How Consumption Volatility Affects the Risk-Free Rate and Equity Premium
Summary
The document explains the standard consumption-based asset pricing intuition for how lower consumption growth volatility affects interest rates and the equity premium. Investors value smoother consumption, so higher uncertainty can increase precautionary saving. Greater demand for safe assets raises their prices and lowers risk-free rates, while reduced demand for equities lowers their prices and raises expected stock returns, widening the equity premium.
In a simple Lucas tree setup, the equity premium is proportional to risk aversion times consumption growth variance. Reversing the argument, lower volatility reduces precautionary saving, tends to lift the risk-free rate, and can raise stock prices enough to lower expected equity returns and the premium. This is a stylized result that depends on the model assumptions; the document does not explore richer preferences, changing investment opportunities, or empirical evidence.
Key ideas
- Investors may save more when consumption growth is more uncertain to smooth future consumption.
- Greater precautionary saving can raise safe asset prices and lower risk-free rates.
- Lower demand for equities can reduce stock prices and increase expected equity returns.
- In a simple Lucas tree model, the equity premium rises with risk aversion and consumption growth variance.
- The direction of the effects reverses when consumption volatility falls, subject to the model’s assumptions.
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Full text
# Asset Pricing: What happens to the Risk-Free rate and the Equity Premium? # 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 ## Answer by Phun (score 2) https://quant.stackexchange.com/a/27622 When investors make decisions in the market, they maximize utility. The utility in turn depends on consumption volatility. An overall goal of the investors is to smooth consumption, so they can consume in each point in time roughly the same amount. 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 $$ dC_t = C_t(\mu_C dt + \sigma_C dW_t^C) $$ Edit: Just to be clear. The above explanations hold the other way around as well. So, if consumption volatility goes down, investors don't wish to save as much as before. Therfore the interest rate goes up and, since they invest more in stocks, stocks get more expensive and subsequent expected stock returns go down. This results in a lower ERP. ## Answer by Christian S. (score 0) https://quant.stackexchange.com/a/26038 Short answer: Individuals have a preference for smooth consumption streams. To be able to smooth consumption if consumption is more volatile, individuals save more driving down risk-free rates.
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