Equilibrium moment restrictions on asset returns: normal and crisis periods

Abstract

Empirically, the covariance between stock returns varies with their volatility. We seek a robust theoretical explanation of this. With minimal assumptions, we model stochastic properties of equilibrium returns which result from the interaction between inter-temporal traders and noisy, price-sensitive short-term traders. The inter-temporal traders can have arbitrary investment rules, preferences and information. In all cases we find a set of restrictions between second moments of equilibrium returns. With two assets there is also a bound on the correlation between asset returns. Estimation with second moments of global stock returns supports our theoretical framework. Higher volatility in at least one market can increase comovement among markets. With globalization, covariances between two stock markets can also affect covariances between two other stock markets. We also find that the changes in trader behavior between normal and crisis periods lead to changes in the moment restrictions between asset returns.

Keywords:

• short-term traders
• asset returns
• equilibrium moment restrictions
• correlation bound

Notes

1. The SSE index is a market capitalization-weighted stock index of SSE of China with the base date on December 19, 1990. The index was launched on July 15, 1991. Stocks in this market are divided into two classifications: A shares and B shares. Trading in A shares was initially restricted to domestic investor only while B shares are available to both domestic and foreign investors. After reforms in 2002, foreign investors can trade A shares under Qualified Foreign Institutional Investor program. A shares are quoted in Renminbi (RMB) while B shares are quoted in USD. It is currently composed of 894 constituents.

2. X_it+1 can also be seen as the net supply by momentum traders in Kelsey, Kozhan, and Pang (2011). These momentum traders base their trade at time t+1 on the previous price change from P_t−1 to P_t Thus, X_it+1 is non-random at time t+1.

3. With mean-variance preferences where b_ij is the covariance between returns on asset i and asset j and τ is the common preference trade-off between mean and variance of return sensitive traders. Then the hth trader's demands are , where is the m×1 vector of h′ s expected returns. ι is the m×1 unit vector. B is the covariance matrix of returns at time t+1 as believed by each trader.

4. Without loss of generality, if A was singular we have redundant assets and can consider a smaller set of markets.

5. In addition, the expectation errors or the noise trading shocks could take a more complicated form e.g. the expectation errors or the noise trading shocks may be auto-correlated.

6.

7. The BSE 100 index is a broadbased stock index. It has 1984 as the base year and was launched in 1989. It is composed of top 100 constituents listed in the BSEs in Mumbai, regarded as the oldest stock market in Asia.

8. The AKM composite index is a market value-weighted index of capitalization of all stocks listed in the Russian stock market with the base date on September 1, 1993.

9. The monthly variance of returns is its variance across weeks in the month. Thus, our sample variances and covariances are calculated without nonoverlapping data periods.