Equilibrium in Securities Markets with Heterogeneous Investors and Unspanned Income Risk

We provide the first closed-form solution for the equilibrium risk-free rate and the equilibrium stock price in a continuous-time economy with heterogeneous investor preferences and unspanned income risk. We show that lowering the fraction of income risk spanned by the market produces a lower equilibrium risk-free rate and a lower stock market Sharpe ratio, partly due to changes in the aggregate consumption dynamics. If we fix the aggregate consumption dynamics, the Sharpe ratio is the same as in an otherwise identical representative agent economy in which all risks are spanned, whereas the risk-free rate (and the expected stock return) is lower in the economy with unspanned income risk due to an increased demand for precautionary savings. The reduction in the risk-free rate is highest when the more risk-averse investors face the largest unspanned income risk. In numerical examples with reasonable parameters, the risk-free rate is reduced by several percentage points.