 Earlier we see the implication of standard deviation of a portfolio upon the diversification and in that case we studied a case of equal risk and return in the case of changing correlations. Let take another case in the similar direction. We have now different risk and return but changing correlations that is the second case in our discussion. We have two assets asset 1 and 2, we have corresponding expected rate of returns and these two assets have equal weights. We have co-variances and variances of these individual assets. We have five cases, we have individual coefficient of correlation for individual case, similarly we have standard deviation of asset 1 in individual case and standard deviation for asset 2 individual case. Similarly, we have co-variances in these five cases. Using all these data we determine the portfolio riskiness that is standard deviation of a portfolio in all of the five cases. Now what these values basically say, we can say that with perfect positive correlation the riskiness of a portfolio is basically the weighted average of the standard deviation of the individual assets as we have seen in these five examples. And with the changing weights we see that with the changing weights with perfect positive correlations changes in the portfolio riskiness occur in a linear fashion that we have also seen earlier. At number 3 we have seen that with perfect negative correlation the riskiness of a portfolio is not zero because the assets have equal weights but unequal standard deviation. So these are the observations that we can see in our computations. In the right below panel there is a graph that depict five portfolio combinations portfolio A, B, C, D, E and here is individual asset 2 and individual asset 1. You see that there is no change in expected rate of return for the portfolio because the assets have equal weight so all of the portfolios lie along the horizontal line and all of these portfolios are earning an equal rate of return which is 15% in this case. The second case we have the constant correlation with changing weights. In this case the changing weights of two assets while holding the correlation constant a set of combinations can be derived that basically trace an ellipsey starting at asset 2 going through a 50-50 point and ending on asset 1. We see the implication of this observation in the lower half of the screen. Here is asset 1 that is individual asset here is asset 2 that is individual another individual asset when we combine these assets in equal proportion we have a combination of certain portfolios and we see there are portfolios G, H, I, J and K and if we go on combining these assets in certain percentages we have a complete curve in this shape as a complete curve we can drive by simply varying the weights of individual assets in a smaller proportion or smaller increments starting with 100% investment in asset 2 which is case F and changing weights as follows then ending with 100% investment in asset 1 which is case L that is our assumption then we already know that the standard deviation for portfolio F and I because there is only one asset in each these two portfolios and then the various pairs with a constant correlation yield the following risk-retrain relationship the data we have used earlier that data can be manipulated and we have certain values these values we can see on the screen and we have certain cases from F to L F, G, H, I, J, K and L we have a weight 1 for one asset and weight 2 for another asset so in this way we have seven combinations using seven different weight proportions then for each combination of each portfolio we have derived expected return and expected riskiness of that particular portfolio. Using these combinations we have derived a diagram this is that the graphs curvature that is here this curvature is basically subject to the correlation between two assets or the portfolio of assets we have certain correlations here the first correlation is plus 1 and that is the perfect positive correlation and that is here we see that combination lies along a straight line between two assets from asset 1 to asset 2 if we see another correlation that is 0.5 though this is a positive correlation yet it is not a perfect correlation then we have seen that the curve is to the right of the curve that has a 0 covariance and that lies here then we have a third combination of portfolio where we have correlation of negative 0.50 and here the curve is to the left of the curve which has plus 0.5 correlation and this is the curve where the correlation is 0.5 here is that is the curve then we have a perfect negative correlation where the correlation is negative 1 if we see this diagram we see that the graph here is basically the two straight lines touches at the vertical lines with a certain combination and here is the point where we can say that this is the no risk point that is any portfolio touching here is basically the risk-free portfolio going into deeper we have certain observations we can see that where the correlation is 0 and that is the point here and this is the basically low risk level at that low risk level the investor at L point and that is here with a return of 10 percent and a risk level of 7 percent and that is the portfolio L the investor can increase this return up to 14 percent by investing in portfolio J he can reduce his riskiness to 5.8 percent so shifting from portfolio L is only possible if he invest 40 percent of his investment in asset 2 then he can buy a portfolio J and here at portfolio J his rate of return would be 14 percent that is 4 percent up from the previous 10 percent and his risk level will be decreased to 5.8 percent from 7 percent so here we can see a little benefit of diversification using this particular relationship between these two portfolios we can see that the diversification benefits are basically very critically dependent on the correlation between two assets that is we have just seen that from zero to earlier to a lower level we have a benefit of diversification this means that with low zero or negative correlation portfolios with lower risk then either single asset can be drawn this means that at zero correlation or less than zero correlation or a perfect negative correlation we have multiple assets in a particular portfolio and the riskiness of all these assets will be much lesser than a single asset which in our case is asset 2 that is forming the portfolio F and asset 1 that is forming the portfolio of L this ability to reduce risk is basically the essence of the diversification now we can see the individual interpretation of these graphs with two we have first perfect positive correlation where R is plus 1 with two perfectly correlated asset it is only possible to create a two assets portfolio with risk return along a line between either single asset so in this combination either we can invest wholly in asset one or we can invest fully in asset two then we have a case there where there is no correlation this means that we with uncorrelated assets it is possible to create a two asset portfolio with lower risk than the riskiness of an individually single asset and this is the case with our portfolio H I J and K where the correlation is equal to zero this ability to reduce riskiness of a portfolio is termed as the diversification and this is the essence of diversification as we can see on the diagram the curvature of the diagram is showing that as we go to the left of the straight line our riskiness of the portfolio goes on decreasing another case where we have a positive correlation yet not perfect positive correlation with the correlated assets it is possible to create a two assets portfolio between the first two curves and it lies between the plus one correlation and the zero correlation and in our example it is showing as a red curve where the correlation is 0.5 though it is a positive but not perfect positive correlation then we have a negative correlation in our case we have a negative 0.5 correlation this is to the more left of the curve which has zero correlation in our case with negatively correlated asset it is possible to create a two asset portfolio with much lower risk than the earlier single this statement basically says as we as we move away from perfect correlation away from positive correlation away from zero correlation as we are moving away from perfect correlation perfect positive correlation our riskiness in the portfolio is going on decreasing then the riskiness of a standalone asset now in the last we have a case where we have perfectly negative correlation that is the correlation equal to minus one this is the case in which with perfectly negatively correlated assets a two asset portfolio can be created with almost no riskiness as we have seen in earlier cases that as we move towards the left from our perfectly positive correlation by moving more towards the left we reaches to the vertical line and that is basically the y-axis towards this line our riskiness goes on decreasing and any portfolio on the line when this line touches to the vertical line where the riskiness is almost zero this means that any portfolio lying on this line will have no riskiness and by this means we mean that the standard deviation of such portfolios is zero and we can see on the screen that such graph basically would be the two straight lines touching at the vertical line with some combination and any combination on this line with have zero standard deviation this means we are creating risk free portfolios for the investor at this line