# Asset Pricing: What happens to the Risk-Free rate and the Equity Premium?

What would a standard asset pricing model predict for the risk-free rate and the equity premium, if the volatility of consumption growth fell?

My gut feel is that the equity premium should fall, but I cannot justify why this would be the case.

Note: By 'standard asset pricing model' the question is implying one with the following intuition:

• Assets are priced according to their covariance with consumption growth. Assets which pay off when consumption growth is high (in good times) are risky, hence they will need to have high expected returns for investors to want to hold them

It is clear, the more volatile consumption growth the more uncertain is future consumption. Investors want to hedge this risk by saving money today to have extra money tomorrow to keep their level. The saving is done by putting more money in the bank account and less money into stocks. Therefore, stock (bond) prices go down (up) and thus expected returns (interest rates) go up (down). This increases the Equity Risk Premium. In the simplest case of a Lucas Tree model the Equity premium is given by $$ERP = \gamma \sigma_C^2$$ Where $\sigma_C$ is consumption growth volatility and $\gamma$ the risk aversion. The consumption process is defined via $$dC_t = C_t(\mu_C dt + \sigma_C dW_t^C)$$