Okay, so what have you learned so far from this discussion? Well, if you thought that these are the expected return on volatilities for the US and Japanese equity markets and you could form these portfolios by mixing them together. What's the main insight? If this is the data that's given to you and you can combine these portfolios. Well, clearly you should never hold a 100% US portfolio, right? As I have illustrated, holding a mix of the Japanese and US equity market strictly dominates a 100% invested US portfolio. So now let's add. The UK equity market into the mix, okay? So, we can plot the equity, the UK could have markets average return and volatility in the same graph, all right? And in this graph it's represented by the star, so as this is from the given data. So now, what happens when we combine all three equity markets into a portfolio? Okay? So the lighter line shows you the possible combinations that we can create by holding a mix of the US and the Japanese equity markets. The blue line now shows you the minimum variance frontier, the local support folios that deliver the minimum variance, right? From holding the three markets, the US, the Japanese, and the UK market. Now you see that the minimum variance frontier expends in two ways, right? The wings expand outward, right? And the minimum variance portfolio, sort of the left most point, shifts a little bit towards the left. You can barely see it on this graph, why? Well, remember, adding another security into the mix that is less than perfectly correlated. Increases the amount of risk you can diversify away and reduces the portfolio risk, right? So for example, when you could achieve this expected return by the holding a mix of the US and the Japanese market for this given little of risk by combining the three equity markets together. Now you can actually achieve a higher reward, a higher expected return for the same level of risk. Now what if we add the rest of the G5 countries into the mix? Let's now add also Germany and France into the picture, okay? So now, what do you see, well, what you see is the minimum varies frontier expands even further, right? There is a considerable leftward shift in the front here, the minimum variance portfolio has shifted leftward. And the wings have become even wider, providing a greater reward, greater expected return, for the same amount of risk. You also see something interesting, something really interesting, right? Notice that all individual equity markets, they are inside the frontier. They lie inside the frontier, not on the frontier, they are inside the frontier, what does that mean? Well, that means all individual assets are dominated all individual equity markets, in this sense, are dominated by the diversified portfolio combinations that lie on the frontier. In other words, no individual market should be held by itself, right? Diversification helps remove the specific risk, the equity market specific risk and reduces the all risk of the portfolio. All right, so in this lecture, we look at how diversification works using equity market data from G5 countries. Your take away should be that there are large diversification benefits when there are low correlations in markets. You also learn the definitions of minimum variance frontier and what an efficient portfolio is. The key insight here is that individual assets lie inside the frontier or another words diversified portfolios always do better than individual assets. Because diversification removes or eliminates the idiosyncratic risk and reduces the overall risk of the portfolio.