# Why is the equity premium not arbitraged away?

The Equity Risk Premium Puzzle concerns the observation that equity returns are generally greater than bond returns.

The puzzle is well known and widely studied, what is keeping investors from shorting bonds and buying equity? Why hold bonds at all when their expected rate of return is clearly lower. Are there other advantages to holding bonds?

• Hi A.L. Verminburger, welcome to quant.SE! Your question is not on-topic here, see the help for what is – Bob Jansen Jul 26 '14 at 13:11
• I did consider posting it on "Personal Finance & Money" (and please do move it there if appropriate), but wanted to get an answer from people working on the quantitative finance side (also more likely to be involved in trading). – A.L. Verminburger Jul 26 '14 at 14:14
• I don't believe it's inherently bad and I also don't believe it fits very well on Personal. I do believe it can be fixed, the equity premium puzzle is on-topic IMO. Lemme try. – Bob Jansen Jul 26 '14 at 14:46
• I hope you like it. – Bob Jansen Jul 26 '14 at 15:51
• @BobJansen: Good salvage. – Tom Au Jul 30 '14 at 14:27

It's complicated. A satisfactory solution is not known.

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:

1. The utility function $u(c) = c^{1-\gamma}$ is wrong.
2. A richer model of consumption is needed implying other consumption data over more periods.
3. Risk seeking behaviour should be explained on the individual level, not as an economic average.
4. People fear financial meltdowns more extreme than we have seen.
5. The effect isn't really there. Stock returns will be lower in the future.
6. Other markets didn't haven't had these returns. The American stock market is an anomaly.
7. Regulations favour bonds.
8. 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.

Actually, the historical returns, going back to the 1920s, took place in two different ways over two distinct time periods; 1980-present, and 1925-80. This is a more important premise than the fact that stocks have an average total return of 10 percent over the past 80-odd years, and bonds have an average total return of only 5 percent a year over that time.

In 1980, bond yields started in the mid-teens, and average annual bond returns from 1980 to about 2010 were in the low double digits, closely matching that of stocks. So arbitrage strategies might not have worked during that time.

It was in the period from 1925 to 1980, when bond returns averaged more like 3 percent and stock returns were in the low double digits, that your arbitrage strategy might have worked. But there were few arbitrage strategies employed before 1980.

More to the point, the stock and bond markets from about 2010 to the present appear to be more like those of the mid-20th century than those of 1980-2010. Now is as good a time as any to try arbitrage strategies and see what happens.

• Up-voted, nice change of perspective. – Bob Jansen Jul 30 '14 at 15:12
• @BobJansen: Thank you That was out of my 2004 book "A Modern Approach to Graham and Dodd Investing" (Chapter 14). Except for the 2010-present part, which came after 2004. – Tom Au Jul 30 '14 at 18:04

The Equity Premium Puzzle is not that Equities have higher returns than Bonds.

Bonds always have lower required return than Equity, because they present promised cashflows with senior claims over equity shareholders.

The Equity Premium Puzzle is, that Equities have abnormally higher returns than Government Bonds, which means real investors require a higher return on equity relative to a riskneutral investor. This means, real investors are strongly riskaverse.

Explanation on why real investors would be strongly riskaverse is part of ongoing research, where I do quote Wikipedia "no one solution is generally accepted by economists".

• The statement 'the reason for the EPP must be high investor's risk aversion' is not correct. The fact that unreasonably high risk aversion is needed to reconcile the risk of equity and bond returns, indicates that the resolution of the EPP must lie outside the realm of risk preferences. Also, your answer confuses corporate bonds (wich have senior claims over shareholders) and government bonds (which do not have shareholders); the EPP refers to the latter only. – Kiwiakos Jul 27 '14 at 8:44
• @Kiwiakos Thanks for your input. I dont think Governments have no equity though, the equity is rather owned by the public implicitly. – emcor Jul 28 '14 at 12:29

Equity risk premium, in theory, should not be zero, given the relative risk profiles of equities and bonds. The equity risk premium puzzle refers to how difficult it is to explain the magnitude of historical equity risk premium using standard economic models. I recommend that you take a look at chapter 2 of this document (written by Antii Ilmanen); it provides an excellent discussion on the theories and practice of equity risk premium.