What if we have a very large number, n, of assets that we can combine to get in a portfolio? Think of n being very very large. What happens to the portfolio risk then? Think about it. Well, let me first remind you the portfolio variance formula for the two asset case. All right, I know you sort of know it by heart now. So what is a portfolio variance if you have two assets with weight w1 and w2. Remember it's the weight of the first one squared times the variance of the first one + weight of the second one squared times the variance of the second one. And then we have to account for the covariance terms. 2 times weights times the covariance of the integer. Now think of what happens when we have N assets. Okay, and let's say that each has a weight wi, okay? What does the portfolio variance look like? So this is going to look ugly so don't get intimidated by the mess. I'll try to illustrate the intuition. So what do we have? Well, we have all the variance terms which is going to be i going from one to N, wi squared, sigma i squared. These are the variance terms. All right and then we have to pair wise covariance terms. How many of those we're going to have? Well, i going from 1 to N, j going from 1 to N i not equal to j, wi, wj sigma ij. I know it look ugly, but what do we have? Well, we have n of these variance terms, and then we have all the pairwise covariances n (n- 1) covariance terms. And this is the total portfolio list. Well think about what's happening here. As N increases the relative importance of the variance terms become relatively much smaller, less important compared to all the pairwise covariance terms. Now, the fact that most things in the economy are affected by some common factors. Right, for example, if there is a recession, people aren't employed, that will probably affect the fortunes of a lot of businesses at the same time. These covariance terms are going to be what sort of constitutes most of the portfolio risk. And this is what we call, the systematic risk. Right, all the common terms. All the common risk to all of the assets, okay. So what does this say? Well this says that we can decompose risk into two sources, right? One that we call the systematic risk, right? Or the market wide risk. And the other, the idiosyncratic risk, right? So, market wide risk, or systematic risk, is due to economy wide factors that effect everything. Right, recessions or monetary policy. Whereas idiosyncratic risk or sometimes what we call firm-specific or unique risk is the part that affects only the fortunes of a single business, right? For only precisely for specific reasons. So this source of specific risk is, by definition, aren't related to any other businesses or any other assets. And what happens is when we combine securities into a portfolio, the first specific risk is unique to that asset. Tends to cancel out and gets eliminated, right? This is what we call being diversified way away, right? So is there a limit to how much diversification we can achieve, right? Well, yes. Why? While we can diversify away the idiosyncratic risk, right, by holding large diversified portfolios, we can't get rid of the systematic risk, right? Because that affects everything. Right? So it will affect all the companies in how they move with the market. Okay, so this graph is from one of my favorite books. It's a classic called, A Random Walk Down Wall Street. And what it's shows is, is how much risk can be eliminated when you increase the number of stocks in your portfolio. You see what happens is is focus on the solid line, right? As you increase the number of stocks in your portfolio from, let say, ten to about 50, all right? Most of this risk here gets diversified away, right? That is the idiosyncratic risk That gets diversified away by putting together different stocks from different businesses. Now even when you add many, many more securities, many more stocks into the portfolio however, there's a limit right? The amount of risk doesn' t change. Right? Because that is the systematic risk that can't be diversified away, okay? So at some point all of the systematic risk is eliminated and the systematic risk that affects all of the stocks remains. What the graph also shows is how adding in international stocks to the mix can help eliminate even more risk. For the same dotted line in this graph shows you the extent of risk that is being eliminated as you increase the number of stocks. Including the international stocks in your portfolio now helps you eliminate even more risk more quickly. Now, given these concentrations, why would you want to diversify? Well, why would you want to wear a helmet? It's the same idea. So let me recap. So I tried to illustrate in many different ways to you, the basic notion of diversification. Which is, when assets are imperfectly correlated, combining them into a portfolio reduces total variance. The risk of the portfolio. By eliminating the unsystematic or the idiosyncratic risk, of the individual securities that are in the portfolio. And what we are left with, in a well diversified portfolio is the systematic risk that can be diversified away
