What is the significance of the Hansen-Jagannathan bound?
Fundamental theorem of asset pricing tells us that if there is no arbitrage, there must exist a positive random variable $M$ (also called stochastic discount factor) such that for any return $R$, we must have $1 = E[M R]$. Thus to understand asset prices, in cross-section as well as across time, we need to understand which factors affect $M$. 

Now, the most basic Hansen-Jagannathan bound can be stated as

$$
\left| \frac{E[R^e]}{\sigma[R^e]} \right| \leq \frac{\sigma[M]}{E[M]}
$$

where $R^e$ is an excess return. The main contribution is that it allows us to say something about moments of $M$ (right-hand side), which is unobservable, in terms of moments of returns (left-hand side), which can be (in principle) observed. Specifically, given the observed Sharpe ratio (say, around 0.4), the bound tells us that the SDF must be at least just as volatile.

The problem is that standard consumption-based models for $M$ have problems matching this volatility. For example, your basic Lucas-tree economy with CRRA utility leads to $M = \beta \exp(-\gamma \Delta c)$, where $\Delta c$ is log consumption growth. Since $E[M] \approx 1$, we have $\sigma[M] \approx \gamma \sigma[\Delta c]$. If $\sigma[\Delta c]$ is let's say around 1% yearly, we need to set $\gamma &gt; 40$, an unrealistically high value of risk aversion.