Risk Aversion and Equilibrium Optimal Portfolios in Large Markets

We consider a simple continuous-time economy, populated by a large number of agents, more risk averse than the log agent, with hetero- geneous risk aversion densely covering an interval. Even though the dividend is a geometric Brownian motion, the equilibrium investment opportunity set is stochastic and optimal portfolios are highly non- trivial and non-myopic. We present closed form asymptotic expres- sions for the optimal portfolios when the horizon, or the volatility of terminal dividend becomes large. The non-myopic component of the optimal portfolios is always positive and monotone decreasing in time. For each moment in time, there is a threshold risk aversion such that the non-myopic component is increasing (decreasing) in risk aversion for risk aversion below (above) the threshold. The threshold risk aversion is monotone decreasing in time and approaches the value of two at the terminal horizon. The phenomena we obtain are markedly different from the corresponding results in 2-agent economies.