Welcome back, today we are going to talk about Naive Diversification. Well, when we think about portfolio diversification, we think about these proverbial definition where there's a farmer that goes to the market and the farmer needs to sell eggs in the market. The real question for the farmer is how to bring these eggs to the market in the most possible efficient way, in the safest way. We are told through the proverbial definition of diversification, we are told that the best way to go is not to hold the eggs in a single basket, but instead to spread the eggs across many baskets. Of course the intuition is that if you have all the eggs in one single basket if something wrong happens to the baskets if the basket fails you then all the eggs are gone. Now, if you spread the eggs across multiple baskets if any one of them goes wrong, well, that's not a big deal because you are going to just lose a few eggs. So that's the concept of diversification. Having a well balanced allocation of eggs to baskets or if you will having a well-balanced allocation of your dollars to different assets or securities. Now, it is fair to say that the holding a well-balanced portfolio is not the end objective of diversification. It's nothing but the main goal to achieve a greater objective which in itself is holding a well rewarded portfolio. So in other words, nobody should be excited about say as an investor to be holding a well balanced portfolio with dollars spread across many securities. The reason why we lack it is because by doing so we remove, we diversify away some of the unrewarded risk within the portfolio and that allows us to achieve the highest possible reward per unit of risk. So let's keep that in mind, what we care about either high reward per unit of risk and that can be achieved through a well-diversified portfolio. The story of the farmer going to the basket, gives us a very simple way to think about what it means to hold a well-diversified portfolio. Now, if we want to think carefully about what the proverb says we have to start asking a few questions. The first of those questions is, how do we measure the number of baskets in the first place? Well, I know this question sounds very straightforward. I mean you just look at how many securities somebody is holding and then counting the security and that telling you how many baskets quote and quote they are holding. It turns out that it's a little more complicated than this, and to see why it's a bit more complicated just let take a look at this very simple example. So on this simple example we are looking at a portfolio invested in 30 assets, let's think about them as 30 stocks. As we can see on the graph, we have different weights allocated to these 30 assets. Now, clearly the nominal number of securities or stocks held in that portfolio is 30 as we said and that is clearly shown on the picture. But on the other hand, only very few securities is actually get a meaningful allocation. So when you're looking carefully at the graph what you see is you know stock number one, maybe stock number three, number four, number five, maybe seven and eight, but that's about it. Everything else get a very small tiny fraction of investor wealth. So in this case even though the nominal number of stocks is clearly 30 the effective number of stocks is much lower, it's certainly less than 10 in this particular case. So what we have to do is we have to think about a mathematical procedure, a way to think about how we count this number of baskets while taking into account the fact that some baskets or some stocks will get a higher fraction of investor wealth and that's why they have an impact. Well, it turns out that there is one very simple mathematical equation that allows us to measure what we call the effective number of constituents in a portfolio. This equation is provided here, it's telling us that these effective number of constituents or ENC in short, is given by the reciprocal of the sum of the squared weights. In other words, you take the portfolio weights that we call W i, you square them you get W i squared. You sum them up and then you take the inverse of that, one divided by the sum of the portfolio weights. Now you may wonder why this mathematical equation giving us a meaningful quantity or meaningful estimate of the effective number of stocks or the effective number of constituents in a portfolio. Well, to see this It's actually useful to look at a couple of very extreme cases and that will allow us to better understand the mechanics of this equation, how it works in reality. Okay, the first example or the first extreme case that we're going to look at is the case of a fully concentrated portfolio. So in this case, let's assume that all the wealth of the investor is invested in stock number one. So W1 is one or 100 percent if you will. Then all the Wi are zero, for i equals to n. All the investor wealth is invested in stock number one. Well in this case, we know that even though we are presumably looking at the universe of n stocks, the effective number of constituents in this portfolio is actually one. Well, let's check that this formula actually gives exactly the correct answer, well that's exactly what it does. So W1 is one, one squared is one, plus Wi squared or zero, so zero squared is zero. So the sum of the weight square is equal to one and one divided by one is also equal to one. So as a result we get that the ENC is indeed equal to one. So the equation that we have introduced actually give us the proper number of stocks, the proper number of constituents which is one in this example. Let's take a look at the other extreme example, which is the most balanced possible portfolio, we think about it as the equally weighted portfolio. Where the weight allocated to each stock is one over n. So if you have a 100 stocks you allocate one percent to each one of these a 100 stocks. While in this case if Wi is equal to one over n, well then Wi squared is one over n square. Now, you sum them up and they are n such terms, so you get n times one divided by n squared. So that gives you one over n and then you take the reciprocal of one over n and then you get n. So in this particular example, what we find is that the highest possible effective number of constituent at most, at best is actually equal to the nominal number of constituent which is equal to n, and that can be achieved if and only if we're holding an equally weighted portfolio. By the way, the statement is pretty interesting because it gives us a different way to think about the equally weighted portfolio. We think about it as an hope portfolio where we weight stocks as a function of one divided by n. Well, we can also think about that portfolio as the output of an optimization process. In other words, an equally weighted portfolio is a MAX ENC portfolio. You're trying to maximize the effective number of constituents, then you get the equally weighted portfolio. So in conclusion what we see is that, the ENC number is anywhere between one and n. One is the minimum, n is the maximum, one is achieved for a fully concentrated portfolio, n is achieved for an equally weighted portfolio, and other portfolios fall somewhere in between. Well precisely, let's just take a look at a practical, real-world example of a portfolio to see what we get. Well, in this case we're looking at the S and P 500 index. So it's a portfolio that contains 500 stocks that happens to be the largest stocks in the US. Those stocks are weighted as a proportion of their market cap, proportionally to their market caps, so the largest cap stocks get a higher allocation. Well, if you're looking at the nominal number of stocks which is the green line, where we see is as expected it's 500. Actually, it's not always exactly 500 as you can see on the graph sometimes it's 499, there are periods of time when the stock get it out of the index for various reasons and before new stocks come in there's a little bit of time where the so-called 500 index doesn't contain 500 stocks, maybe 499. Now, more interestingly if you're now looking at the blue line, the blue line gives you a sense of where the effective number of constituencies. In this case what we see is that, on average the effective number of constituents is about a 100. So in other words, it's five times lower than the nominal number of constituents. Well, this is clearly telling us that these cap weighted index which is the S and P 500 index, is not as well balanced as an equally weighted portfolio would be. Equally weighted portfolio would give you effective number of constituents equal to 500. So that is an interesting insight to which we will come back to eventually. That interesting insight is telling us that the cap weighted index tend to be a concentrated portfolio, because there's a disproportionately large allocation to the largest gap stocks. Wrapping up, well intuition and portfolio theory suggests that we should hold well-diversified portfolios. The naive approach diversification suggests that we should evenly balance dollars across portfolio constituents. Now, there happens to exist a quantitative measure of how well balanced any portfolio is, that quantitative measure is known as effective number of constituents or ENC in short. When we apply this measure to some well-known cap weighted indices market benchmarks such as the S and P 500 Index, we come to the conclusion that such indices are not necessarily well-balanced, they tend to be fairly concentrated.