 The returns on any risky assets are generally uncertain and their uncertainty is measured by the standard deviation of the returns. On the other hand, the returns on risky assets are certain with zero standard deviation. The rate of return earned on risky assets is known as risk-free rate of return or RFR. Now let's see what happens when a risk-free asset is combined with risky assets through the Markowitz portfolio model. At first we have a covariance with a risk-free asset. We know that covariance between two assets is the sum of their individual main deviations. And for a moment if we assume that in our equation the Ri refers to the return on a risk-free asset, then we know that the returns of risky assets are certain. This means that the standard deviation of returns on risky assets and their main deviation both are equal to zero. This means the covariance of risky assets with any risky asset will always be equal to zero. This means that correlation between any risky asset and the risky asset would then also be equal to zero. What will happen if we combine a risk-free asset with a risky portfolio? We have two effects of this combination. The first effect would be on the expected returns. We know that the expected returns of a portfolio that includes a risk-free asset with the collection of some risky asset is basically the weighted average of the two return streams. This means that the equation would be equal to weight of risk-free asset into risk-free rate plus 1 minus weight of risk-free asset into the expected return on market portfolio. If we analyze this equation, we can understand that the expected return on portfolio has a linear relationship with the expected return on the market. The second impact in this connection is to be considered on the standard deviation of the expected return. The expected variance for a two-asset portfolio as we know that it is the sum of the weight of weight 1 of the standard deviation of risk of an asset 1 plus the weight 2 of the standard deviation of the asset 2 and the covariance between these two assets. Now if we substitute the risk-free asset for the security 1 and any risky asset portfolio which is portfolio M for the asset 2, this means that if we replace asset 1 with the risk-free asset and asset 2 with the risky portfolio, the equation will remain the same but with the replacement of the expressions. We know that both the standard deviation on risk-free rate of return and the correlation between risk-free asset and a risky asset is equal to 0 when we place 0 in the second equation, we will basically have the standard portfolio riskiness which is equal to 1 minus risk weight of risk-free asset into the standard deviation of the market portfolio. And the result is also the same as we have seen in our previous slide that this portfolio of the standard deviation is basically in this particular case is also the linear proportion of the standard deviation of the riskiness of the market portfolio. Now after these two workings when we have a risk return relationship between the portfolio's riskiness and the portfolio's expected return, if we manipulate a little mathematical equation, we have a conclusion in the equation that the expected return on portfolio is basically the summation of risk-free rate and the risk premium of the investor per unit of risk. If we interpret this equation, it will say that investors who allocate their money between a riskless asset and a risky portfolio M, they can expect two things. Number one, a return equal to the risk-free rate and number two, a compensation for the number of risk units. These are the two rewards that an investor should expect when he combines a risk-free asset with a risky portfolio. And this particular expression confirms the investment theory which says that investor basically performs two functions in the capital market for which he expects to be rewarded for the others to use their money in exchange for the risk-free rate. This means that whenever an investor comes into the capital market, he needs a compensation for the amount he gives to other in exchange of a risk-free rate of return. And second, the investment he puts in risky assets, he bears the risk that the returns promised with that particular investment will not be repaid to him. So, these are the two rewards, reward first as a risk-free rate of return and reward two as a risk-free premium for which an investor comes into the capital market. Now, what happens if we combine a risk-free asset with a risky portfolio in this continuation, we see that a linear combination between the expected return and this riskiness for such a portfolio helps to draw a graph of possible portfolio returns and risk. And this graph shows a straight line between the two assets. And that straight line we see here which starts from RFR at the vertical axis and it goes on here. That line is termed as a CML or the capital market line. Basically the equation which we have seen earlier is called as the CML equation. This CML is in fact a straight line who has an intercept at risk-free rate and its slope is equal to the expected risk-free premium. So, that is the slope which is basically equal to the expected risk-free premium per unit of market riskiness. This CML defines the relationship between expected return on a market portfolio and the riskiness of the market portfolio. And the slope of the CML shows the investor's overall attitude towards the risk. And now for a short while, if we assume M as a market portfolio which comprises of a single risky asset to minimize the risk-free premium. This means that M is the market portfolio containing all risky assets held everywhere in the market. So, this means that this market portfolio will receive highest level of expected return in excess of the risk-free rate. So, if we draw this up to this, then the difference from this to this will be the excess return, excess over the risk-free rate and that is the excess return. And this excess return is measured in terms of the riskiness of the market portfolio. In this way, we see that the earlier equation which is basically explaining two things, the risk-free rate and the market risk premium per unit of riskiness, that equation is basically the CML or the capital market line. Now if we draw a straight line using various combinations starting from a risk-free asset and combining with the many risky assets along with this market Markovitz efficient portfolio. For instance, we have this combination from RFR to the portfolio A that RFR A is the combination that is lays alongside the market portfolio. Then any combination on this line would dominate the portfolio possibilities that fall below it. For example, if we see another combination that is RFR to B, then RFR to B is a portfolio that is above the portfolio of RFR A. The portfolio of RFR B is dominating the portfolio of RFR because at this portfolio the expected return is higher having the similar or same level of riskiness. If we continue to draw lines similar to RFR A or RFR B at the efficient frontier with the increasingly higher slope, then we will reach at a point that will be tangent to the market portfolio. Then we will say that the portfolios along line RFR M will dominate all other possible combinations that investors could form. For example, at point C, this point C could be said, let's say while investing 50% of the assets in risk pre-securities and the remaining 50% of the amount in portfolio M. So, we will have an other efficient frontier alongside the Markovid's efficient frontier. While adding these lines, we will have a complete straight line with all the feasible combinations that the investor can set. This means that our CML basically is now an other efficient frontier that combines the market efficient frontier of the risky assets with the ability to invest in the risk-free security. So, in this way, we will be having two efficient frontiers. The first is the Markovid's efficient frontier with the risky assets and the other is the CML that is containing a risk-free assets. Now, what will happen if we put the leverage into our investment? Then to attain a higher expected rate of return, then the return which is available at portfolio M or the market portfolio excepting a higher riskiness, what will happen? This means either we need to invest along the efficient frontier beyond the point M and here we have at point D. This means that we can extend our investment to point D or we can add leverage to our portfolio while borrowing funds at the risk-free rate or the RFR rate. Then we will invest in risky portfolio at M. In fact, this is depicted at point E. So, in this case, through the leverage, we will be extending our expected rate of return with the higher standard aviation or the riskiness of the portfolio. Now, what is the effect of this leverage on our risk return combination for the portfolio investment? If we borrow at the risk-free rate, then it will have two effects. First, let's see on the expected return on market portfolio. We see that expected return on market portfolio will be increased in a linear fashion alongside the CML because this return increases in proportion to the borrowing because the investor must be interested at the risk-free rate on the borrowing that we see here that as we have earlier seen that expected return on a market portfolio is the weighted average return on the risk-free rate and weighted average return on the market portfolio. With the portion of borrowed capital, our portfolio return will increase in linear proportion to the borrowing. Now, let's see what is the effect on the portfolio riskiness of the leverage. We see that the effect is as same as the effect of leverage on the expected return of the portfolio. It moves in linear fashion with the borrowing proportion. So, we can say that both the return and the risk of the portfolio increase in a linear fashion alongside the CML in the presence of the leverage.