# Risk aversion and risk-free rates

New to finance. When I read textbook like Financial Economics by Bodie, I encountered the following idea, namely, higher risk aversion is associated with higher risk premium and lower risk-free rate.

I understand the risk premium part. But I cannot convince myself about the risk-free rate. Higher risk premium does not necessarily imply lower risk-free rate. I thought higher risk aversion should be associated with higher risk-free rate. My reasoning is that as more and more people get highly risk averse, more and more people will prefer risk-free assets to risky assets, the demand for risk-free assets increases. Simple supply and demand story tells us that higher demand of risk-free assets leads to higher prices or higher risk-free rates in this case (of course, if we hold supply constant). I do not know where my reasoning goes wrong.

• Higher risk premium means low risk-free interest rates. – Dhamnekar Winod Oct 5 '18 at 12:26
• The prices and yields of fixed income securities vary inversely. So "higher demand of risk-free assets leads to higher prices or (equivalently) LOWER rates in this case". – noob2 Oct 5 '18 at 12:53
• @noob, that's what I mean to say. – Dhamnekar Winod Oct 6 '18 at 6:37
• Could you make that an answer @noob2? – Bob Jansen Oct 13 '18 at 9:24

As we know from Bonds 101, the prices and yields of fixed income securities vary inversely. So your statement should be amended to read "higher demand of risk-free assets leads to higher prices or (equivalently) LOWER rates in this case".

For example suppose Tbills are priced at 98. Now risk aversion increases and there is a stronger demand for Tbills. The price might go to 99, so now they have a yield of barely 1.01% (I invest 99 and get back 100 a year later), instead of 2 point something previously.

• very nice. I was confused by the supply-demand argument until your clarification. thanks. – mark leeds Oct 13 '18 at 16:50

Higher risk aversion implies lower risk-free rate, because when people are risk averse they demand more risk free assets or lower risk assets. Increase in demand for risk free assets, with their supply fixed leads to a lower price (interest rate).

Just demand and supply!

I find that statement from Bodie weird. The most basic financial economics models have higher risk aversion implies higher risk free.

Assume there is a Lucas tree economy and that the utility is CRRA. Then the fundamental asset pricing formula implies that for any asset:

$$0 = \log E_t \bigg [R_{t+1}\beta \bigg ( \frac{C_{t+1}}{C_t} \bigg ) \bigg ]$$

Take the risk free and assume that $$R_{t+1}$$ and $$C_{t+1}/C_t$$ are jointly log normal:

$$r^f_{t+1} = 1/\beta + \gamma E_t(g_{t+1}) - \frac{1}{2}\gamma^2 \sigma^2_g$$

where $$g_{t+1} \equiv C_{t+1}/C_t$$.

So higher risk aversion implies higher risk free, unless the Jensen inequality term $$\left(\frac{1}{2}\gamma^2 \sigma^2_g\right)$$ starts dominating. As volatility of consumption growth is usually small this is not the case and usually we have:

$$\frac{\partial r^f_t }{\partial \gamma} > 0.$$

This is true in this simple model as well as more generally. Intuitively, the higher the risk aversion the higher the precautionary savings motive of agents so higher demand for safe assets. In general equilibrium regardless of whether the assets are in zero or positive net supply this leads to higher risk free rates.

• It would be great to get a comment from whoever downvoted this. This answer is the only one that explains theoretically the relation between risk aversion and risk free rate. – phdstudent Oct 18 '18 at 7:50
• (Not the downvoter) Regarding your last paragraph: Wouldn't a higher demand for risk safe asset increase their price and thus decrease their return? – Bob Jansen Dec 16 '18 at 18:01