Hello, welcome back. In this lecture, we're going to look at an example of portfolio diversification, using data from International Equity. I will illustrate the insights for you graphically, and in the process we will also define some terminology like mean-variance frontier. What is a minimum variance portfolio, or an efficient portfolio? You will see that these are the building blocks of modern portfolio theory and mean-variance investing, that we will look at in a more formal way in the next module. Then, you will also learn how to derive these portfolios analytically but for now, I want you to just focus on the intuition. Let's start with some data. So in the first table, what you see is the average annual return and volatility for the equity market in G5 countries. You have the US, UK, France, Germany and Japan. You see that the average annual return in each of these markets was around 14 to 15% but the volatility varies considerably across 5 countries, right? So for example, the US Equity market had annual volatility of about 15% compared to let's say 23% in Japan and 24% in UK. Now, the second table gives you the correlation matrix, right? Remember, these are pair wise correlation coefficients and they tell you how two markets co-move together. So notice that the correlations are not very high, so when you look at the table the highest correlation is between Germany and France. All right, about 0.6 and the lowest is between US and Japan, which is only about 0.27, right? So, as we say, as we have seen before the low correlations between these markets suggest that there should be plenty of opportunity for diversification. So let's first focus on just two markets. Let's just focus on the US and Japan only. Suppose these were the only two markets available to you. How would you construct a portfolio that consists of the US and the Japanese equity markets? What would be the possible combinations? So let's let W. Be the weight or the share of the US market in our portfolio. The share of US markets in the portfolio and then using our expected return and various formula, right? We can compute it portfolio expected return and portfolio volatility for different combinations and plot the portfolio return and volatility. And expected return and volatility space like this, right? So here, on the X-axis I have the portfolio volatility, so that you can it see better and on the Y-axis, I have the portfolio expected return. So how do we find the portfolio expected return? Remember, the portfolio return is the weight in the US times it's expected return of the average return plus 1 minus weight in the US. Which is essentially the weight of the Japanese market times the expected return of the Japanese market. What about the variance? Well, we now have to find the variance of a two asset portfolio, right? The portfolio variance is going to be the weight in US squared times Variance of the US market, the world total squared. Plus the weight of the Japanese market squared, which is 1- w US, times the variance of the Japanese market, right? So now, I have to they can checkout the covariance or the correlation term, right? It's 2 times the weights. Times the correlation coefficients between US and Japan times the volatilities. Right? So, by varying these weights, you can obtain different expected return and volatility combinations and then plot these in this expected return and volatility space, right? So, what do you see? Well, the square here represents being 100% invested in the US market, in which case. You have a volatility of 15.4%, and an expected return, an average return of 13.6. What is the triangle? Well, the triangle represents being invested 100% in the Japanese market, in which case you get a volatility of 23% and average return of 15%. Now the blue line shows you the different portfolio combinations that we can create when we put together the two markets, right? When we invest in the US and Japanese market together. Now already you can notice something really important. Notice already that being invested 100% in the US does not seem like a very good idea at all, why? Well, it is strictly dominated by another portfolio, that is made up of the Japanese and the US equity market together, right? While where you could achieve the same. Right? You would have the same volatility but a higher expected return if in fact you combine the US and the Japanese market together instead of being just invested in the US market. Now why is this? Well, remember the coalition between the US and Japan in our data is only 0.27, all right? So these local relation between the US and the Japanese equity markets allows us to diversify some of the risk away, okay? I also want to point out that here we have the weights constraint to be between 0.0 and 1.0, right? So we have to have positive basic weights on each market, what if we allow short ceiling? Remember short selling from the previous course? So in other words, what if we allow short selling of the US market, allowing W to be negative, which is essentially you're short selling. And combining it with the Japanese market, essentially having a weight in the Japanese market that is greater than one, right? What would that look like? So this locus of portfolios, let me draw it for you, so this is locus of portfolios that we put together, right? Has a special name, we called this the Minimum Variance Frontier. Minimum Variance. Frontier, why do we called that? Well, because it is the [INAUDIBLE] portfolio's that have the minimum variance for a given level of expected return. Now, for two assets obviously, it's simply the set of all possible portfolio combinations that can be made from this two assets. But you will see that from with multiple assets is actually going to be just different here, all right? So, again, what is the mean-variance frontier? It is the locus of the portfolios In the expected return standard deviation space that have the minimum variance for each expected return. Now, we can also identify some special portfolio along this frontier. Now, for example, I already mentioned that a portfolio that is 100% invested in the US equity markets, right? The square is dominated by the upper part of the parabola, the upper part of that frontier, why? Well, because an alternative portfolio, right, provides a higher expected return for the given level of risk, all right? So we call the portfolios that lie on this upper part of the top part of the frontier, efficient portfolios. So, in fact, you probably would never want to be on the lower side of the front here. You would always want to be on the upper side of the front here. Why would you want to hold this portfolio, when you can actually get on this upper part of the frontier and get a higher return for the same level of risk, right? Finally, we have a special name for the leftmost point on the frontier. We call this portfolio, this combination The minimum variance portfolio, all right? So in the later lectures, you will actually learn how to find these portfolios analytically. Okay so, what is an efficient portfolio? Well, an efficient portfolio is the portfolio that is on the top half of the mean-variance frontier, right? Because it gives an investor the highest expected return, the highest reward for a given level of risk and we also define what's the minimum variance portfolio.