Labor income and income from nontraded assets are important sources of wealth for most individuals, cf. Heaton and Lucas (2000) and Campbell (2006), with a potentially large impact on their consumption and portfolio decisions and, consequently, on the equilibrium securities prices. If income risk can be completely hedged by appropriate financial assets, it is straightforward to include income risk in standard models of individual consumption and portfolio choice and in equilibrium asset pricing models. However, a typical individual’s labor income is only weakly correlated with traded financial assets and, thus, it has a large unspanned and unhedgeable component, which complicates these models tremendously. This paper provides the first closed-form solution in the literature for the equilibrium risk-free rate and the equilibrium stock price in a continuous-time economy with heterogeneous investor preferences as well as unspanned income risk.
The equilibrium has the following properties. Lowering the fraction of income risk which is spanned by the market leads to a lower covariance between the dividends of the stock and aggregate consumption and, consequently, a lower stock market Sharpe ratio. The equilibrium risk-free rate decreases due to a higher demand for precautionary savings. If we fix the aggregate consumption dynamics, the Sharpe ratio is the same while the risk-free rate (and the expected stock return) is lower than in an otherwise identical representative agent economy in which all risks are spanned. The reduction in the risk-free rate depends on the magnitude of all investors’ unspanned income risk and their risk aversion. The reduction is highest when the more risk-averse investors face the largest unspanned income risk. Even though our closed-form solution hinges on very specific assumptions about the preferences, the dividend process, and the income processes, our results suggest that unspanned income risk may in more general settings play an important role in explaining the risk-free rate puzzle produced by the standard representative agent models.