
their wealth would otherwise permit. During good times competition drives
investors to funds that use more leverage, because they have higher profits. As
leverage increases price fluctuations become heavy tailed and display clustered
volatility, similar to what is observed in real markets. Previous explanations
of fat tails and clustered volatility depended on "irrational behavior", such
as trend following. Here instead this comes from the fact that leverage limits
cause funds to sell into a falling market: A prudent bank makes itself locally
safer by putting a limit to leverage, so when a fund exceeds its leverage
limit, it must partially repay its loan by selling the asset. Unfortunately
this sometimes happens to all the funds simultaneously when the price is
already falling. The resulting nonlinear feedback amplifies large downward
price movements. At the extreme this causes crashes, but the effect is seen at
every time scale, producing a power law of price disturbances. A standard
(supposedly more sophisticated) risk control policy in which individual banks
base leverage limits on volatility causes leverage to rise during periods of
low volatility, and to contract more quickly when volatility gets high, making
these extreme fluctuations even worse.
