<p>From my research online, when learning CAPM with <span class="math-container">$n$</span> risky assets and a risk free asset with return <span class="math-container">$r_f$</span>, I always see the conclusion that in equilibrium, the <strong>market portfolio = tangency portfolio</strong> from mean-variance (Markowitz) theory.</p>
<p>However I don't really understand <strong>how</strong> the price mechanism will work to reach equilibrium.</p>
<p>This is the story in my head so far:</p>
<p>We have <span class="math-container">$n$</span> assets with pre-defined expected returns <span class="math-container">$\mu_i$</span> and standard deviations <span class="math-container">$\sigma_i$</span>. We also have a risk-free asset with return <span class="math-container">$r_f$</span>. We assume we have an economy of rational investors, who all seek to maximise their expected utility, which we assume only depends on standard deviation and expected return.</p>
<p><strong>Question 1.</strong>  Are these <span class="math-container">$n$</span> assets already distributed within the economy? They are randomly allocated among investors (who will become potential buyers/sellers)?</p>
<p>Now from this information <span class="math-container">$(\mu_i, \sigma_i)$</span>, which we assume all investors agree upon, all investors will conclude that the optimal weights for risky assets is <span class="math-container">$T = (w_1^*, w_2^*, ..., w_n^*)$</span>, i.e. the weights for the tangency portfolio, as this has the highest sharpe ratio, which investors are looking to maximise. Then, they will decide based on their own risky preferences an <span class="math-container">$\alpha$</span> to make an efficient portfolio <span class="math-container">$\alpha \text{Tangency} + (1-\alpha)\text{Risk free}$</span>.</p>
<p><strong>Now</strong>. <strong>All</strong> investors in the economy <strong>demand the portfolio T</strong>. This will drive up the price of assets in <span class="math-container">$T$</span> with high weight and cause the price of assets in <span class="math-container">$T$</span> with low weights to fall. This will cause the expected return of the high-weight assets, which is inversely proportional to the price, to fall.</p>
<p><strong>Question 2.</strong> Won't this change in expected returns cause a change in the tangency portfolio from <span class="math-container">$T$</span> to some new <span class="math-container">$T^*\ne T$</span>? And will this cause a change in the standard deviation of asset returns?</p>
<p>Again, all investors are now chasing <span class="math-container">$T^*$</span>. Driving the prices of those high-weight assets up, and the prices of low-weight assets down.</p>
<p><strong>Question 3.</strong> How do we know this cycle will ever come to an end? And more surprisingly, how is it that the market portfolio will converge to the original <span class="math-container">$T$</span> when the prices will keep changing as investors chase this optimal portfolio?</p>
<p>To me, it seems like equilibrium will be more like a situation where there is no longer an optimal portfolio, and investors are indifferent between all portfolios (i.e. they all have a return correlation of 1).</p>

CC BY-SA 4.0
2020-10-16T11:18:36.367Z
The question you're asking is not about the CAPM specifically. Your question is really about the process through which market price converges to equilibrium price. Convergence to equilibrium price is not explicitly modeled in typical asset pricing models (of which CAPM is probably the most basic). Also, the statement "...All investors in the economy demand the portfolio T. This will drive up the price of assets in T..." is not correct---consider e.g. a Robinson Crusoe/representative investor CAPM.
Michael
CC BY-SA 4.0
2020-10-17T09:46:48.527Z
Hi @Michael 
I'm still not sure if i understand why the price of T would not increase if all investors demand it?
user523384
CC BY-SA 4.0
2020-10-17T12:07:56.003Z
Consider a representative investor CAPM (an example of a Robinson Crusoe economy)---"all investors" now is just the representative investor. It's pretty clear that the statement "...if the representative investor demands the portfolio T...this will drive up the price of...T" does not hold in general.
Michael
CC BY-SA 4.0
2020-10-17T12:26:48.447Z
Sorry, I should have said more than one investor in the economy. Would it be true now?
user523384
CC BY-SA 4.0
2020-10-18T00:32:46.900Z
No, not really. Just because all economic agents demand a certain commodity, it need not drive up the price. The issue is excess demand, i.e. demand relative to supply, not just demand. In any case, there is a process which price equilibriates ("Walrasian auctioneer") that's not explicitly modeled in asset pricing models.
Michael
CC BY-SA 4.0
2020-10-18T02:06:32.450Z
Thank you for the "Walrasian auctioneer" reference. I think that makes sense. So the equilibrium price can be achieved more so by investors providing their entire demand curve to suppliers, and thus a price is selected which satisfies total demand in the economy.

Then, whatever prices (and thus returns) are decided, this determines the market portfolio (and also the tangency portfolio) which is drawn in a mean-variance plot?
user523384
CC BY-SA 4.0
2020-10-18T02:08:00.770Z
Would you be willing to provide a short answer to the original post? It would help me understand your answers collectively and I'd be happy to accept it..
user523384
<microeconomics><financial-economics><asset-pricing><portfolio-theory>
How is equilibrium reached in CAPM such that the tangency portfolio = market portfolio?
user523384
# How is equilibrium reached in CAPM such that the tangency portfolio = market portfolio?

(Asked by: username_0 on 2020-10-15)

From my research online, when learning CAPM with $n$ risky assets and a risk free asset with return $r\_f$, I always see the conclusion that in equilibrium, the **market portfolio = tangency portfolio** from mean-variance (Markowitz) theory.

However I don't really understand **how** the price mechanism will work to reach equilibrium.

This is the story in my head so far:

We have $n$ assets with pre-defined expected returns $\mu\_i$ and standard deviations $\sigma\_i$. We also have a risk-free asset with return $r\_f$. We assume we have an economy of rational investors, who all seek to maximise their expected utility, which we assume only depends on standard deviation and expected return.

**Question 1.** Are these $n$ assets already distributed within the economy? They are randomly allocated among investors (who will become potential buyers/sellers)?

Now from this information $(\mu\_i, \sigma\_i)$, which we assume all investors agree upon, all investors will conclude that the optimal weights for risky assets is $T = (w\_1^\*, w\_2^\*, ..., w\_n^\*)$, i.e. the weights for the tangency portfolio, as this has the highest sharpe ratio, which investors are looking to maximise. Then, they will decide based on their own risky preferences an $\alpha$ to make an efficient portfolio $\alpha \text{Tangency} + (1-\alpha)\text{Risk free}$.

**Now**. **All** investors in the economy **demand the portfolio T**. This will drive up the price of assets in $T$ with high weight and cause the price of assets in $T$ with low weights to fall. This will cause the expected return of the high-weight assets, which is inversely proportional to the price, to fall.

**Question 2.** Won't this change in expected returns cause a change in the tangency portfolio from $T$ to some new $T^\*\ne T$? And will this cause a change in the standard deviation of asset returns?

Again, all investors are now chasing $T^\*$. Driving the prices of those high-weight assets up, and the prices of low-weight assets down.

**Question 3.** How do we know this cycle will ever come to an end? And more surprisingly, how is it that the market portfolio will converge to the original $T$ when the prices will keep changing as investors chase this optimal portfolio?

To me, it seems like equilibrium will be more like a situation where there is no longer an optimal portfolio, and investors are indifferent between all portfolios (i.e. they all have a return correlation of 1).

**Tags:** microeconomics, financial-economics, asset-pricing, portfolio-theory

> **Comments:**
> **username_1**: The question you're asking is not about the CAPM specifically. Your question is really about the process through which market price converges to equilibrium price. Convergence to equilibrium price is not explicitly modeled in typical asset pricing models (of which CAPM is probably the most basic). Also, the statement "...All investors in the economy demand the portfolio T. This will drive up the price of assets in T..." is not correct---consider e.g. a Robinson Crusoe/representative investor CAPM.  
> **username_0**: Hi @username_1  I'm still not sure if i understand why the price of T would not increase if all investors demand it?  
> **username_1**: Consider a representative investor CAPM (an example of a Robinson Crusoe economy)---"all investors" now is just the representative investor. It's pretty clear that the statement "...if the representative investor demands the portfolio T...this will drive up the price of...T" does not hold in general.  
> **username_0**: Sorry, I should have said more than one investor in the economy. Would it be true now?  
> **username_1**: No, not really. Just because all economic agents demand a certain commodity, it need not drive up the price. The issue is excess demand, i.e. demand relative to supply, not just demand. In any case, there is a process which price equilibriates ("Walrasian auctioneer") that's not explicitly modeled in asset pricing models.  
> **username_0**: Thank you for the "Walrasian auctioneer" reference. I think that makes sense. So the equilibrium price can be achieved more so by investors providing their entire demand curve to suppliers, and thus a price is selected which satisfies total demand in the economy.  Then, whatever prices (and thus returns) are decided, this determines the market portfolio (and also the tangency portfolio) which is drawn in a mean-variance plot?  
> **username_0**: Would you be willing to provide a short answer to the original post? It would help me understand your answers collectively and I'd be happy to accept it..  

---

## 0 answers

