[MUSIC] So let's see how we move from the implication of the two-fund separation to what's going on at the market level. Okay, so to do that, we're going to look at the very simple example that summarizes a little bit the situation in terms of individual allocation and tries to make the link with what should happen at the market level. So, let's have a look at this big table. There's a lot of information here. Let start at the top. We are considering here the allocation performed by three different investors. These three different investors are labeled 1, 2 and 3. They appear in the upper part of the first table. These three investors are going to be different in terms of their wealth and they are also going to be different in terms of their tolerance to risk. All of them will follow this idea that we have exposed before of optimal diversification, so each of them is going to choose a portfolio, which is following the two-fund separation. Just a composition between the risk-free asset and the tangency hold for you. But because they have different tolerance to risk, they will choose different relative repetition between the two fonts. So investor 1 has the largest wealth. He has 1000. Investor 2 has 500, investor 3 has 50. Now, their different level of risk tolerance, is here represented by the fraction of wealth they invest in the risky assets. This is the third line in this top table. We see that investor 1 invests more than 100%, so he has a leverage position. He borrows money and invests more than 100% in the risky portfolio. He is probably the most risk tolerant, he has the less level of risk aversion, his portfolio is going to be the most risky among the three investors here. Investor 2 invests 60% of his wealth in the risky asset. He is more risk averse. He has less tolerance for risk. He is seeking a portfolio with a lower level of risk which has a consequence, will have a lower level of expected return. Investor 3 has a wealth of 50, as we said before, and he invests a 100% in the risky assets. So he's less risk averse than investor 2, but more risk averse than investor 1. So, in this example, there are four stocks available for investment. And the left column with the label T portfolio weights represents the relative contribution of each of these four stocks to the tangency portfolio. Okay, so this is the optimally diversified portfolio of risky asset that is on the green efficient frontier and tangent to the red line, where we also include the risk-free asset. So these four assets have a repartition. Here 50 % in stock 1, 10% in stock 2 and 20% in both stock 3 and 4. Of course the sum here of these weights is a 100. Because the investors have different amount of wealth, even if they choose the same portfolio, they will invest different amounts in the risky asset. And this is the middle table. Amount in risky assets. For example, the first investor who invests 120% of his wealth in the risky asset is going to put 1200 in the risky asset. Out of this 1200, 50% is in Stock 1. So this mean that he invests, actually, 600 in monetary terms in stock 1. All the elements in this table are computed in a similar way. Let's look at another example. Investor 3, excuse me. Invest 100% of this wealth, his wealth is 50, and he puts 10% in stock 2. So his monetary investment in Stock 2 is going to be exactly 5. Okay, 10% of the 50 invested in total. And we can do this for all the amount invested in all the risky assets. Now, let's have a little bit of imagination and imagine that these three investors are all the investors in the world and these four assets are all the assets in the world. What can we conclude from this little exercise? Well, the amount invested in the risky asset, the 600 for example, in stock 1 by Agent 1, 150 by Investor 2 in stock 1, and 25 by Investor 3 in stock 1. This represents the total wealth invested in Stock 1. So, the sum of these three amounts is actually the total market capitalization of Stock 1, assuming of course that the three investors are the only investors in the world. So what we can do is look at the market capitalization, the value of Stock 1, the overall money invested in that security. And we see that this sum is 775. We can do the same thing for Stock 2, Stock 3 and Stock 4. This is the next to last column on the right hand side, market portfolio sum across individual. Now this market capitalization, we can also look at the weight, the fraction that each stocks represent in the total market. How we would do that, for example we can take 775 which is the monetary value of Stock 1, and divide the 775 by the sum of the monetary values of the four stocks. This will represent the weight in the market of Stock 1. And this weight is computed in the last column, market portfolio weights. And we see that we have 50% for Stock 1, 10% for Stock 2, 20% for Stock 3, 20% for Stock 4. And if you look carefully, these market portfolio weights, which represent the market capitalization in relative term of each of the securities in the market, they're exactly identical, to the portfolio weight in the tangency portfolio. So, this is the main implication in terms of aggregation of these individual decisions. The tangency portfolio, the weights that define the tangency portfolio are actually exactly the same in equilibrium as the relative market capitalization of each securities. So actually, when we look at this efficient frontier, this portfolio that is at the intersection, or excuse me, the tangency between the green line and the red line, we've call it the tangency portfolio. If all the investor in the market follow this rule of optimally diversifying, this portfolio, it is going to be what we call the market portfolio, it represents the relative share of each stock in the economy. [MUSIC]