Short answer
It's complicated. A satisfactory solution is not known.
Long answer
A satisfactory solution is not known and research is ongoing. That doesn't mean there is nothing interesting to say about it.
The phrasing in the question is not entirely correct:
First off all, there's is no risk free arbitrage between bonds and stocks. Both are risky and it's impossible to construct a risk free long/short position in both. So it's not possible to arbitrage anything away.
Second, as @emcor notes: there seems to be an excess risk premium in the risk adjusted returns. In order for the premium to disappear requires choosing parameters in the standard risk-return frameworks that are inconsistent with findings from behavioural finance and human behaviour.
In order to give quantitative answers, wee need a model for the relation between risk, return and behaviour. I'll be heavily borrowing from prof. Cochrane notes (prof. Cochranes course on Coursera starts with this subject).
Some facts
Over the period 1927 to 2002 (note that this timespan includes the Great Depression) we have the following return statistics:
Bond | Stock - Bond
Mean annual % return: 1.1 | 7.5
Standard Deviation: 4.4 | 20.8
So approximately for every \$100 you borrowed you would have made \$7.5. However, this strategy is risky in the short run as the volatility is huge and not obvious for those living in 1927.
What does an utility maximizing agent do?
Maximize his utility, off course! This objective can be modelled as follows: Let $u(c)$ denote the utility of consuming \$$c$ and let us restrict ourselves to 2 time periods, $t$ and $t+1$, then in the optimum we have
$$u'(c_t) = \mathrm{E}\left[\beta u'(c_{t+1})R_{t+1}\right]$$
where $\beta$ denotes the discount for consuming in the future and not now and $R_{t+1}$ the return from $t$ to $t+1$. This equation states that the marginal utility of spending one dollar now should be equal to spending one dollar in the future. For now we use
$$u(c_t) = c^{1-\gamma}$$
where $\gamma$ is the coefficient of risk aversion.
Putting these together
We can combine theory and fact to check whether the equity risk premium is justified. Cochrane derives that for the chosen utility function $\gamma = 53$ should hold. This implies that someone earning 30k/year would pay $\approx$ \$9.430 to avoid a 50/50 bet on \$10.000. This seems wrong... This also has some crazy implications for the interest rates, see the notes.
Conclusion
Established economic and financial theory does not have the answers to this question. Possible explanations are:
- The utility function $u(c) = c^{1-\gamma}$ is wrong.
- A richer model of consumption is needed implying other consumption data over more periods.
- Risk seeking behaviour should be explained on the individual level, not as an economic average.
- People fear financial meltdowns more extreme than we have seen.
- The effect isn't really there. Stock returns will be lower in the future.
- Other markets didn't haven't had these returns. The American stock market is an anomaly.
- Regulations favour bonds.
- Holding bonds by banks is a favour to their clients.
The size of these effects is subject of ongoing research.
To conclude: whatever the reason is, people seem to really like bonds. This could be caused by an extreme preference to low volatility, a seemingly irrational utility curve or other advantages not captured in the mean and standard deviation.