Institutional Investors and Stock Market Volatility
In this paper, the authors, Xavier Gabaix, Parameswaran Gopikrishnan, Vasiliki Plerou and Eugene Stanley, propose a theory that explains the fat tail behavior of stock returns (or the excess stock market volatility) and trading volumes.
They introduce power law distributions with different tail (or Pareto) exponent. In their notations, if r is the return, the probability of the return to be larger than a positive x or smaller than –x is asymptotically close to the inverse of x raised at the power of the tail coefficient “eta”:
P(| r|>x) ~ x^(-eta)
If the tail coefficient is low then there are fat tails in the distribution of returns.
If returns follow such distributions then events such as the crash of 1987 or LTCM should not be uncommon.
They present four facts:
1. High frequency stock returns are distributed according to a power law with a tail coefficient close to 3
2. Trading volumes are distributed according to a power law with a tail coefficient close to 1.5
3. The price impacts are distributed according to a power law
4. Sizes of large investors are distributed according to a power law with a tail coefficient close to 1 (Zipf’s law).
They proceed to build micro-foundations for these features:
A large investor receives some private information on the value of a stock and trades with some liquidity providers in a competitive market. A liquidity provider sells the shares with a permanent and a transitory price impacts. The risk-averse liquidity provider then buys back the shares on the market to rebuild its inventory. With first-order risk aversion (the provider cares about the volatility), the price impact function is a square root of the quantity of shares traded.
The large investor (the fund) receives mispricing information but is exposed to model risk (the information could be wrong). He cares about maximizing his expected return but also about minimizing model risk. The authors derive the optimal trading policy (the quantity of shares traded given the fund size and the perception of mispricing) for the fund and show the traded volume and price returns follow power-law distributions.
Their main result is that if the price impact is a square root of the trading volume and if the size large institutional investors follow the Zipf’s law then volumes and returns follow power-law distributions with tail exponents of 1.5 and 3 (facts 1 and 2).
Gabaix, Xavier, Gopikrishnan, Parameswaran, Plerou, Vasiliki and Stanley, H. Eugene Eugene, "Institutional Investors and Stock Market Volatility" (October 2005). NBER Working Paper No. W11722 Available at SSRN: http://ssrn.com/abstract=837165
Also published in the QJE, May 2006.
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