 Once we have understood the computational process of covariance and correlation, then it becomes easy for us to understand the computational process of standard deviation of a portfolio riskiness. It is the Markowitz who derived the general formula of standard deviation for determining the riskiness of a portfolio. This standard deviation formula for a portfolio is basically the function of two factors. The first is the average of the individual asset's variances. And the second factor is the covariance is between all pairs of assets in the portfolio. So we can also say that the portfolio standard deviation covers two items. The first is the individual asset's variances and the second is the covariance is between all of the assets in the given portfolio. So what is the impact when we add a new asset in the existing portfolio of assets? There would be two effects while adding a new security to the portfolio of assets. The first is on the asset's own riskiness means that there will be an effect on the riskiness of the new asset that is adding to the portfolio. And second is the covariance between the new asset's returns and the returns of every other asset in the portfolio. This means that relative weight of these numerous covariance is substantially greater than the asset's unique variance. This means that a portfolio with many assets, if this is the case, the more is true for this particular case. Hence we can say that important factor to consider while adding a new asset in the portfolio of existing assets is that the average covariance of new assets will be with the portfolios of all other assets. So the change will be on the covariances of the new asset's return with the returns of every other asset in the portfolio. Now how to calculate portfolio's standard deviation? Due to the assumptions used to develop Markovitz's portfolio model, any portfolio of assets can be featured by two characteristics as we have also seen earlier. And these features include expected rate of returns and the expected standard deviation of returns. So these two are the features that are used in order to determine a portfolio's standard deviation. The correlation measured by covariances, these affect the standard deviation. So along with the individual asset's expected returns and their standard deviations, there are also other factors that is the covariances and the correlation that determine the riskiness of a portfolio. Lower correlation reduces portfolio risk but it does not affect the expected returns. There are certain cases, let's take an example. If we have equal risk and return but changing correlation, let's see what happens. There are certain assumptions. We have two assets of portfolio. We have expected return on asset 1, 20% and expected return on asset 2, 20%. Both of the assets have similar riskiness of 10% that is standard deviation. Both are equal weighted assets that is 50% in each. Then the standard deviation of both the assets is equal to 0.10 or 10%. We have five correlation of these two assets equal to 1.5, 0.0, minus 0.5 and minus 1. In this data, we have determined five covariances in these five cases. For case A, we have covariances of 0.01, for B, we have covariances of 0.005. We have zero covariances point C, we have negative 0.005 covariances for D and for case E, we have negative covariances of negative 0.01. When we determine the individual portfolios return in all these five cases, we come to certain observations and these observations can be seen on the screen. For portfolio A, the standard deviation is 0.1, for portfolio B, the standard deviation is 0.08, for portfolio C, it is 0.07 and 0 is for portfolio E. Now, what these values say for standard deviation of portfolio A, as we have a perfect positive correlation, therefore the portfolios riskiness is basically the weighted average of the individual assets standard deviation. So as we have a positive correlation, so we have no benefit of diversification in this case and in case of portfolio B, again, we have a positive correlation. But there only a change in the assets riskiness, but no change in the expected return of the portfolio. Same interpretation can we have for the standard deviation of portfolio C and standard deviation of portfolio D, but for the portfolio E, we have a different interpretation because we have perfect negative correlation and a negative covariance between the two assets in portfolio E. So we have exactly case where the individual variances terms are offsetting each other and therefore there is no riskiness for this portfolio. This means that the standard deviation of this particular portfolio is 0. Therefore we can say that this portfolio E is basically a risk-free portfolio. To understand the concept of diversification, we have a graph here. In the left panel, we see that there is a full diversification because there is the case of risk-free portfolio and we know that perfect negative correlation gives a combined mean returns for the two assets over time equal to the each asset's mean. And that we say here, we have over the line mean returns for asset A and for below the line we have returns for asset B. We see that there is a mean combined return for these two assets and that is basically the portfolio's return. Any return above and below the individual assets mean are completely offset by the return for these other assets and this we can see here. The increase in the return of asset A are offsetting the decrease in the returns of asset B. So any increase in an asset is offsetting the decrease in other assets but we see that the straight line shows that there is no variation in the portfolio's return. There is variation only in the individual asset's return. So as the returns of the portfolios are showing no variation, so there is no risk for the portfolio and that is what the diversification can offer benefits at maximum to an investor. On the right side, we see there is no diversification as seen through the straight line because the assets with no perfect correlation have no effect on the portfolio returns but they can only reduce its risk. We have five portfolios A, B, C, D and E and all these fives are offering a rate of return of 20% to the investor. So there is a common return for every portfolio but the variation is only in terms of standard deviation. So combining these two pictures, we can say that only assets with perfect negative correlation can eliminate the risk and meet any portfolio risk-free.