 We are doing introduction to financial risk management and the module we are doing is mayor of returns and risk. In last lecture we discussed how returns are being calculated and we learned that there are many ways in which returns are being calculated and looked at. Now we'll move to another main category that is how risks are being calculated. So risks can be calculated again in many multiple ways and many ways of directing and looking into this spectrum. Like there is a view point in which how it is being taken care. So we have variance that leads us to calculation of the standard deviation. Then we need to learn about covariances, correlation and that for then portfolio risk. As I've told you, you might have seen standard deviation correlation in your statistical knowledge, but here we are applying them in the context of pure finance and portfolio management. Various and standard deviation of single acid. First we'll discuss about single acid and then we'll move to portfolio of two and more acid. So if we have a population of which we are calculating that is the whole data is available, then the formula is sum of the deviation square divided by the number to which it relates. But if we have and when we have to calculate its standard deviation, we take a square root of that. When we have a sample data, that means the whole population cannot be calculated. You have thousands or hundreds of entry and take a sample out of it. The formula remains a bit similar with the only difference that in denominator we deducted with one. Wherever the sample is talked about, normally there is a sample from the numbers. One minus is taken to take care of the sample factor or population to relate to it. And then for the same way we take, we get the variance. We'll take the standard square root and then we get the value. It has some important assumptions. When we take the standard deviation, we have few important assumptions which we need to specifically know. A, the returns are normally distributed. That means there is no skewness, no specific noise. And another important assumption is markets are informationally and operationally efficient. When I say informationally and operationally efficient, that means the information related to the stock or the bond is reflected in its price. Variance of portfolio of assets. When we have more than one asset in our portfolio, then how it is being calculated, they are triggers in a new element that is covariance. Covariance is actually the relationship between the two assets with each other. So we are going to learn about this as you can remember we have once talked about diversification. Diversification was the point that when you want to add assets, your returns will be on an average size whereas risk can fall down. If you can recall, we already spread out investment so that risk falls down. So here we see how that mechanism actually works. When we combine more than one asset, we will get an average of returns. But the average of risk will be reduced. The covariance and the correlation will play a role to bring down our risk. So here we can see the variance of a portfolio is we take the weights and the covariance this way and we take some of that. That will capture the risk of the whole portfolio. We'll do that with figures. We do that in examples here. Our important point that we'll be spending a little more time on it. For example, assume that investor, you decide to invest some money in an index. Because index is a wide area and you are spreading your investment. So now you decided that I will invest in an 80% index more like S&P 500 US or KSC 100 index like S&P 500. And remaining 20% in MSCI emerging market index. Emerging markets are developing markets where return potential is high. At the same time, risk level is a little high. Expected return is 9.93% for S&P 500 and 18.2% for emerging market. Here you can see the huge difference of expected return in both of them. How much gap? Because the risk of emerging markets is more. Now let's see the risk. The risk is 16.21% for S&P 500 and 33.11% for emerging market. So the return of emerging markets was more but the risk level was very high. If I invest all the money in the emerging market, that will mean that I am taking a lot of risk. What will be the portfolio expected return given that covariance between the S&P and emerging is 0.005. Now this covariance will determine the benefit of diversification. Hope we have understood this question. Then we will move to the discussion of its solution. Here what we have? 0.8 refers to 80% invested into the return of the S&P that is 0.99.93. We will take it as 0.093 because we are applying the factor. Then we have 20% investing in the emerging market to 0.2 multiplied by 0.18-20. This gives us return of 11.58. You can just see it's just average. We have invested 80% there and 20% there and we have summed it up. Which is normal understanding that we have added this. But this will not be the case in the matter of risk. We will not say that weight multiplied by risk of 1 then weight multiplied by risk of 2 and add that. No. Then there is no diversification benefit in the risk factor. As we have seen, we take weight of 1, then 2, square of these and then 2 multiplied by weight of 1 into weight of 2 and covariance. If you remember the formula of mass, then it is A2 plus B2 plus 2AB. It is on the similar pattern. So we have 0.8 square into 16.21, the risk of that. Then 20% square with multiplied by risk of that. And 2 into weights of the factor. So end result is the variance of 0.2 to 1. When we take its standard deviation, then standard deviation comes out to be 15.1. So if we take its simple average, then you can see that it was just the 33th of the emerging market and the 16th of the S&P. But the return of our portfolio was 15, which is a very lower number. This is the beauty of the diversification. This is its main particular benefit. Now let's look at it graphically. We have plotted the returns and the risks on the axis. You can see two points, S&P 500 and emerging markets. You can see that the return of S&P 500 is less and its risk is less. The return of emerging markets is on the higher side as well as its risk. But there is another third dot which is a portfolio that is even before S&P. And this is a bulge. This bulge is basically the achievement of the diversification. That's the beauty of the diversification that it works on our risk level. When we made this portfolio because its correlation was less and its covariance was less, our portfolio's return is higher than S&P 500 whereas risk is lower than the S&P 5. So this is the main diversification benefit. We'll be starting more of it. We'll do more practice of it in next modules. So hope you'll understand about this thing. Thank you for the day.