What mechanisms does the market use to brining an asset back to the market line, as defined by CAPM?

Question

The Capital Asset Pricing Model (CAPM) model states that, on efficient market, expected return of an asset should be given by a linear function of its volatility (as measure by standard deviation of returns).

$ avg(R_i) = R_f + \beta_i \cdot (R_m - R_f), $

where $\beta_i$ is the asset "beta" (which measure "riskiness" or volatility of an asset), $R_m$ is the expected return of well diversified / optimised market portfolio and $R_f$ is return of a risk-free asset. The given formula, if plotted, determines the so called "market line".

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Then we say, that because of inefficiencies on the market, there might be assets that are above or bellow the market line. The assets that are above the line are "too good". They provide "too high" expected return for their volatility. So, market players are going to buy this asset and, therefore, the price of the asset is going to grow! So, we can say, that at the moment this asset is underpriced and this is an inefficiency that we can use.

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What is not clear to me, is what are the mechanisms for the "correction" of the deviation from the market line. As I have just describe, the price is going to grow but, why should it change the relation between the expected return and volatility of the asset? To bring the asset back to the marker line, the expected return should decrease and / or the volatility should increases. But so far we just have the growing price!

A similar logic is applicable to the points that are bellow the marker line. They are over-priced, so market players are going to sell it and we need to do so as well. But, how does decreasing price is going to bring the asset back to the market line?

Answer 1

Supply and Demand

That's a great question and your question has a simple answer. @Daneel showed you the ``maths'' behind it.

It boils down to: if an asset has too high returns than the CAPM predicts, then the asset is super attractive. An asset has a certainty riskiness (measured by its (market) beta) which warrants a certain future return (the CAPM formula, i.e. the SML). However, if the real return of an asset is above this theoretical return, then investors are going to buy the asset like crazy. Wouldn't you like to own an asset which pays you more than it should (according to its risk)? As @Daneel and @noob2 say, the rest is Econ 101: $$\text{Demand increases} \implies \text{Price increases} \implies \text{(Expected) future returns decrease}$$ Investors continue to buy the asset until the price has increased so much that the real expected future return equals the CAPM expected return. Then, everything is in equilibrium and (financial) economists are happy.

Conversely, think of an asset which has too low returns. Nobody wants this asset and demand drops. This means the price decreases and real returns increase until they equal the CAPM implied expected future returns.

Price and Returns

Returns are simply defined as payoff tomorrow divided by price today. That's what a return is. The higher the price today, the lower the expected future return.

The inverse relationship between price and return (yield) is particularly emphasised for bonds, applies equally to other assets though.

The Real World

Firstly note that according to the CAPM, an asset's volatility is meaningless. That's not the right measure of risk. It all depends on the (market) beta, the covariance between asset return and market return.

However, the (expected) return of assets in the real world is not predicted by the CAPM. The CAPM is widely rejected. Academics (and practitioners) have known this for decades. The paper from Fama and French (1992) is sometimes called the ``beta is dead'' paper. They essentially show that the CAPM does an awful job at explaining stock returns.

An influential paper was written by Frazzini and Pedersen (2013). They find that stocks with higher betas have lower (!) returns (not higher returns, that's the complete opposite of the CAPM!) The authors explain their findings by margin constraints.

Have a look at the first panel of their Figure (1). They sort US stocks into ten groups of increasing betas. Stocks with low betas (the left bars) have much higher returns than stocks with high betas (the right bars).

Momentum

In the comments, you raise these questions to @noob2.

Do you say that securities that cost more tend to have smaller expected return per day? Or you say that securities that have exhibit price increase recently tend to have a smaller expected return per day? If it it is a know fact, does it have a name? Does it have an explanation? Why should it be like this?

We have already cleared that @noob2 correctly pointed to an increase in today’s price which lowers expected future returns. But what about your second questions?

Stocks which recently (say last 6-12 months) had an increase in price are called ``recent winners''. Empirical studies have shown that recent winners continue to have high future returns. Conversely, recent losers continue to drop in price and have low expected returns. This pattern is called momentum. Some of the seminal papers about the topic were written by Jegadeesh and Titman (1993) and Carhart (1997). There's some debate about short-term reversals (stocks that dropped last month go up this month) and long-term reversals (stocks that went up during the last three years have low future returns).

What is the reason for this momentum pattern? That's the jackpot question! Behavioural finance points to investors extrapolating returns and bias of recent information. Rational models could say that firms own valuable growth options. Recent price surges increase the value of these growth options and tilt the firm value away from low-risk assets-in-place to riskier growth options. Thus, recent winners are riskier and deserve to have higher returns. This is a rational explanation for momentum. There are many more potential explanations for momentum and the jury is out there to determine which side is right. Interestingly, momentum exists in many asset classes (Asness, Moskowitz and Pedersen (2013)) but not for Japanese stocks (Fama and French (2012)).

Answer 2

For simplicity let us assume we are considering a single investment period, that is from $t$ to $t+1$. Let $S_i(t)$ be the price of the asset $i$ at time $t$. Then the return of the asset between $t$ and $t+1$ can be decomposed into its price return and any dividends paid: $$R_i=\frac{S_i(t+1)-S_i(t)}{S_i(t)}+D_i$$ where $D_i$ are the dividends paid during the period. Assuming dividends are known, the expected return at time $t$ is: $$\begin{align} E(R_i)&=\frac{E(S_i(t+1))-S_i(t)}{S_i(t)}+D_i \\ &=\frac{E(S_i(t+1))}{S_i(t)}-1+D_i \end{align}$$ If the return $R_i$ is attractive enough compared to the volatility, then investors will be interested in purchasing the asset. The price to pay to purchase it is simply equal to $S_i(t)$. Because the demand of the asset will increase with respect to its supply, market dynamics will push the price $S_i(t)$ up to $S_i^\prime(t)$: $S_i^\prime(t)>S_i(t)$. Hence: $$\frac{E(S_i(t+1))}{S_i^\prime(t)}<\frac{E(S_i(t+1))}{S_i(t)} \quad\Rightarrow\quad E(R_i^\prime)<E(R_i)$$ That is the expected return will be lower.

Answer 3

There are two slightly different kinds of market "error" to correct here; and the mechanisms how this happens depends on which you're dealing with. These are separate from the whole question of if/how useful the CAPM is as a model in the first place.

Stochastic Errors

The problem here isn't that any asset is "above" or "below" the market line. In theory, that can't actually happen (if you believe in the CAPM). The real problem is they lie above/below above an estimate of the market line, that contains inevitable sampling errors relative to the true market line. This can happen because different markets and assets have good and bad years, that are not usefully representative of their behaviour.

Half of the observations used to compute the market line were "lucky" and half "unlucky" in your sample. compared to their true levels. Some of them might even just have been lucky/unlucky at the right/wrong time, generating (spurious) correlations with other assets that affect the calculation of their beta.

Wind the clock forward, providing new data, and all these lucky stories need to stay lucky (and vice versa) to keep the estimate of the market line unchanged. More likely, the market line adapts to a different set of random errors in the new sample! Even if momentum effects continue to generate positive returns on average, the market line will correct unless momentum stays as strong in the future as it was in the past. That's a heroic ask.

The key point being that you don't necessarily need an explicit "reversion to the mean" process to actively correct CAPM's error. A lot of them will come out in the wash, simply given a "reversion back to basic randomness".

Fundamental Errors

The problem above involves investors mis-measuring market behaviour. But the market concensus can also make explicit mistakes about its investible assets. Nobody ever said that efficient markets are infallible. The market will make mistakes, all of the time. They're "efficient" if the nature of those errors is unpredictable. And correcting these kinds of market error does require some kind of mechanism.

The missing link here, in terms of how the market's correction of consensus errors relates to CAPM is the implicit return in the valuation of most (but not all) financial assets. This obviously moves assets up or down the y-axis in traditional CAPM terms. It can also move them a little bit around the x-axis of vol as well (due to convexity effects); but this is secondary, complicated, and people still argue about it.

So a simple worked example. It's Dec31 1999. 10y real yields are 4% (ie TIPS, Treasuries closer to 6.5%); and S&P PE is >30x, ie a <3% earnings yield. Both stock and bond markets have drunk the Kool-Aid about strong US growth persisting; which is too good to be true. Consensus doesn't know it at the time, but markets are going to spend the next 5 years adjusting to a "normal" that's more like ~2.5% real growth priced.

So for bonds - in ballpark terms, you get a -200bp * -8y (roughly the duration of a 10y) / 5 years = +320bps a year revaluation +450 running real yield = +7.7% forward real returns.

So for stocks - also ballpark - earnings do a little better than GDP, say 3.25% real. 100 EPS goes to 117 after 5 years. De-rated from 30x to 20x, the market is down 22%. Chuck in 5 years of dividends at 1% (of the purchase price in 2000), you end up in a similar place to the 17% decline in the S&P index over this period. Or -3.65% real forward a year.

Et voila, you get a very very different market line to the one you would get if you modelled what investors thought they were going to get on 31/12/99. Which itself would be very different looking back on 31/12/99 at the period leading up to then, and building a market line based on the 1990s.

The essence of the problem here, even assuming CAPM is correct, is that there is a different market line for every type of return you can describe. What investors historically got (in hindsight); what they will actually get in future; what they think they will get beforehand; and what they think they would need to get to switch are four related but distinct concepts. As such, if the associated return values are not identical, you have multiple CAPM market lines.

Answer 4

Edit: my answer below assumed that when the OP asks "there might be assets that are above or bellow the market line", he's referring to the Capital Market Line. In fact OP's question seems to be about the Security Market Line, so the answer below is somewhat redundant. Regard the below as an "alternative" view of the CAPM model...

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Derivation of the Capital Market Line

Let me add a couple of points that I feel are missing (basically, the rather long intro below aims to explain why it should never be possible for any one asset to be above the Capital Market Line in the CAPM model).

In the CAPM world, we have the following three base features:

• We assume the existence of some "risk-free" asset (could be a government bond free of credit risk): the return of the risk-free asset is denoted $r_f$, and is assumed non-stochastic (that's why it's risk-free)

• Risky assets, i.e. stocks, have stochastic returns (denoted $r_i$)

• Investor's only measure of "risk" is the volatility of these stochastic returns. The volatility is the standard deviation, denoted $\sigma_i$

Furthermore, the key feature of the CAPM model (in my opinion) is the fact that when one constructs a portfolio that is a combination of the risk-free and risky assets by allocating proportions of one's wealth into each asset, the expected return of the portfolio is simply the weighted average of the individual assets' expected returns, whilst the standard deviation of the portfolio can benefit from the fact that: $Var(\omega r)=\omega^2 Var(r)$.

Assuming three risky assets and one risk-free asset, and denoting our portfolio as $\Pi$, we have:

$$\mathbb{E}[r_{\Pi}]=\omega_1\mathbb{E}[r_1]+\omega_2\mathbb{E}[r_2]+\omega_3\mathbb{E}[r_3]+\omega_4r_f$$

Whilst the Standard deviation of the portfolio would be:

$$\sigma_{\Pi}^2=\omega_1^2Var(r_1)+\omega_2^2Var(r_2)+\omega_3^2Var(r_3)+2\omega_1\omega_2Cov(r_1, r_2)+2\omega_1\omega_3Cov(r_1, r_3)+2\omega_2\omega_3Cov(r_2, r_3)$$

(note that the sum of all weights $\omega_i$ has to add up to 1).

Because of the above linearity of returns but non-linearity of the Standard Deviation, it is possible to allocate weights to each asset such that for a given level of expected return (anywhere between the lowest-return asset and the highest return asset in the portfolio), the standard deviation of the portfolio at that selected level of expected return can be lower than any individual asset at that selected level of expected return.

I try to show this on the plot below:

Above, the risk-free asset has a return of 5%, whilst the risky assets have expected returns & standard deviations: $\mathbb{E}[r_1]=10\%$, $\sigma_1=15\%$, $\mathbb{E}[r_2]=15 \% $, $\sigma_2=20\%$, $\mathbb{E}[r_3]=20\%$, $\sigma_3=35\%$.

The risky assets are the three red balls.

The small red, yellow and grey balls are portfolios that consist of various weights of only two of the three risky assets.

The larger blue balls are portfolios that consist of various weights of all three risky assets. (Note that the portfolio that consists of all three assets has better risk-reward profile than the portfolios that consist of only two assets or only the individual assets).

The blue line connecting the risk-free asset and one of the portfolios that consists of all three risky assets is the Capital Market Line (CML): this line is simply a linear combination of the risk-free asset and that one specific portfolio (the larger blue ball on which the line is tangential). We can see from the graph that it is not possible to achieve a more optimal portfolio for any given level of risk than the corresponding point on the CML line.

Why this lengthy introduction? Because in the CAPM model, the CML line is always a combination of the risk-free aset and all other risky assets. Therefore if any one asset becomes "miss-priced" (i.e. $S_i(t)$ decreases so that the future expected return $\mathbb{E}[r_i]$ increases, or $S_i(t)$ increases so that $\mathbb{E}(r_i)$ decreases), the whole CML line will move (because it "contains" all assets).

So it should never possible in CAPM for any individual asset to be above the CML line!

Otherwise Daneel's answer clearly explains how changes in current stock price $S_i(t)$ affect the future expected return.

Derivation of CAPM from CML line & Efficient Frontier

The idea that the portfolio consisting of the entire market (i.e. all securities) is the optimal portfolio for any given risk-tolerance, can be used to derive the CAPM model as follows:

Suppose that the marginal contribution of each individual asset to the optimal portfolio (with expected return $\mathbb{E}[r_M]$ and risk $\sigma_M$) is:

• Return: $\mathbb{E}[r_i]$
• risk: $\frac{\sigma_{iM}}{\sigma_M}$

Define the Risk-Return Ratio for any one asset as $RRR_i:=\frac{\mathbb{E}[r_i]-r_f}{\sigma_{iM} / \sigma_M}$. Because the optimal portfolio cannot be improved any further, the $RRR_i$ of all assets must be the same and equal to the overall $RRR_M$ (otherwise if any one asset had a superior $RRM_i$ we could add a bit more of that asset into the portfolio to improve the overall $RRR_M$). So we can write:

$$RRR_i=RRR_M \longleftrightarrow \frac{\mathbb{E}[r_i]-r_f}{\sigma_{iM} / \sigma_M} = \frac{\mathbb{E}[r_M]-r_f}{\sigma_{M}} $$

Re-arranging:

$$\mathbb{E}[r_i]-r_f=\frac{\sigma_{iM}}{\sigma_M^2}(\mathbb{E}[r_m]-r_f)$$

Above, $$\frac{\sigma_{iM}}{\sigma_M^2}:=\beta_i$$

So finally we can write:

$$\mathbb{E}[r_i]-r_f=\beta_i(\mathbb{E}[r_m]-r_f))$$

Removing the expectation, we can get the following formula:

$$r_i-r_f=\beta_i(r_M-r_f)+\epsilon_i$$

Where $\epsilon_i$ is some random noise term with $\mathbb{E}[\epsilon_i]:=0$

Additional Thoughts on Stock Returns & Earnings

One additional point that came to my mind (not specifically related to CAPM, but that might help to understand Stock returns conceptually): stock price is strongly linked to reported earnings, because these are in turn linked to the dividends (when earnings beat estimates, stock price goes up, when earnings disappoint, stock price goes down).

Prior to earnings, the market just tries its best to "estimate" the actual earnings that will be reported. Suppose that once earnings get reported at time $t_E$, the market forces will quickly establish the "fair" stock price that exactly reflects those earnings as $S(t_{E})$. Prior to earnings at time $t<t_E$, the stock price $S(t)$ will fluctuate as the market buys or sells the stock.

The ultimate "return" that will become apparent right after earnings get reported will be $\frac{S(t_E)-S_t}{S_t}$ (so it will depend on which $S(t)$ one has bought the stock at): but in the simple fraction above, $S(t_E)$ is "exogenous": i.e. the stock price $S(t)$ prior to $t_E$ does not affect the earnings that will materialize in any way. These earnings are completely independent of the stock price and managed by the company management and employees.

So to sum it up, I conceptually treat the earnings that are reported every year as a lottery ball that will be drawn at time $t_E$ out of a black bag and will reveal the earnings. Prior to that, supply and demand will drive the price of $S(t)$. When the black ball is revealed and the market will quickly establish the true value of $S(t_E)$, your "return" for that year is then clearly a function of the level at which you purchased $S(t)$ prior to $S(t_E)$ being revealed. And then the game starts again for the earnings to be reported the following year, etc...

Volatility of earnings (i.e. how stable the company is) should then in some way translate into the standard deviation of returns on that stock, and this should be naturally reflected in the CAPM beta.

Answer 5

This is the main part of the question, as I understand it:

"As I have just describe, the price is going to grow but, why should it change the relation between the expected return and volatility of the asset?"

There is only one way to price an equity security, and that's using the basic Dividend Discount Model. The free-cash-flow model, residual income model, and abnormal earnings models all use accounting methods and mathematical manipulation to link earnings/cash flow back to the dividends an investor receives on their stock. This is shown "Valuation Models: An Issue of Accounting Theory" by Stephen H. Penman, the co-author of the abnormal earnings model.

So as a reminder, here's the price of a stock:

$P_i=\sum_{t=1}^{\infty} D_t*d_t$

Where $D_t$ is the dividend at time t and $d_t$ is the discount rate at time t, and $P_i$ is the price of security i. How is the discount rate calculated? Well, CAPM says it is calculated using the formula you provide:

$avg(R_i) = 1/d_t = r_f + \beta_i *(r_m-r_f)$

This gets into the idea that security prices, when taking into account risk premiums, are martingales. They have expected return built-in. Prices go up when people think the market is inaccurately estimating future expected returns (discount rates) or when people think the market is inaccurately estimating future dividends.

Your understanding of how CAPM affects price is incorrect. If the expected return is "high", that means the price is too "low". That is, the expected return will revert down to its equilibrium, thus reducing the rate at which you discount future cashflows and increasing your price. On the flip side, if expected return is "low", that means the price is too high, because you're not discounting your cashflows enough. Thus, as expected return reverts up to its equilibrium, the price will drop, because your discount rate is growing.