Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk

The question turns around

The previous three papers all looked from the same place: there is a person with money to invest, what should they do? Markowitz drew them a curve, Roy picked a point on that curve, and Tobin showed that the point comes out the same for everyone.

Sharpe looks at the same world but turns the question around: if everyone really does behave this way, where do prices settle?

It sounds like a small difference, but it is not. The previous three pages explained what an investor ought to do; this page explains what the market itself will do. We are moving from a normative theory to a positive one, and the biggest break in the series happens right here.

The step that has to be taken

In Tobin, everyone holds the same risky portfolio. Now let us take one more step.

Imagine for a moment that you pool the holdings of every investor in the market. The sum of their risky assets has to equal every risky asset that exists, because there is no other possibility: whatever share one person sold, someone else must have bought. If everyone holds the same portfolio and those portfolios add up to the whole market, then that common portfolio is the market itself.

The market portfolio stands on the left of the equation, and the wealth-weighted aggregate demand of investors on the right. This equality is not a definition but an equilibrium condition: it holds only when prices are in the right place. If prices are wrong, demand does not come out equal to supply, and prices go on moving until the market clears.

The symbols keep the same meaning throughout the page:

One warning about a collision: the summation sign ∑ in the equation above and the name of the covariance matrix Σ are the same Greek letter. The first means "add up" and carries a counter beneath it; the second is the name of a matrix.

The figure below actually runs that calculation. There are dozens of investors; each applies Tobin's rule according to their own risk tolerance, and none of them looks at the others. Their demands are added up one by one and compared against the supply in the market.

Let me say first what to look for. Open up the spread of tolerance and the investors fan out along the line; concentrate the wealth in a few hands and the size of the points changes. When you give everyone the same tolerance, all the points gather in one place, and that place turns out to be the market portfolio itself. However you move the controls, the point where the line touches the frontier does not budge.

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Figure parameters

Number of investors40Spread of risk tolerance0.60Wealth inequality0.30Risk-free rate%3.0Number of assets16Average correlation0.45

The number under the figure gives the gap between the investors' total demand and the supply in the market. It sits somewhere around zero, too small to see, which is to say that the equilibrium really does hold.

The conclusion is this: whatever the distribution of investors, the tangency portfolio does not change. The market portfolio is born of supply, and supply does not care who likes how much risk. This is how Tobin's separation theorem finds its counterpart at the level of equilibrium.

Under which assumptions

You have seen the equilibrium; now I have to tell you which world you saw it in.

This setup assumes four things, and none of the four has any counterpart in the real world:

• Homogeneous expectations. Everyone looks at the same μ and the same Σ. But people think different things about the same asset, and that is largely why a market exists at all.
• A single period. The investor decides today and everything ends at the close of the period. There is no changing your mind along the way, no rebalancing, no looking at different horizons.
• A frictionless market. Transaction costs, taxes, minimum trade sizes, limits on liquidity — none of it enters the account.
• Unlimited borrowing. Everyone can borrow and lend as much as they like at the same rf. The leveraged investors in the figure rest entirely on this.

Sharpe says himself that these assumptions are not realistic. The value of the theory comes not from its realism but from what it produces: once the four are accepted, the two results below follow of necessity.

One more thing needs adding: Lintner (1965) and Mossin (1966) arrived at the same result independently of one another. The series runs through Sharpe, so I do not work the others through separately, but it would be unfair to credit the result to one person alone.

The measure of risk changes

The first result is this: an asset's risk is not its own variance.

The reason has to be looked for in the equilibrium itself. If everyone holds the market portfolio, then nobody carries a single asset on their own. However volatile that asset may be by itself, most of its fluctuation is damped inside the portfolio by the opposite movements of other assets. What remains is only the part of the asset that moves together with the market.

The number that measures this is beta:

In the numerator stands the asset's covariance with the market portfolio, in the denominator the variance of the market. An asset with a beta of 1 moves just as hard as the market; an asset with a beta of 0.4 does not swing as much as the market — however choppy it may look on its own.

This is not a change of definition but a consequence of the equilibrium. Risk that can be destroyed by diversification is not priced, because everyone has already diversified it away.

Two planes

Seeing that the measure of risk really has changed takes looking at the same assets from two planes. In the figure below, the horizontal axis of the left panel shows the asset's own standard deviation, and the horizontal axis of the right one shows its beta. The vertical axis is the same in both: expected return.

Try the controls here as well. Cut back how much the assets move together and the scatter on the left opens right up. Then turn the common movement up: the two panels draw closer, because once there is nothing left to diversify, σ and β say almost the same thing.

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Figure parameters

Risk-free rate%3.0Number of assets16Average correlation0.45One asset's own risk1.00×

In the left panel the points stand scattered; the dashed line fitted to them offers no explanation, and in fact admits that it cannot explain. Two assets with the same total risk can have expected returns points apart; more than that, a riskier asset can pay less.

In the right panel the assets line up on a single line. And that line is not one fitted to the points either; it comes straight out of the risk-free rate.

If I had drawn the right panel on its own I would have shown nothing. What gives it meaning is that the same points scatter in one plane and fall into line in the other.

The Security Market Line

The equation of that line is this:

Both ends of the line come from the market. The place where it crosses the vertical axis is the risk-free rate: an asset with a beta of zero earns rf. Its slope is the market risk premium, which shows what carrying one unit of beta is worth.

From here on, finding an asset's expected return takes knowing a single number about it. There is no need to know its own variance or its covariances with the other assets; its beta is enough.

What gets priced is not all of the risk

The most direct way of seeing why goes through splitting each asset's total variance in two: the part explained by the market and what is left over.

In the figure below the assets are ordered by total variance, that is, in the order the expectation "riskier pays more" would put them in. Raise one asset's own risk to three times its level: that asset's bar stretches on the light side, but its expected return stays very nearly where it was. At the same time, in the two-plane figure the point slides to the right and goes on sitting on the line.

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Figure parameters

Risk-free rate%3.0Number of assets16Average correlation0.45One asset's own risk1.00×

The bars grow steadily from left to right. The expected return, on the other hand, does not grow; it zigzags. You can see what that zigzag follows in the dark part of the bars.

Here I have to make a correction. The textbooks say "idiosyncratic risk is not priced"; in a finite market that sentence is not exactly true. The market portfolio holds a certain amount of that asset too — with N assets, roughly 1/N of it — and nobody can diversify that part away. The numbers in the figure show it: while total variance rises by well over a hundred percent, the expected return moves by a few percent. Not zero, but next to nothing beside the risk that was added.

In a real market N runs into the thousands. That is the limit in which the textbook sentence becomes true.

What changed

Markowitz drew a curve and said "the good portfolios are here". Roy picked a point on that curve. Tobin showed that the point is the same for everyone.

Sharpe said that common point is the market itself, and from then on he asked about the price rather than the portfolio. Where he arrives is a pricing equation rather than a portfolio recommendation.

Read one after another, the four papers come out as a single uninterrupted argument: from a curve to a theory of equilibrium.

What comes next

Beneath this whole structure sits an assumption that has never once been questioned: μ and Σ are known.

On the Markowitz page that was a note in the margin. Here it undermines a structure four papers deep, because the equilibrium prices, the betas and the line itself all derive from those two objects. If the two were estimated, then everything built on them carries the error of that estimate.

In 1989 Michaud measures exactly this: optimisation does not reduce the error in the inputs, it amplifies it.

That will be the next item in the series.

The expected returns and covariances implied by the equilibrium prices were fed to a tangency solver that never sees the pricing equation, and on 900 randomly generated valid inputs the weights it returned were compared against the supply in the market. The largest deviation was 9.57e-14, against a tolerance of 1e-9.