\(\gamma\)
10.0
Coefficient of relative risk aversion
\(\sigma_g\)
2.0%
Std. dev. of log consumption growth
U.S. data
Equity premium6.9%
Risk-free rate1.0%
\(\sigma\)(market)16%
Market Sharpe ratio0.43
Fixed: \(\mu_g = \rho = 2\%\).
The joint \(r^f\) and ERP puzzle
The model's predicted \(r^f\) and ERP as \(\gamma\) varies (navy), with U.S. data (star) for reference.
Mehra–Prescott (1985)
Lucas one-tree, CRRA, i.i.d. lognormal consumption growth
\(\mathrm{ERP} \;=\; \gamma\,\sigma_g^2\)
\(r^f \;=\; \rho + \gamma\mu - \tfrac{1}{2}\gamma^2\sigma_g^2\)
At \(\gamma = \)10.0:
ERP0.40%
\(r^f\)20.00%
Hitting ERP \(=\) 6.9% requires
\(\gamma = \)172.5
which gives
\(r^f = \)−248%
The Hansen–Jagannathan bound: \(\sigma(M)/E[M] \geq S\)
Any positive SDF must lie above the gold line. CRRA only gets there at high \(\gamma\).
Hansen–Jagannathan (1991)
Any positive SDF pricing a return with Sharpe ratio \(S\)
(no preferences needed)
(no preferences needed)
\(\dfrac{\sigma(M)}{E[M]} \;\geq\; S\)
For CRRA: \(\dfrac{\sigma(M)}{E[M]} = \sqrt{e^{\gamma^2\sigma_g^2}-1} \;\approx\; \gamma\sigma_g\)
At \(\gamma = \)10.0:
CRRA \(\sigma(M)/E[M]\)0.20
Bound \(S\)0.43
CRRA reaches the bound at
\(\gamma = \)20.6