<p>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. </p>

CC BY-SA 3.0
2015-02-24T17:17:25.013Z
So the CAPM is a partial equilibrium model because the risk free rate and the market return are exogenous?
Constantin
CC BY-SA 3.0
2015-02-28T17:59:40.030Z
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.
jmbejara
CC BY-SA 3.0
2015-02-24T00:53:04.307Z
2015-02-25T18:50:44.467Z
2015-02-25T18:50:44.467Z
BKay
<p>In general, you can distinguish theory for a given market with off-equilibrium and on-equilibrium theory. </p>

<h3>Off-Equilibrium Theory</h3>

<p>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.</p>

<p>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. </p>

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

<h3>On-Equilibrium Theory</h3>

<p>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. </p>

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

CC BY-SA 3.0
2015-02-26T22:49:40.523Z
2015-02-26T22:49:40.523Z
FooBar
<p>See <a href="https://ssrn.com/abstract=4751502" rel="nofollow noreferrer">An Analytic Solution to the Mean-Variance Equilibrium: Is the Market Beta a Valuable Tool?</a></p>
<p><strong>The CAPM equation is equivalent to the semi-clearing condition</strong>, where the feasible prices of the primitive securities are restricted in a one-dimensional space and the total value of the market portfolio is a free variable.</p>
<p><strong>Market Settings</strong>: Given <span class="math-container">$N$</span>
primitive securities (stocks) and <span class="math-container">$I$</span> investors, <span class="math-container">$N\geq2$</span>, and <span class="math-container">$I\geq2$</span>. Let
<span class="math-container">$X_{i}$</span> be the payoff of stock <span class="math-container">$i$</span> at time 1, then the payoff vector of <span class="math-container">$N$</span>
stocks is <span class="math-container">$\mathbf{x}=[X_{1},X_{2},\cdots,X_{N}]^{\prime}$</span>. Let
<span class="math-container">$\boldsymbol{\eta}=\operatorname{E}(\mathbf{x})$</span>, and <span class="math-container">$\mathbf{\Omega
}=\mathrm{var}(\mathbf{x})$</span>, we assume that the first two moments
<span class="math-container">$\boldsymbol{\eta}$</span> and <span class="math-container">$\mathbf{\Omega}$</span> are known, and the variance matrix
is positive definite, <span class="math-container">$\mathbf{\Omega}&gt;0$</span>, such that there is no redundant
security. Defining <span class="math-container">$\mathbf{p}=[P_{1},P_{2},\cdots,P_{N}]^{\prime}&gt;0$</span> to be the price vector
of stocks (<span class="math-container">$P_{i}$</span> is the market value of stock <span class="math-container">$i$</span>, not the market price per
share of stock). If <span class="math-container">$\mathbf{1}$</span> is a conforming vector of ones, then the total market payoff
is <span class="math-container">$X_{M}=\mathbf{1}^{\prime}\mathbf{x}$</span>, with mean <span class="math-container">$E=\operatorname{E}
(X_{M})=\mathbf{1}^{\prime}\boldsymbol{\eta}$</span>, variance <span class="math-container">$Q=\mathrm{var}
(X_{M})=\mathbf{1}^{\prime}\mathbf{\Omega1}&gt;0$</span>, and total market value
<span class="math-container">$P_{M}=\mathbf{1}^{\prime}\mathbf{p}&gt;0$</span>. Let the return of <span class="math-container">$N$</span> stocks be <span class="math-container">$\mathbf{r}=[R_{1}
,R_{2},\cdots,R_{N}]^{\prime}$</span>, then <span class="math-container">$R_{i}=X_{i}/P_{i}$</span>, and
<span class="math-container">$$
\mathbf{r}=\mathbf{P}^{-1}\mathbf{x}
$$</span>
where <span class="math-container">$\mathbf{P}=\mathrm{diag}(\mathbf{p})$</span>, the diagonal matrix of price
vector <span class="math-container">$\mathbf{p}$</span>. Let
<span class="math-container">$$
\boldsymbol{\mu}=\operatorname{E}(\mathbf{r})=\mathbf{P}^{-1}\boldsymbol{\eta
}\qquad\mathbf{V}=\mathrm{var}(\mathbf{r})=\mathbf{P}^{-1}\mathbf{\Omega
P}^{-1}
$$</span>
Since the first two moments of payoffs are known, the following three
variables, price vector <span class="math-container">$\mathbf{p}$</span>, expected return vector <span class="math-container">$\boldsymbol{\mu
}$</span> and variance matrix <span class="math-container">$\mathbf{V}$</span> 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 <span class="math-container">$X_{0}&gt;0$</span> and return
<span class="math-container">$R_{0}&gt;1$</span> (risk-free interest simple rate is <span class="math-container">$R_{0}-1$</span>).</p>
<p>Let's define market beta (also beta value, or beta coefficient) of stock <span class="math-container">$i$</span>
to be
<span class="math-container">$$
\beta_{i}\equiv\frac{\mathrm{cov}(R_{i},R_{M})}{\mathrm{var}(R_{M})}
$$</span>
the classic CAPM formula
<span class="math-container">\begin{equation}
\mu_{i}-R_{0}=\beta_{i}(\mu_{M}-R_{0})\qquad i=1,2,\cdots,N \label{E:CAPM}%
\end{equation}</span>
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).]
<span class="math-container">$$
P_{i}=\frac{1}{R_{0}}\left(  \eta_{i}^{\,}-\frac{E-P_{M}R_{0}}{Q}Q_{i}\right)
\qquad i=1,2,\cdots,N \label{E:CAPM-P}
$$</span>
Which is a system of linear equations on asset prices. we find that its solution space is one-dimensional.</p>
<p>If all investors use the mean-variance criterion, the following statements are equivalent</p>
<ol>
<li>The CAPM formula</li>
<li>The market portfolio is equal to tangency portfolio</li>
<li>Price vector of risky securities follows
<span class="math-container">$$
\mathbf{p}=\frac{\boldsymbol{\eta}}{R_{0}}+x\mathbf{\Omega1}\qquad
x\in\mathbb{R}
$$</span></li>
</ol>
<p>The semi-feasible equilibrium solution <span class="math-container">$x$</span> is affected by the endowment and/or preference of any individual investor. If <span class="math-container">$x=0$</span>, the formula <span class="math-container">$\mathbf{p}=\frac{\boldsymbol{\eta}}{R_{0}}+x\mathbf{\Omega1}=\frac{\boldsymbol{\eta}}{R_{0}}$</span> is the discounted expected pricing.</p>
<p>CAPM equilibrium may coexist with arbitrage opportunities.</p>
<p><strong>Summary</strong>: The mean-variance equilibrium does not exclude arbitrage opportunities. It is a misinterpretation that the expected return of an asset is determined by its beta, because the market betas and expected returns are endogenous and jointly determined by mean-variance equilibrium. We point out that the market beta is a redundant variable given the expected return of market portfolio, and the market beta is not the time series beta of the market model.</p>

CC BY-SA 4.0
2020-04-29T09:33:31.273Z
2024-07-15T07:34:19.393Z
2024-07-15T07:34:19.393Z
Chen Deng-Ta
<p>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?</p>

<asset-pricing>
Why are asset pricing models equilibrium models?
Constantin
# Why are asset pricing models equilibrium models?

(Asked by: username_0 on 2015-02-22)

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?

**Tags:** asset-pricing

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## 3 answers

### Answer by username_1 on 2015-02-24. Score: 2

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.

> **Comments:**
> **username_0**: So the CAPM is a partial equilibrium model because the risk free rate and the market return are exogenous?  
> **username_2**: 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.  

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### Answer by username_3 on 2015-02-26. Score: 0

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?


---

### Answer by username_4 on 2020-04-29. Score: 0

See [An Analytic Solution to the Mean-Variance Equilibrium: Is the Market Beta a Valuable Tool?](https://ssrn.com/abstract=4751502)

**The CAPM equation is equivalent to the semi-clearing condition**, where the feasible prices of the primitive securities are restricted in a one-dimensional space and the total value of the market portfolio is a free 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 mean-variance equilibrium does not exclude arbitrage opportunities. It is a misinterpretation that the expected return of an asset is determined by its beta, because the market betas and expected returns are endogenous and jointly determined by mean-variance equilibrium. We point out that the market beta is a redundant variable given the expected return of market portfolio, and the market beta is not the time series beta of the market model.


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