# Does risk aversion of utility function cause the existence of positive interest rate?

In standard macro model, it is usually time preference that causes positive interest rate. But is there anything to do with risk aversion of utility function that causes existence of positive interest rate?

Yes and no; it depends on which interest rate you look at.

You are right in that risk aversion affects interest rates, but the direction can go both ways. In what follows I look at an economy with risky (stochastic) and non risky assets and risk averse agents. For the thought experiment, we increase the volatility of the risky asset, "increasing its riskiness".

### Risky Asset

Despite having the same expected return, due to Jensen's inequality, the risky asset gives less utility to the agents. The risky asset will need to pay a higher interest, a risk premium, to attract investors.

### Safe Asset

We have increased the aggregate risk of the economy by making the stochastic asset more volatile. Agent's added utility from insurance (through safe assets) has increased. Demand for safe assets at the old interest increases. Holding the supply of safe assets fixed, this means that the safe asset's interest rate will go down.

To understand this fully, you might want to write down a two period model with two different assets, one standard normal with standard deviation $\sigma$ and one deterministic. Examine how interest rates for both assets change if you increase $\sigma$.

• I think stochastic agent death can also influence the safe asset rate. This can be through a lower beta but I don't think it has to. It depends on what the utility of being dead is. – BKay Feb 26 '15 at 15:12

Another way to look at it is this. The utility function of the representative agent affects the SDF, but we have some conditions that the resulting SDF must satisfy in typical economies.

A stochastic discount factor (SDF) exists if and only if there is no arbitrage. (It is unique when markets are complete.) A key point is that a SDF must is a strictly positive process. Recall the definition of a SDF.

A stochastic discount factor is an adapted stochastic process $\pi$ where $\pi_0 = 1$; $\pi_t > 0$ for all $t$; for each time $t$, $\pi_t$ has finite variance; for any basic asset $$P_{it} = E_t \left [ \sum_{s=t+1}^T D_{is} \frac{\pi_s}{\pi_t} \right ],$$ for all $t \in \mathcal T$.

Now, this gives us the relationship $1 = E_t\left [ \frac{\pi_{t+1}}{\pi_t} R_{t+1} \right ]$. Then, the risk-free rate is just \begin{align} E_t\left [ \frac{\pi_{t+1}}{\pi_t} R_{t}^f \right ] = 1\\ R_t^f = \frac{1}{E_t\left [ \pi_{t+1}/\pi_t \right ]}. \end{align}

So, in some ways, we can just say that the positivity of the risk-free rate is just due to the assumption of no arbitrage.