Suppose markets are perfectly efficient and asset prices reflect all available information. Under this assumption one expects current prices to be non-biased estimators of future prices. It seems to me that this should impose some upper bound on the returns one expects to receive from holding assets. In particular, I would expect the returns to equal the discount rate of other market participants, as the market needs to at least compensate for deferring consumption.

However, most samples of large-cap stocks have systematically performed better than this, generally generating double-digit returns. Of course, there is some suvivorship bias here, but it seems plausible that when one accounts for this, it still exceeds discount rates. Why is this? What explains these high returns? It seems that either investors must be systematically mistaken with their expectations, or that there are other factors that explain these returns.

Possibly loss aversion might result in the discount of assets with disproportional downside risks, but this explanation should not apply in the age of algorithmic trading, as this aversion is unlikely to be programmed into trading software.

To clarify my question,

Efficient markets: $p_{today}=E(p_{future})$

High returns: $p_{future}-p_{today}>>0$

Which seems to imply either systematic irrationality ($E(p_{future})<p_{future}$), or that something else goes on that explains high returns.

  • $\begingroup$ I really do not understand your statement. Ex-ante conditional on the right set of state variables all stocks shall have the same expected return. Further to that if there is any evidence is that small stocks have outperformed large caps on average. $\endgroup$ – phdstudent Jun 7 '18 at 16:22
  • $\begingroup$ Prices today reflect expectations of future prices, future prices seem to systematically outstrip expectations as we see large price increases over time. Why is this? Are expectations systematically wrong, or is there some other explanation? $\endgroup$ – user526463 Jun 7 '18 at 16:54
  • $\begingroup$ @phdstudent see my clarification. $\endgroup$ – user526463 Jun 7 '18 at 17:03
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    $\begingroup$ A very warm welcome to Quant.SE and thank you for your question. See my answer below. $\endgroup$ – vonjd Jun 7 '18 at 17:11

What you describe is known as the Equity Premium Puzzle - and it really is, as the name says, a real enigma:

"The equity premium puzzle (EPP) is a phenomenon that describes the anomalously higher historical real returns of stocks over government bonds."

Source: https://www.investopedia.com/terms/e/epp.asp#ixzz5HlCdHS2Z

A good first introduction can be found (as always) on Wikipedia: https://en.wikipedia.org/wiki/Equity_premium_puzzle

  • $\begingroup$ Thanks for this, and for your welcome! Really interesting enigma. From wikipedia I see risk aversion, consumption variability, loss aversion, statistical illusions as explanations of this phenomenon. Do you have any idea which explanation is most popular? $\endgroup$ – user526463 Jun 7 '18 at 17:13
  • $\begingroup$ Probably the most common explanations are Habits, fears of Long-run risk or disasters. But that would entail a whole different question. My answer below also sheds some light on your question. $\endgroup$ – phdstudent Jun 7 '18 at 17:17
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    $\begingroup$ Just a comment, the equity premium puzzle is about the level of the returns nothing to do with what OP was asking regarding why expectations are lower than realizations. $\endgroup$ – phdstudent Jun 7 '18 at 17:23
  • $\begingroup$ @phdstudent: Well, technically you are correct, but I think they are closely connected: the expected returns for equities over bonds would be much lower compared to their actual returns. $\endgroup$ – vonjd Jun 7 '18 at 17:27
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    $\begingroup$ @phdstudent is right. Read my answer below to have a clarification on why prices on average should increase without requiring any equity premium puzzle $\endgroup$ – fni Jun 10 '18 at 17:10

Suppose markets are perfectly efficient and asset prices reflect all available information. Under this assumption one expects current prices to be non-biased estimators of future prices.

It is a common mistake to think that market efficiency implies $P_t = E_t[P_{t+1}]$! In general, the correct statements are:

  • $P_t = \frac{E_t^Q[P_{t+1}]}{R_f}$ where $Q$ is the risk-neutral measure (which is different from the physical one!) and $R_f\approx 1$ the risk-free rate

  • $P_t = E_t[M_{t+1}P_{t+1}]$ where $M_{t+1}$ is a stochastic discount factor

You can rewrite the second statement as: $$P_t = E_t[M_{t+1}]E_t[P_{t+1}] + Cov_t[M_{t+1}, P_{t+1}] = \frac{E_t[P_{t+1}]}{R_f} + Cov_t[M_{t+1}, P_{t+1}]$$ For a risky asset with $Cov_t[M_{t+1}, P_{t+1}]<0$, we have that $P_t < E_t[P_{t+1}]$. The price of a risky asset should grow on average!

The equity premium puzzle is a distinct phenomenon and has to do with the fact that it is almost impossible to find a parametrization of the Stochastic Discount factor that:

  1. Depends on consumption growth (or other real variables)
  2. Is consistent with sensible levels of relative risk-aversion (say 1-5 and not 50-80)
  3. Is consistent with the relatively high (5-8%) US Market risk-premium
  4. Is consistent with the relatively low US risk-free rates

In particular you can rearrange the previous equation, noticing that the expected gross return is $E_t[R_{t+1}] = \frac{E_t[P_{t+1}]}{P_t}$, to obtain: $$E_t[R_{t+1}] - R_f = -Cov_t[M_{t+1}R_{t+1}] = -\rho_{M,R} \sigma_M \sigma_R$$ A common Stochastic Discount Factor suggested by the macroeconomic literature has the following form $M_{t+1} = \beta \left(\frac{C_{t+1}}{C_t}\right)^{-\gamma}$. It turns out that it is very hard to make this SDF consistent with asset returns because consumption growth is not very volatile and would require a coefficient of relative risk-aversion $\gamma$ that is implausibly high and would be inconsistent with risk-free rates.

  • $\begingroup$ Indeed! That’s why returns matter more than prices! $\endgroup$ – phdstudent Jun 10 '18 at 17:15

We know that:

\begin{equation} R_{t+1} = \frac{P_{t+1} + D_{t+1}}{P_t} \end{equation}

After some algebra and taking logs we can write the returns as: \begin{equation} r_{t+1} = k + \rho (p_{t+1} - d_{t+1}) - (p_t - d_t) + \Delta d_{t+1} \end{equation}

where is constant $\rho = \frac{P/D}{1+P/D}$.

or: \begin{equation} (p_t - d_t) = k + \rho (p_{t+1} - d_{t+1}) - r_{t+1} + \Delta d_{t+1} \end{equation}

Solve the equation above forward: \begin{equation} p_t - d_t = constant + \sum^\infty_{j=1} \rho^{j-1}(\Delta d_{t+j} - r_{t+j}) \end{equation}

Now solve for return at $t+1$: \begin{equation} r_{t+1} = k - (p_d - d_t) + \sum^\infty_{j=1} \rho^{j-1} \Delta d_{t+j} - \sum^\infty_{j=2} \rho^{j-1}r_{t+j} \end{equation}

Take expectations as of $t$ and $t+1$: \begin{equation} E_t[r_{t+1}] = k - (p_d - d_t) + \sum^\infty_{j=1} \rho^{j-1} \Delta E_t[d_{t+j}] - \sum^\infty_{j=2} \rho^{j-1}E_t[r_{t+j}] \end{equation}

\begin{equation} r_{t+1} = k - (p_d - d_t) + \sum^\infty_{j=1} \rho^{j-1} \Delta E_{t+1}[d_{t+j}] - \sum^\infty_{j=2} \rho^{j-1}E_{t+1}[r_{t+j}] \end{equation}

Subtract the former from the latter:

\begin{equation} r_{t+1} -E_t[r_{t+1}]= (E_{t+1}-E_t)\sum^\infty_{j=1} \rho^{j-1} \Delta [d_{t+j}] - (E_{t+1}-E_t)\sum^\infty_{j=2} \rho^{j-1}[r_{t+j}] \end{equation}

So you can see that returns would be different from expected returns either due to news about cashflows: \begin{equation} CF_{news} = (E_{t+1}-E_t)\sum^\infty_{j=1} \rho^{j-1} \Delta [d_{t+j}] \end{equation}

or due to news about discount rates: \begin{equation} DR_{news} = (E_{t+1}-E_t)\sum^\infty_{j=2} \rho^{j-1}[r_{t+j}] \end{equation}

So indeed prices reflect all future price and dividends expectations. It's unexpected news about the future that make them change.

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    $\begingroup$ Thanks for this! Impressively thorough! But my question is "suppose prices reflect all future price and dividends expectations, then why are returns so high?" I think my question is precisely one about the equity premium puzzle. $\endgroup$ – user526463 Jun 7 '18 at 17:27
  • $\begingroup$ I am not sure I can follow your line of thought. Do you have a source is this just ad hoc? $\endgroup$ – vonjd Jun 7 '18 at 17:28
  • $\begingroup$ @vonjd this the Campbell-Shiller decomposition of returns. Once you have gone through that you might want to look at Campbell-Vuolteenaho. user526463 for the eqt premium puzzle you should look at:(1) Mehra and Prescot, (2) Campbell and Cochrane, (3) Bansal and Yaron and (4) Rietz. $\endgroup$ – phdstudent Jun 7 '18 at 17:30

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