CAPM and beta

Everything so far has been advice to one investor. This chapter asks what happens if everyone takes it, and the answer is a theory of prices rather than of choices.

The move from choice to equilibrium

Markowitz draws the division himself:

My work on portfolio theory considers how an optimizing investor would behave, whereas the work by Sharpe and Lintner on the Capital Asset Pricing Model (CAPM for short) is concerned with economic equilibrium assuming all investors optimize in the particular manner I proposed.

He calls these "part one and part two of a microeconomics of capital markets." Chapters 1 to 6 were part one. This is part two.

The argument, in four steps

Step 1 — everyone holds the tangency portfolio. Chapter 6's two-fund separation. Given the same inputs and the same riskless rate, every investor's risky holding has identical proportions.

Step 2 — therefore the tangency portfolio is the market. If every investor holds risky assets in the same proportions, and all assets are held by somebody, those proportions must be the proportions in which assets exist. The tangency portfolio is the market portfolio — every asset weighted by its total value.

That step is the whole trick, and it is worth pausing on. It converts an unobservable theoretical construct into something with a definition anyone can check. Nobody has to compute a tangency portfolio; they only have to look at what the market holds.

Step 3 — so the only risk that matters is risk shared with the market. An investor holding the whole market has already diversified away everything that can be diversified away. Chapter 3's formula said what survives: the covariance terms. What a security contributes to the risk of a portfolio that already contains everything is its covariance with that portfolio — not its own variance.

Step 4 — in equilibrium, expected returns line up with that contribution. If an asset offered more return than its covariance justified, everyone would want more of it, and its price would rise until it did not. Sharpe states the result:

This shows that in equilibrium there is a linear relationship between the expected returns on securities and their covariances with the market portfolio.

Beta

The covariance is rescaled for convenience. Sharpe describes beta as "a scaled measure obtained by dividing a security's covariance with the market portfolio by the variance of the market portfolio":

$\beta_i = \frac{\sigma_{iM}}{\sigma_M^2} = \frac{\rho_{iM}\,\sigma_i}{\sigma_M}$

The second form is the useful one for reading betas. Beta is the correlation with the market, times the ratio of the two volatilities. So a high beta comes either from high correlation or from high volatility, and those are different situations — a point chapter 8 does a great deal with.

The market's own beta is 1, since its covariance with itself is its variance. A riskless asset's beta is 0.

And beta adds up. Sharpe notes that "portfolio expected returns and covariances with the market portfolio are simply value-weighted averages of the corresponding measures for the component securities", so the relationship "holds for all portfolios as well as for all securities." A portfolio's beta is the weighted average of its holdings' betas — unlike variance, which is not a weighted average of anything. That linearity is most of why beta became the working measure.

The security market line

Putting it together, with a riskless asset available:

$E(R_i) = r_f + \beta_i\left[E(R_M) - r_f\right]$

Expected return is the riskless rate plus beta times the market's excess return. Plotted against beta, every asset should lie on one straight line — the security market line.

Sharpe's own assessment of its standing:

Many would argue that this relationship is the most important single conclusion derived from the CAPM. It shows that expected returns will be linearly related to market risk, but not, as often believed, to total risk

That final clause is the chapter's main claim and the next chapter's title. Two assets with identical volatility can have different expected returns, and two with different volatilities can have the same.

Two lines that are easy to confuse

	Capital market line	Security market line
Horizontal axis	standard deviation $\sigma$	beta
What lies on it	only efficient portfolios	every asset and portfolio
What it describes	the menu available to an investor	the return every asset must offer

The security market line is the stronger statement. The capital market line says what is achievable; the security market line says that everything, including an inefficient single stock, must sit on one line. That is a claim about prices, and it is testable — which is chapters 8 and 10.

What CAPM claims, stated precisely

Worth separating, because the model is argued about loosely.

It does claim that expected returns are linear in beta, that the intercept is the riskless rate, that the slope is the market's excess return, and that no other characteristic of a security affects its expected return.

That last clause is the strong one. CAPM says size does not matter, valuation does not matter, profitability does not matter, past returns do not matter — once beta is known, nothing else is priced. Chapter 10 is about what happened when that was tested.

It does not claim that high-beta assets will outperform. It is a statement about expectations. The Behavioural finance subject's gap between expected and realised outcomes applies here with full force.

And it does not claim that beta measures how much you can lose. Nothing in the derivation mentions loss, drawdown or ruin. Beta is a covariance, scaled.

Why the conclusion is reasonable even though the model is not

Chapter 10 reports that CAPM fails empirically, so it is worth saying what survives.

The central intuition is sound and almost model-free. Chapter 3 showed specific risk vanishes as holdings are added. Anybody willing to hold a diversified portfolio can shed that risk at no cost. A risk that can be shed for free cannot command a premium, because someone will always undercut the price of bearing it. So compensation attaches to what cannot be diversified away.

You can accept that and reject CAPM. The question CAPM answers — which undiversifiable risks are there, and how are they measured — is where it is vulnerable. It says there is exactly one, and that beta measures it. Chapter 10's factor literature says there appear to be several.

So read this chapter's result in two layers. The layer that only shared risk is paid for is robust and is the useful one. The layer that shared risk is one-dimensional and captured by beta against a market index is the contested part.

Working the problem

Two shares, both 30% volatility. Betas 1.3 and 0.5.

Why CAPM says they differ. Using $\beta = \rho_{iM}\sigma_i/\sigma_M$, and taking the market at 15% volatility:

$\rho_1 = \frac{1.3 \times 15}{30} = 0.65, \qquad \rho_2 = \frac{0.5 \times 15}{30} = 0.25$

The two shares are equally volatile and differently connected. The first moves with the market about two and a half times as tightly as the second.

Now put each into a large diversified portfolio. Chapter 3's result: the own-variance contribution gets divided by $N$ and disappears, while the covariance contribution does not. The 30% volatility is mostly irrelevant to a diversified holder of either share; what survives is the part that moves with everything else. For the first share that is 0.65 of its volatility, for the second 0.25.

So they are not equally risky to the investor CAPM describes, even though they are equally volatile in isolation. The first adds more than twice as much undiversifiable risk, and must offer more return or nobody holds it in preference to the second.

With $r_f = 5.5\%$ and a market return of 11%:

$E(R_1) = 5.5 + 1.3(5.5) = 12.65\%, \qquad E(R_2) = 5.5 + 0.5(5.5) = 8.25\%$

What would have to be true for this to make sense.

The first share's business is tied to the economy as a whole — a bank, a cyclical manufacturer, a commodity producer. When the market falls it is usually because something is wrong economy-wide, and this company is hurt by the same thing.

The second share's volatility comes from something idiosyncratic — a single regulatory decision, a drug trial, a court case, a concentrated customer. It moves a great deal, and for its own reasons. A pharmaceutical company awaiting an approval is the clean example: enormous volatility, almost none of it related to the market.

And the condition on the investor. All of this requires that the holder is diversified. For someone whose entire wealth is in one of these two shares, the 30% volatility is the whole story and the betas are irrelevant. CAPM does not describe that person — which is not a subtlety but the model's explicit assumption, and the Corporate finance subject's treatment notes the same mismatch for a promoter holding one company.

The honest caveat on the numbers. Those two expected returns are what the model says, not what the shares will deliver, and chapter 8 shows that when this prediction is tested the line comes out flatter than CAPM requires — so a beta of 0.5 has historically earned more than 8.25% would suggest, and a beta of 1.3 less than 12.65%.

The point

If every investor holds the tangency portfolio, it must be the market portfolio, and then the only risk anyone bears that cannot be removed is covariance with the market. Scaling that covariance by the market's variance gives beta, which — unlike variance — is a weighted average across a portfolio's holdings. In equilibrium expected return is $r_f + \beta[E(R_M) - r_f]$, and every asset must sit on that security market line, which Sharpe calls the most important single conclusion of the model and which relates return to market risk "but not, as often believed, to total risk". The robust layer is that freely diversifiable risk cannot be paid for; the contested layer is that the undiversifiable part is one-dimensional and measured by beta.