After years of mathematical finance I am still not satisfied with the idea of a risk premium in the case of stocks.

I agree that (often) there is a premium for long dated bonds, illiquid bonds or bonds with credit risk (which in fact they all have). I can explain this to my grandmother. My question is very much in the vein of this one. Yes, investors want to earn more than risk free but do they always get it? Or does the risk premium just fill the gap - sometimes positive sometimes negative? Finally: Do you know any really good publications where equity risk premium is explained and made plausible in the case of stocks? Does it make any sense to say "with $x\%$ volatility I expect a premium of $y\%$"? Sometimes stocks are just falling and there is risk and no premium. What do you think?


4 Answers 4


If you have the mathematical sophistication, you should review the original papers referenced on the Equity Premium Puzzle page, particularly Mehra and Prescott (1985). Note, however, that contrary to other opinions on this page, the puzzle is NOT that there is an equity risk premium. On the contrary, the puzzle is that the premium had been so high, at least empirically up to the point in time those papers were written. More recently, it has been in vogue to claim that the risk premium has recently been too low, or perhaps even nonexistent. Some also question whether there is a monotonic relationship between risk and reward in the markets (see, e.g., low volatility anomaly). But no serious thinker believes there should be no risk premium at all.

The explanation you can "tell your grandmother" is that all stocks are issued by companies to raise money. Since companies fail a lot more often than governments, particularly governments such as the US that can control their own currency, they must at least promise to investors to pay them more than what they can get "risk-free" from the government. In practice, of course, some companies will fail, and sometimes entire economies will "fail" in the sense of underperforming expectations, and actual returns to investors may not match what was promised. That does not change the fact that what was promised is always greater than what could be gotten "risk-free" from a large, stable government. That is the equity risk premium.


This is the equity premium puzzle. (See that article for references.)

My thoughts are that individual investors are rational to be risk-averse and demand a premium for bearing a type of market risk that cannot be diversified away. This risk is actually worse and more insidious than it appears, because "personal" circumstances tend to correlate in undesirable ways with the market: when times are booming and people have steady incomes and money to invest, it's likely to go into the market at a high, and not much premium is demanded; conversely, when times are a little tougher, with wages and income tight and stagnating, and costs of education, healthcare, gas etc. skyrocketing, people are likely to forgo even a substantial equity premium in favor of a conservative rainy day fund, simply because they feel that money subject to market risk is not likely to be there when they are likely to need it the most, (in the event things get even worse.) In other words, rational investors may be concerned that market crashes correlate with mass layoffs, or other possible losses to their own businesses or livelihoods.

It is puzzling to me that even long-term bonds are to all appearances at negative real yields, when these are more market-determined than affected by central bank policies anyways. It either reflects profoundly low long-term expectations for economic growth, or extreme long-term risk aversion, and the two are almost impossible to tell apart. In fact, substantial losses will be incurred by bondholders if interest rates revert to the mean or rise even moderately, but investors in the long bond must be willing to pay this price for some protection in the event of a Japan-like secular bear market with perpetual ZIRP, and if the bonds do lose value in the medium term, (and they most likely will,) that will most likely be because the economy will have recovered, and with it the stock market, and many other opportunities to recoup those losses.

Despite the Orwellian proclamations of the "new normal" by the elite economic punditry, and even though health care and education are outrageously out of reach of the "middle class" and are still subjected to double-digit cost increases every year, no, they have not always been so exorbitantly expensive, and in fact there was a time when a young man could work his way through college by a summer job, and even see the doctor once in a while, and not even that long ago, there was even a time when you could teach children thrift by the virtues of compound interest without being scoffed at. And even though they say we've always been at war with Eastasia, I believe that tomorrow will bring an even newer normal, and the economy will in time heal. I have to conclude that there is at present a substantial equity risk premium.

  • $\begingroup$ Thank you for pointing to the wikipedia article and your other comments. $\endgroup$
    – Richi Wa
    Commented Jul 6, 2012 at 10:35
  • $\begingroup$ Another indication of an elevated equity risk premium is the dried-up IPO market and the booming junk bond market. Because of the premium demanded by investors, companies are finding it too expensive (in terms of dilution) to issue equity, especially when they can issue junk debt for investment-grade interest rates. $\endgroup$
    – JL344
    Commented Jul 7, 2012 at 2:40
  • 1
    $\begingroup$ The equity risk premium puzzle is not that it exists at all, just that it has been so large in the past (as of when the papers were written). $\endgroup$ Commented Jul 9, 2012 at 15:52
  • $\begingroup$ @JL344 What you say is not true. There is still a positive spread between junk and investment grade rated debt. Also, you are suggesting that it has become more popular for private (pre-IPO) companies to raise money via the corporate bond markets than via equity, which I'm not sure is true. $\endgroup$ Commented Jul 9, 2012 at 16:00
  • $\begingroup$ @TalFishman Of course you're right in that companies owned privately are no doubt more likely to borrow privately, too, but in any case at lower than normal interest rates. I didn't mean to refer to the spread per se, but to the ability to borrow at rates only highly rated companies can in normal times. And I still feel that a higher than usual equity premium is being demanded by investors today in public and private markets. $\endgroup$
    – JL344
    Commented Jul 9, 2012 at 17:33

Another observation that the connection between return and risk is not that straightforward (and in contradiction to modern portfolio theory!) is the low-volatility anomaly.

It turns out empirically that stocks that have low-volatility or low-beta show higher returns than high-volatility or high-beta stocks.

See also this question and answers:
Why does the minimum variance portfolio provide good returns?

See also my new answer here: https://quant.stackexchange.com/a/25724/12

  • 1
    $\begingroup$ Thank you for this link. I know minimum variance - and yes, it often does a really good job. $\endgroup$
    – Richi Wa
    Commented Jul 6, 2012 at 10:35

This one is far from straight-forward, although bear with me. It is possible to infer from first principles an ERP reasonably close to normative consensus expectations.

The attached from Howard Marks at Oaktree is a classic: "Everything you wanted to know about the equity risk premium (and much more)". The simple point is that there are four different equity risk premiums out there. These are variously swapped around, usually with little regard for the fact they measure different things.

So let's take as a not-obvious given that we are actually talking about the same ERP! There is then the dilemma of why it seems to be so large. Others have highlighted Mehra and Prescott's work on the "Equity Risk Premium" puzzle.

I would also throw in Robert Shiller's seminal work on Dividends and Volatility. In a nutshell, equity markets are much more volatile than they "should" be. So whatever "premium" anyone receives is compensation for a "risk" that doesn't make sense in the first place.

And as @vonjd has observed, there is then the awkward fact of the low-volatility anomaly. IE investors seem to get compensated more for "safer" stocks. The asset allocation team at GMO actually have an interesting justification for that: that for leverage-constrained investors, lottery stocks behave like call options with an associated option-like premium for the long gamma. https://spectruminvestors.files.wordpress.com/2011/11/rethinkingrisk_betapuzzle_1111.pdf

However, AQR then break this down into "betting against beta" and "betting against correlation". They seem to find that it's not just the rewards for volatility that are the "wrong" way round; but investors also seem to get paid for diversification. IE higher returns as well as the traditional diversification benefits, in terms of lower risk! https://www.aqr.com/Insights/Perspectives/Betting-Against-Correlation

Suffice it to say the whole subject is in a pretty big mess right now. Very little of the traditional framework seems to stand up to scrutiny.

However, do not abandon all hope quite yet:

  • no-arb conditions require that stock = cash + future
  • no-arb conditions require that future = call - put
  • black-scholes requires that both the call and the put are priced assuming an expected return of zero.
  • So anyone who buys a stock is synthetically holding cash, buying a call, and selling a put even if none of them ever think about it thus.

There can only be one return distribution for which it is optimal to buy 1.000 calls and sell 1.000 puts. Under any other distribution, there is a superior alternative allocation that the investor should be following.

There's an upside scenario where the call is ITM, put OTM; a flat scenario where both ATM = OTM; and a downside one where the put is ITM, call OTM. With equal probability, what magnitude of market move would you need up vs down to make +1 call -1 put the optimal log-wealth maximising strategy? Anything else, and I shouldn't be buying the stock but doing something else.

Because the strategy is long calls, short puts, it will have an expected return >0. And since this is the implied excess return on the future, you need to add in cash to get to the stock return. IE being ex-cash, this excess return is the risk premium!

On the current Dec S&P500 options strip, with a little calculus and iterative gradient descent, the current ~17% implied vol requires a 2.98% annualised ERP to justify itself. Anything else, and I should be doing something different to just vanilla buying the market. This 3% is remarkably consistent with traditional asset allocation "rules of thumb".

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  • $\begingroup$ thank you for this evolved answer .. it will take me a while to fully digest it. $\endgroup$
    – Richi Wa
    Commented Sep 3, 2019 at 13:21
  • $\begingroup$ In your argument why is "Because the strategy is long calls, short puts, it will have an expected return >0" true? True in the real world measure of in the risk neutral one? And "stock-cash = call - put" ... does this explain it all? $\endgroup$
    – Richi Wa
    Commented Sep 3, 2019 at 15:51
  • $\begingroup$ Hi, the risk neutral is zero (lest there be arb). I’m interested in what expectations about the underlying’s futures (that is its RP) would make +1C -1P the optimal exposure to that underlying. That is the incentive that the market has to offer to hold the underlying. $\endgroup$
    – demully
    Commented Sep 4, 2019 at 18:48
  • $\begingroup$ a side question -- i have heard people referring to first principles here and there. What are the principles? How many are they? Any source/material would be highly appreciated. thanks! $\endgroup$
    – AK88
    Commented Sep 5, 2019 at 1:49

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