[MUSIC] We're now going to extend the investment universe a little bit. We're going to introduce a new financial asset in our example. This asset has a special characteristic, it will have no risk. We're going to call this asset the risk free asset. One way of interpreting what this represents is to think of investing in, for example, a treasury security. If you lend money to a government, you have a guaranteed expected return in the sense that you know exactly what amount you're going to receive at the end of the investment. However, there is no uncertainty. Some might say that there is a risk that the government falls in default, and therefore this create a potential risk. But let's, for the sake of our example, consider the situation where there exists such a financial asset that has a given expected return, usually a relatively low one, but absolutely no risk. In our graph, this asset would be located on the far left of the graph, actually right on the y axis with the level of standard deviation equal to zero. Here I've chosen level of expected return for the risk-free asset of 2%. This expected return is actually the return that will be realized By this financial security, because there is no dispersion for the distribution of return for this specific asset. So we're going to look now at a similar construction than the one we followed in the previous video. We're going to construct portfolios using not just a, b, and c, but a, b, c, and that new asset. That new asset has a special role here because it has of course no correlation with the other asset and no risk. So let's see if the structure of the diversified portfolio is going to look similar or different from what we've obtained in the previous construction. So here all these green dots are portfolios randomly chosen by allocating between the three risky asset and the risk free security. What we can see is that again there is a limit to the effective diversification, in the sense that all these portfolios reduce the risk of the portfolio by mixing the securities together through the effect of correlation. But there appears to be a limit. And this limit here is depicted by this red line and a black line going down. We can see that this new envelope, this new efficient frontier which includes our risk-free asset, start at the risk-free rate level. This makes sense. The portfolio with the lowest possible risk that we can construct is entirely invested in the risk-free asset. Along the red line we see all other portfolios that combine the risky security and the risk-free asset. So this new efficient frontier, the red line, includes the risk-free asset and all the risky security. It doesn't have exactly the same shape as the one we had before, and this is due to the fact that the risk-free asset has a risk equal to zero. Now what we could do is look at the two frontiers together, plotting the one we have now which includes the risk-free asset, and adding the one we had before which just asset a, b, and c. This is what is represented in this next graph. And you can see that there is one point that belongs both to the red line, the efficient frontiers with the risk-free asset, and the green line, the efficient frontier, without risk-free asset. Because it can be constructed with both set of financial asset, we call this point a point of tangency. It belongs to the two curves. This portfolio is rather particular because it can be constructed just by investing in a, b, and c and it also belongs to the efficient frontier with the risk-free asset. From that, we can conclude that this portfolio contains only risky assets. All the other portfolio on the red line actually have a very particular structure. All the portfolio on the red line are combination of that particular tangency portfolio and the risk-free assets. We will discuss more on this on the next video on the two separation theorem. [MUSIC]