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I have several times heard scholars refer to asset pricing models (such as the CAPM) as a type of equilibrium model. Why exactly is this the case? Does this simply mean that equilibrium is a necessary condition we need to accept for the model relationship to hold? Can this be shown formally?

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An equilibrium asset pricing model is one in which the asset prices jointly satisfy the requirement that the quantities of each asset supplied and the quantities demanded must be equal at that price. It is as opposed to a partial equilibrium model where the price of the asset (or at least some assets) are determined outside of the model.

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  • $\begingroup$ So the CAPM is a partial equilibrium model because the risk free rate and the market return are exogenous? $\endgroup$ – Constantin Feb 24 '15 at 17:17
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    $\begingroup$ CAPM (e.g., Sharpe-Linter or Black's) is an equilibrium model in the sense that supply equals demand. The assumption of mean-variance investors leads to the two-fund separation theorem. After that, it's easy to show that in the aggregate, the tangency portfolio is the market portfolio. That key point at the end---that the tangency portfolio is the market portfolio (thus, markets clear)---is the reason why CAPM is an equilibrium model. $\endgroup$ – jmbejara Feb 28 '15 at 17:59
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In general, you can distinguish theory for a given market with off-equilibrium and on-equilibrium theory.

Off-Equilibrium Theory

Off-Equilibrium theory basically means that the market is not cleared using prices, supply does not equal demand. This type of theory needs to give a good story why this is the case.

One very typical example is old Keynesian theory: due to stickiness of both wages and prices, in particular the labor market is not cleared: supply dominates demand.

The story for non clearing wages is a mixture of long-term wage contracts, efficiency wages, observed downward rigidity of wages and similar.

On-Equilibrium Theory

Asset markets however are typically assumed to be in equilibrium. The reason being that there exists a huge machinery (the whole financial system) that immediately (and mostly automatically) buys and sells assets when it predicts future changes in supply or demand.

We just don't think that asset prices are fixed at off-equilibrium values. What could possibly cause them to?

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See CAPM: Absolute Pricing, or Relative Pricing? or Arbitrage Opportunity, Impossible Frontier, and Logical Circularity in CAPM Equilibrium

The CAPM formula holds under the partial equilibrium of purely risky assets, which is equivalent to the condition that the market portfolio is the tangency portfolio. Since the general solution to asset prices in CAPM has only one dimension, given all individual investor's endowments and mean-variance preferences, the condition of CAPM equilibrium turns out to be an equation of only one variable.

Market Settings: Given $N$ primitive securities (stocks) and $I$ investors, $N\geq2$, and $I\geq2$. Let $X_{i}$ be the payoff of stock $i$ at time 1, then the payoff vector of $N$ stocks is $\mathbf{x}=[X_{1},X_{2},\cdots,X_{N}]^{\prime}$. Let $\boldsymbol{\eta}=\operatorname{E}(\mathbf{x})$, and $\mathbf{\Omega }=\mathrm{var}(\mathbf{x})$, we assume that the first two moments $\boldsymbol{\eta}$ and $\mathbf{\Omega}$ are known, and the variance matrix is positive definite, $\mathbf{\Omega}>0$, such that there is no redundant security. Defining $\mathbf{p}=[P_{1},P_{2},\cdots,P_{N}]^{\prime}>0$ to be the price vector of stocks ($P_{i}$ is the market value of stock $i$, not the market price per share of stock). If $\mathbf{1}$ is a conforming vector of ones, then the total market payoff is $X_{M}=\mathbf{1}^{\prime}\mathbf{x}$, with mean $E=\operatorname{E} (X_{M})=\mathbf{1}^{\prime}\boldsymbol{\eta}$, variance $Q=\mathrm{var} (X_{M})=\mathbf{1}^{\prime}\mathbf{\Omega1}>0$, and total market value $P_{M}=\mathbf{1}^{\prime}\mathbf{p}>0$. Let the return of $N$ stocks be $\mathbf{r}=[R_{1} ,R_{2},\cdots,R_{N}]^{\prime}$, then $R_{i}=X_{i}/P_{i}$, and $$ \mathbf{r}=\mathbf{P}^{-1}\mathbf{x} $$ where $\mathbf{P}=\mathrm{diag}(\mathbf{p})$, the diagonal matrix of price vector $\mathbf{p}$. Let $$ \boldsymbol{\mu}=\operatorname{E}(\mathbf{r})=\mathbf{P}^{-1}\boldsymbol{\eta }\qquad\mathbf{V}=\mathrm{var}(\mathbf{r})=\mathbf{P}^{-1}\mathbf{\Omega P}^{-1} $$ Since the first two moments of payoffs are known, the following three variables, price vector $\mathbf{p}$, expected return vector $\boldsymbol{\mu }$ and variance matrix $\mathbf{V}$ are equivalent, in the sense that any one of them is known, the other two are determined. We assume that there is a risk-free bond, with payoff $X_{0}>0$ and return $R_{0}>1$ (risk-free interest simple rate is $R_{0}-1$).

Let's define market beta (also beta value, or beta coefficient) of stock $i$ to be $$ \beta_{i}\equiv\frac{\mathrm{cov}(R_{i},R_{M})}{\mathrm{var}(R_{M})} $$ the classic CAPM formula \begin{equation} \mu_{i}-R_{0}=\beta_{i}(\mu_{M}-R_{0})\qquad i=1,2,\cdots,N \label{E:CAPM}% \end{equation} can be rewritten equivalently in form of payoff [See Eq (1) in Jensen and Long (1972, p.153), or Eq (7.23) in Fama and Miller (1972, p.296).] $$ P_{i}=\frac{1}{R_{0}}\left( \eta_{i}^{\,}-\frac{E-P_{M}R_{0}}{Q}Q_{i}\right) \qquad i=1,2,\cdots,N \label{E:CAPM-P} $$ Which is a system of linear equations on asset prices. we find that its solution space is one-dimensional.

If all investors use the mean-variance criterion, the following statements are equivalent

  1. The CAPM formula
  2. The market portfolio is equal to tangency portfolio
  3. Price vector of risky securities follows $$ \mathbf{p}=\frac{\boldsymbol{\eta}}{R_{0}}+x\mathbf{\Omega1}\qquad x\in\mathbb{R} $$

The semi-feasible equilibrium solution $x$ is affected by the endowment and/or preference of any individual investor. If $x=0$, the formula $\mathbf{p}=\frac{\boldsymbol{\eta}}{R_{0}}+x\mathbf{\Omega1}=\frac{\boldsymbol{\eta}}{R_{0}}$ is the discounted expected pricing.

CAPM equilibrium may coexist with arbitrage opportunities.

Summary: The CAPM formula is not an equilibrium pricing formula (absolute pricing), because it is only a result of partial equilibrium, and it has a one-dimensional general solution. Only when market return is given in advance, the CAPM formula could at best be considered a relative pricing formula. In this case it can only be used to price the portfolio of primitive securities. As relative pricing, it does not make sense to discuss risk in the CAPM formula. Because the relative pricing is based on the equilibrium prices of primitive securities and is realized through an arbitrage mechanism (replication), whereas arbitrage is not affected by risk preference. When the CAPM equilibrium prices are free of arbitrage, the CAPM formula must be a risk-neutral pricing formula.

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