 In this section, I will derive the formula that is used to draw the expected rate of return and risk rate offline, which is also known as the capital allocation line. The first thing we need to do is to relate the portfolios expected return to the proportion that is invested in the risky asset and that proportion is we are assuming that proportion will be given by W. So if we consider the entire investment as one, W is the proportion of that entire investment. So then the proportion which you have invested in the risk less asset will be given by 1 minus W. So firstly, we need to find out the expected rate of return in this situation and we are assuming that the investor has a choice to invest in the risky asset and the risk less asset. We need to invest both in the risky and the risk less asset. As I told you two or three times before, in order to be safe, it is always better to put your money in the risky as well as the risk less assets so that you get some reasonable return and you don't have to take too much risk. So how can we explain the trade-off line in the form of an equation and derive it? What we need to know is that first you need to know the proportion of the total investment that you want to invest in the risky asset and how much you want to invest in the risk less asset. We are representing them from W and 1 minus W. So in this case, the expected rate of return is given by capital ER and that is equal to the proportion of investment invested in the risky asset and the ERS that is the expected value of the return which you are going to get from the risky asset. So you have multiplied the average of the risk less asset by calculating the proportion of the risk less asset. Then we have got 1 minus W is the proportion of the risk less asset that you are investing in the total investment and we didn't take the average with it. You know that it is fixed, it is known, it is predictable, it is given and there is no fluctuation in the risk less asset so we are not taking the mean or average out of it. The absolute value given is the predictable known value that is RF and we are taking it. Because it is known that there is no fluctuation in it so we know that the standard deviation of the risk less asset is zero. So if we develop this particular equation and solve it, simplify it rather then what can we do? You have multiplied RF with 1, RF multiplied with W and W is the expected value of the risk return multiplied so to simplify it, you will get 1 RF with which there is no W so you have simplified it, it will be RF plus W, ERS, right? One W is coming with it, then you have W here you know that it is a negative sign so it will be minus W and then it will be RF. So if we simplify it by taking out the W in common then we have this equation. So this is the equation of the trade-off line in which you can find some information here. This particular value is the slope and this is the intercept. You have always read the equation of our straight line we write Y equals to A plus BX so we have to take Y on the vertical axis and we have to take X on the vertical axis when we read the equation of the basic straight line in maths this B is your slope, right? Rise over run which we call it or we can write it as delta Y over delta X and this is your A, this is the intercept. So the intercept tells you that your line will cut over Y axis on the vertical axis that is the Y intercept. So Y intercept tells you the value of A and the slope of this line tells you the value of B. Similarly, what you have derived expected rate of return equals to RF i.e. risk-free return plus W the proportion of the total investment invested in the risky asset bracket expected rate of return from the risky asset minus RF i.e. risk-free asset. In this particular formula, as I told you RF is the intercept i.e. this is the value of A and ERS minus RF we call this risk-premium risk-premium means I will put my money in the risky asset when I will get more money from risky asset then I will take risk and the value of additional return is called risk-premium to compensate the risk I am taking I will get extra return and we represent the risk-premium with ERS minus RF. So you saw that this particular thing that will give you the value of the slope that your line will be steep, flat how will be the slope to see this, I am going to show you the graph which we have discussed earlier so right over here again I told you that the point that we discussed before on this particular thing again the risk-reward rate of line or capital allocation line we take the expected return on the vertical axis i.e. this is ER and you have taken the standard deviation horizontal or y axis on the x axis and the line you have made we have derived the formula of this line and that formula said the expected rate of return that is equal to RF which is the intercept that we discussed earlier we said on the risk-free asset you are getting 6% interest and there is no risk the risk is zero so we said the risk is zero and return would be 6% we came up with this F portfolio in which you invested 100% investment in the risk-free asset and you got this point this is your intercept which is representing RF along with that we said the slope value of this line as I told you y equals to a plus bx so this is your y and this is your a I am going to write the value of b and then I will write the value of x so the value of b is the risk-premium and what we have derived that is expected rate of return from the risk-free asset return from the risk-free asset this is your slope and with that you have written how much proportion of the total investment you are investing in the risk-free asset you have written w so this is your equation of this straight line this slope which is delta y delta y is the vertical distance over the horizontal distance delta x delta y over delta x this is your value that is the difference between expected rate of return from the risk-free asset minus the risk-free asset or we will say the slope is also the value of risk-premium so this is your risk-premium or from this particular formula you will get the slope of this line the more the difference the risk-free asset of the expected value of the risk-free asset the more the slope of this line the less the slope and the less the difference the flat line will be but where the vertical distance will be cut you will get the value so this is the formula which we use in order to draw the risk-reward paid off line now I am going to give you an example discuss an example with you so that this particular thing can be further explained so in the example we are assuming again that we have a total investment of 1 lakh rupees in which we have to invest some money on the risk-free asset we have to invest some money on the risk-free asset we can also make such a portfolio in which we can put the whole 1 lakh in the risk-free asset or in the risk-free asset right so that we can also make such a portfolio but it is always better that you distribute your money between the risk-free and the risk-free asset then you are in a better position then you are in a more efficient combination so if you are getting 6% interest on the risk-free asset and you are getting 14% return on the risk-free asset so the expected rate of return the equation we made y equals to a plus bx so that can be stated as expected rate of return this is your y on the vertical axis the value of cut point is 0.06 as I have shown you this is your intercept and then 0.14 minus 0.06 which is 0.08 this is your risk premium and this is your slope value so this is the slope this is the intercept and with the help of intercept and slope you can plot the line with ease the risk premium that is the value of 0.08 it will be the difference between the expected rate of return from the risk-free asset minus the return which you are going to get from the risk-free asset basically what we are talking about the risk-free premium it depends on the particular the portfolio how much money you will earn it depends on the risk-free premium the other important thing that you have to keep in mind is how much proportion of your total investment is invested in the risk-free asset if you are not investing in the risk-free asset or you invest very little proportion or the value of W is very small obviously the expected rate of return will not be there but if you put a big proportion of your total investment there are chances that the expected rate of return will be higher as compared to if you invest your entire money in a risk-free asset so these are the two important things which we have to consider when we are planning to allocate funds among different investment opportunities