What is the relation between Relative Risk Aversion and Market Price of Risk

Question

If we assume that the preferences of investors in a market aggregate to display the following utility function

$$u(W)=\dfrac{1}{1-\gamma}W^{1-\gamma},\quad \gamma>0,\quad \gamma\neq1$$

then from$$RRA(W)=-W\dfrac{u''(W)}{u'(W)}$$

$$u'(W)=W^{-\gamma}$$

and

$$u''(W)=-\gamma W^{-\gamma-1}$$

we have that

$$RRA(W)=\gamma$$

if the market price of risk is defined as

$$\lambda=\dfrac{\mu_m-r_f}{\sigma_m}$$

where $\mu_m$ is the expected market return, $r_f$ is the riskless rate and $\sigma_m$ is the market volatility.

Is there any relation between $\gamma$ and $\lambda$?

As investors should require a higher return per unit of risk the more risk averse they are, I would assume that higher $\gamma$ implies higher $\lambda$. However, I am looking for more of a mathematical link between the two if that is possible.

Also, I realize that the answer by quasi in What is the significance of Relative Risk Aversion probably sheds some light on my question, but I did not manage to come any closer unfortunately.

Answer 1

In most economic models the risk aversion coefficient is definitely related to the equity premium.

Assuming utility is CRRA (as you mention):

\begin{equation} U(C_t) = \frac{C_t^{1-\gamma}}{1-\gamma} \end{equation}

Also assume the agent has access to an equity claim and risk free. So that his portfolio follows:

$W_{t+1} = [\alpha_t R_{t+1} + (1-\alpha_t)R_f)(W_t - C_t)$

where $\alpha$ are the weights the investor puts into equity and risk free respectively.

If you make this maximization you get the fundamental asset pricing formula for the case of CRRA preferences (I won't go over the maximization details):

\begin{equation} 1 = E_t \bigg[ R_{t+1} \beta \bigg(\frac{C_{t+1}}{C_t}\bigg)^{-\gamma} \bigg] \end{equation}

Now assume that $R_{t+1}$ and $C_{t+1}/C_t$ are jointly log normal (this is not crucial but allows me to get closed form expressions for the equity premium).

Then you can take logs of the equation above to get:

\begin{equation} 0 = E_t[r_{t+1} + log(\beta) - \gamma g_{t+1}] + \frac{1}{2}[\sigma_r^2 + \gamma^2 \sigma_g ^2 - 2 \gamma \sigma_{r,g}] \end{equation}

where $g_{t+1}$ is log consumption growth.

If you do the same for the risk free rate you get: \begin{equation} 0 = E_t[r^f_{t+1} + log(\beta) - \gamma g_{t+1}] + \frac{1}{2} \gamma^2 \sigma_g ^2 \end{equation}

Subtract both equations to get:

\begin{equation} E_t[r_{t+1} - r^f_{t+1}] + \text{jensen terms} = \gamma \sigma_{r,g} \end{equation}

So the risk premium is proportional to $\gamma$ (and consequently the sharpe ratio).

In most asset pricing models the risk premium will depend on the risk aversion. Different models (utility functions and frictions) might lead to different formulas but almost always there is a $\gamma$ showing up somewhere.