 The Markovitz portfolio throws light on certain other issues like estimation issues that are related with determining the variances and other riskiness measures of the market portfolio, the efficient frontier and the efficient frontier and the investor's utility curves. So far as the estimation issues are concerned, we know that the results of portfolio allocation depends upon accurate statistical inputs for every asset to be a part of the portfolio, their expected return, their standard aviation and the correlation between the two assets among the entire set of assets is to be estimated. So these are the measures that are required to be estimated in order to determine the riskiness and return of a portfolio. Here is an example. For example, for a portfolio of 100 assets, the number of correlations are 4950. So for a portfolio of 100 assets, we need to determine the correlations in the number of 4950 and it is a significant number of estimations. So there may be an estimation risk, which means the potential source of error that arises from these approximations. Using assumptions that stock returns can be described by a single market model. A number of correlations required can be reduced to the number of assets. So there is an assumption and that single as index market model is basically where ri is equal to alpha plus beta of ri into rm plus the error component. In this model, the beta is basically the slope coefficient that relates the security i returns to the aggregate stock market returns. By rm, we mean the aggregate stock market return. So this is the single index market model that can help us in reducing the significant amount of estimations. If all the securities are similarly related to the market and beta of security i is derived for each of the security, then it can be shown that a correlation between two asset is given as this is the formula which is used to determine that said correlation. In this formula, sigma square m is basically the variances of the returns for aggregate stock market. This equation reduces the number of estimates from 4950 to 100. This means that this equation assumes that the single index market model provides a good estimate of security returns. Now come to the efficient frontier. On the screens left panel, you can see that there are certain curves that are combining certain portfolio of assets. Basically, how these curves are drawn with all possible combinations of different two asset portfolios and all possible weights, these curves can be drawn. So this means that to draw such curves, we means any two assets, any two different assets and using different combinations of these two assets, we can draw these types of curves. And on the right panel of the screen, you can see there is a thick curve that is encompassing the all possible portfolios. This thick curve is known as the efficient frontier. So this is basically the envelope curve that contains the all possible combinations that we have just seen on the left panel of the screen. Now how this efficient frontier works? Basically, this efficient frontier signifies a set of portfolio that offers two factors to the investor. It basically at first offers the maximum rate of return for every given level of risk and or the minimum risk for every level of return. So at efficient frontier, the investor can choose any security that offers him maximum rate of return for a given level of risk or another way the investor at efficient frontier can select any security that offers him minimum riskiness for every level of return. Then due to the benefits of diversification among imperfectly correlated assets, the frontier will be the portfolio of investments rather than individual securities that we can see in this diagram here we have portfolio of F and portfolio of B alongside this efficient frontier. But there are two exceptions on this efficient frontier. At the ends of this efficient frontier, we can see only the portfolio carrying a standalone asset. For example, at the maximum end, we have a portfolio that contains only one asset and that is asset two. This asset two is offering the highest most expected return. And on the other end of the efficient frontier, we have another asset that is asset one. This is a standalone asset portfolio. This portfolio of asset one offer the most least riskiness of the portfolio to the investors. Every portfolio lies on the efficient frontier has either a higher rate of return for the equal risk or lower riskiness for an equal rate of return than some portfolio beneath the frontier. This means that any portfolio that lies on this thick line that portfolio will offer either higher return than a given riskiness or lesser risk than a given level of expected return than the portfolios that are available under this efficient frontier. To understand this case, let an example where portfolio A is dominating portfolio C because the portfolio of A is offering the same level of expected return but a lesser level of riskiness. Similarly, the portfolio B is dominating portfolio C because for the same level of riskiness, the portfolio of B is offering a higher rate of return as we can see that C is offering what level of expected return but the B is offering higher level of expected return but both are offering the same level of riskiness. So, that is the fundamental feature of any efficient frontier. An investor can choose any point alongside the efficient frontier based on his utility function which reflects his attitude towards the risk. Remember, no portfolio on efficient front can dominate any other portfolio on the same efficient frontier. All of these portfolios have different returns and risk measures with expected rate of returns that increases with the riskiness. Now, let come to the efficient frontier and the investor's utility curve. An investor's utility curve basically specifies the tradeoff he is going or willing to make between expected rate of return and the standard deviation are riskiness of the portfolio. Slope of the efficient frontier curve decreases steadily at one as one moves upward. This means that with every incremental standard deviation, additional expected return goes on decreases and that is the feature of the utility curve. Interaction of efficient frontier and utility curve basically determine the particular portfolio on the efficient frontier that best suits the individual investor's attitude towards the riskiness of the portfolio he is investing. On the left panel of the screen, you can see there is two types of interacting graph. The first is the efficient frontier and the other six lines are basically the investor's utility curves. So we have two sets of utility curves along with the efficient frontier of investment. We have three sets of utility curves that is u1 prime, u2 prime and u3 prime. Then we have other three sets of utility curves u1, u2 and u3. These u1s to u2 and u3 curves are the strongly risk averse investor's behavior and the curves with prime are showing the behavior of low risk or conservative investor. We see that the curves u1, u2 and u3 are quite steep. This means that the investor will not bear much additional risk in order to obtain any additional return and also the investor is basically equally satisfied towards any risk return combination alongside the curve and specifically the u1 utility curve. The other utility curves u1 prime, u2 prime and u3 prime these are featuring a low risk averse investor. This means such investor is willing to take little more risk for higher expected level of return. Now how these curves can help in determining an optimal portfolio. So first we need to define what is optimal portfolio. Optimal portfolio is such type of portfolio that has highest utility for a given investor and such portfolio lies at the point of tangency between the efficient frontier and one of the investor's utility curve. So we can say that the point where the investor's utility curve is tangent to the efficient frontier at that point the portfolio if located is called as the optimal portfolio because such portfolio at such tangent point offers highest utility to the investor. For example we see that any conservative investor's highest utility at point x where the u2 curve is just touching the efficient frontier. So this is the optimal portfolio for the investor x because it is offering him the highest utility for a given level of riskiness. We see another example where a less risk investors highest utility is occurring at point y which representing higher expected return and higher riskiness than the riskiness of portfolio s. So this y point is basically the optimal portfolio for investor which is at u2 prime because this is offering him the highest utility for highest rate of riskiness and this rate of riskiness and return is much higher than the portfolio x.