 In this section, I will explain how the preferred portfolio is selected. So firstly, we need to understand the concept how the optimal combination of the risky assets are identified. So basically, we look at the capital allocation line and then we look at the efficiency frontier. So we have already discussed these two concepts. Capital allocation line explains all possible combinations of risky asset and risk-free asset which you can take from the amount of investment. Our efficiency frontier is a combination of all those points, those combinations of the assets which are most efficient at a given certain level of risk or volatility which we have defined as a certain risk level and the best of the best combinations we call their combination as an efficient frontier. So if we see that the capital allocation line is tangent to the efficiency frontier we call that particular point as the optimal combination of risky assets. We are talking about risky assets and we are not talking about risk-free assets. So we are assuming that we will be coming up, we will be developing a portfolio in which there are two risky assets and one risk-free asset and these three assets are based on our portfolio. So basically, if we look at this point of tangency of the capital allocation line and the efficiency frontier, it will give us the best risk to reward ratio and that will be considered as the optimal combination of the risky assets. So this is the definition. If we want to look at how we can calculate this through formula, we are assuming suppose if W1 is the proportion of risky asset 1 and W2 is the proportion of risky asset 2, then W1 can be calculated by taking into account the expected return from the risky asset 1 minus the risk-free return. We will multiply this with the variance of risky asset 2's return. Similarly, we will have to consider risky asset 2's expected return minus the risk-free asset and it will be multiplied by the variance of the standard deviation of risky asset 1's return, the standard deviation of risky asset 2's return and this will be further multiplied by the correlation coefficient between the risky asset 1 and risky asset 2. This whole thing will be divided by expected return from risky asset 1 minus risk-free asset multiplied by the variance of risk-free asset 2 plus expected return of risky return 2 minus RF is risk-free return, right? And this will be multiplied by the variance of risky asset 2, risky asset 1 minus, you will subtract the top term. I will add another term and that would be expected return from risky asset 1 minus risk-free asset plus expected return of risky asset 2 minus risk-free asset key return and this whole will be multiplied by, that is, row 1-2, the correlation multiplied by standard deviation of risky return 1, risky return 2 key standard deviation and if you multiply this, you will get a denominator. So, once you have estimated the value of the proportion of the total investment that will go to risky asset 1, the next thing you will be able to come up with is W2 which is the proportion of risky asset 2 and how much total investment you are going to invest in risky asset 2 and that can simply be obtained by subtracting W1 from the 1. So, you will get W2. From here, even with the help of W1 and W2 formula, you can remove the same point which we talked about the tangency point. So, if we want to see it graphically, that is again, for that matter, what we will have to do is, we will have to take into account the risk on the x-axis and the return on the vertical axis. So, then what we do is, we draw the capital allocation line. We have discussed it earlier, its slope, what will be its intercept? Its intercept is always your risk-free return and then we draw the efficiency frontier. So, suppose this is the efficiency frontier and the efficiency frontier point where the capital allocation line is tangent, we have just defined on that point that is that point gives you the optimal combination of the risky asset 1 and risky asset 2. I am labeling it as point t here. So, how much money do you have to invest in risky asset 1 and risky asset 2? That value comes out. Now, the formula of W1 and W2, you can take it out with the help of it mathematically by using the formula, but if you want to see it graphically, so that will correspond to this particular point where the efficiency frontier is tangent to the capital allocation line. Now, for the investor who has options to decide, we can see how he will invest because on the t-point, those combinations are coming that he is investing all his money in risky asset 1 and risky asset 2. But again, in order to properly develop a portfolio, sometimes it is recommended that you put some of your money in the risky asset also. So, for that, what will we do? For that, if we put this particular point, if we label this point as point f or label this point as point t, then any investor who has to decide the portfolio, they will go for any combination that fall on the line between point f and point t. All these points will be recommended. Now, which point or portfolio does one investor prefer? How that particular portfolio, the preferred, all these lines from point f to point t, all these are the preferred portfolios. But how we are going to select, which one of these portfolios we have to select? For that matter, there are certain other things that are taken into consideration by the investment managers, who are the investment advisors or financial advisors. They consider certain other factors also and in that, investors also take their input and to make a final product, what kind of products they like. So, for that matter, there are the other things that are considered in addition to the expected rate of return, the risk-free return, we discussed it as well. Then we counted the standard deviation values. Apart from this, it is important to consider what is your planning horizon. We have discussed the planning horizon. What is it? Next important cycle, important point is that you are sitting on the point of life cycle. You are close to retirement. You have retired. You have started a new job. You are a part of your age. This will play a very important role that you, which is the different preferred portfolios between your entire effort and t, which one of the portfolios you are going to select? And similarly, another important factor that also plays an important role that what is your risk tolerance level? So, all these things, in addition to expected returns, risk volatility, we also have to consider all these things. Now, when we have to choose the optimal combination, now, between F and T, which of the combinations you are going to choose, the wealth of investors or their personal preferences do not count. You do not have to consider every detail of an investor when portfolios are being defined for them. The investors who have so much wealth, these details do not matter. What matters is that the financial advisors consider all these things, which we just discussed, what are the things people like in a different life cycle? What is their risk tolerance? And what combination is better for a high level of risk tolerance? What is the final product for people with low level of risk tolerance? But what they do is, they look at three important factors that you can see. If you look carefully, the formula that we just saw, for W1, W2, we used these three things. They consider the expected rate of returns from the risky return or from the risky assets and the risk pre-assets. The second important thing that we have considered, that is the standard deviations or the third important thing is the correlation between them. So, what is the extent to which they move together? They move together in the same direction or they move together. They move in opposite directions or if they move together in the same direction, then what is the strength of association? So, basically, what we are trying to do is, when we have to do the optimum allocation, we see three important things. We do not see the wealth of your investor, we do not see the basic details of personal characteristics. What we look at basically is primarily, what we see is the expected returns from different risky assets. What is the level of risk attached to it? You will get to know from Sigma. Then we wrote row one, two, symbol. It tells you the correlation that you are pooling the risky assets to make a portfolio. How is the correlation between them? So, these are the important factors that are considered when we go for the optimal allocation and we have a choice of different types of risky assets. And now, we assume for simplicity that we are dealing with a situation in which there are only two risky assets involved.