Okay then, so far we have a handle on defining the risk and expected return of an individual asset. Then having specified those two measures we've worked out how different investors might choose to hold different assets because of their different attitudes towards the risk-return tradeoff. But let's be honest with ourselves. Investors rarely hold all of their wealth in a singe asset. And from a CFO's perspective, companies rarely invest in a single project. Rather assets or projects are generally held as part of a diversified portfolio. Now this makes it very important for us to work at two specific things. Firstly, what happens to expected returns as you combine different assets into a portfolio? Secondly, what is the impact of diversification upon risk? So let's consider these questions in the context of an example. Specifically, let's assume that there are two assets that we are considering investing in. Kellogg's stock in the stock of American Airlines. And we've estimated that Kellogg's stock has a standard deviation of returns of 17.14% per annum, while American Airlines has a standard deviation of 38.53% per annum. Furthermore, Kellogg's has an expected return of 7.41% per annum, while American Airlines has an expected return of 13.13% per annum. Now, pausing for a second, you'll recall from the last session together that we would be unable to say which of these two investments a risk-averse investor would select, as their choice would be dependent on their level of risk aversion. Okay then. Let's assume that we form a portfolio where half of our wealth is invested in Kellogg's shares and half of it is invested in American Airlines shares. So, we have an equally balanced portfolio. What happens to the expected return of the portfolio now that we've combined the assets together? Well it's really quite obvious. The expected return of the portfolio of assets is simply the weighted average expected return of the assets in the portfolio, where the weight reflects the proportion of wealth invested in each asset. So for our portfolio of two assets, we have 50% of our wealth invested in Kellogg's generating 7.14% And the other 50% invested in American Airlines, where it's expected to earn 13.13%. The weighted average therefore is simply 10.27% per annum. So, the expected return for a portfolio is simply the weighted average expected return from the assets in the portfolio. But what about risk? Well it's tempting to risk that risk behaves in exactly the same way. That is that the risk of the portfolio is simply the weighted average of the individual assets. Such that the risk of our portfolio is simply 27.84% per annum. But it's not quite that simple. To illustrate this, let's assume that we're looking at combining two assets with exactly the same level of risk, and in this case the same expected return, into a single portfolio. The risk of each asset reflects the degree of variability and returns around the asset's expected return. So let's allow asset 1's returns to start to vary. See how the range of possible returns is indicated by the black arrow to the left of the asset reflects the risk of the asset. Well now let's allow asset 2's returns to start to vary. You will notice that the returns of the two assets, while equally variable, are not perfectly aligned. That is, they do not move in lockstep together. So when we combine the assets into a portfolio, the risk of the portfolio is actually less than the average risk of the individual assets, because there's the opportunity for returns from each of the assets to either be moving in the opposite direction, or in the same direction but by a lesser or greater amount. So the question is, how do we determine the risk of a portfolio? The answer is given by these two related formulae. The top one simply says that the standard deviation of a portfolio, sigma P, is the square root of W1 squared, that is the proportion of wealth invested in asset 1 all squared. Multiplied by the standard deviation of asset 1 squared, which we know is simply the variance of asset 1's returns, plus W2 squared times sigma 2 squared, plus 2 times W1 times W2, times the covariance between assets one and two. Importantly, that covariance measure, sigma 1, 2 can itself be broken down to Sigma 1 times Sigma 2 times Rho 1, 2. The Greek letter Rho is extremely important when we talk about building portfolios to manage risk. It is what's known as the correlation coefficient. And it measures the degree of relationship between different processes, in this case between the returns of the two assets. We'll come back to that. Now, looking at the formula for sigma p, that is, the risk of the portfolio in the second equation above, we can see that as rho increases, the risk of the portfolio also increases. Let's see why. Rho, the correlation coefficient, can take any value between +1, and -1. When the value of rho for two assets is +1, then the two assets are said to be perfectly positively correlated. When the value of rho is equal to -1, then the returns of the two assets are said to be, and you guessed it, perfectly negatively correlated. When the value of rho is positive, the assets are positively correlated; and when negative, they are negatively correlated. A rho of zero leads to the returns of the two assets being said to be uncorrelated. So what does all this mean? Perfect positive correlation does not necessarily imply that the returns will always match between the two assets, but instead it implies that the ratio of the returns of the two assets will be a constant amount. That is if two assets are perfectly positively correlated, then an increase in the value of asset 1 by X% per annum will always be matched by an increase in asset 2's returns of Y% per annum. Where the ratio of X to Y is constant. When two assets are perfectly positively correlated, there is no chance for risk reduction by diversification, and the risk of the portfolio will always be the weighted average of the individual assets. Just like for expected return. In contrast, when two assets are perfectly negatively correlated, then a return of X% for asset 1 will always be matched by a return of minus Y% for the other asset, where the ratio of X to Y is once again constant. When two assets have a correlation coefficient of zero, they're said to be uncorrelated. This means that if you were to observe a return of X% for asset 1, you'll have no idea as to what the return might be for asset 2. In practice most assets exhibit some degree of positive correlation. Why is that? One of the reasons is that there are general macroeconomic factors that tend to affect the value of all assets in the same direction. For example, a reduction in real demand for goods and services during a recession will tend to be associated with negative returns for most stocks as revenue streams fall across the economy. So how do you obtain a correlation coefficient? Well it's quite simple really. Indeed it's very similar to how we calculated the standard deviation of returns in our first session together. The first thing we do is download each asset's price series into Excel. Once again you can get these from somewhere like Yahoo Finance or another free data service. Secondly, we convert the price series into a return series for each asset. We then estimate the correlation coefficient between the returns of the two assets using the CORREL function, where we highlight the cells containing each return series. When I do this using Yahoo Finance to download adjusted closing prices for both Kellogg's and American Airlines for the 2014 calendar year, I end up with an estimate of rho of 0.1844. So the returns of Kellogg's and American Airlines exhibit positive correlation. Well now lets use this information to estimate the risk of the portfolio we've created at the start of this session where we invested half of our funds into Kellogg's and half into American Airlines. When we do this we see that the risk of the portfolio measured by sigma is 22.48% per annum. But have we achieved a diversification benefit? Well to check that, we need to compare the risk of the portfolio against what the risk would have been if there was no diversification benefit. That is, if the risk was simply averaged between the assets. So, it's a straightforward task to calculate the weighted average standard deviation of the individual assets. In fact, we did this just before and ended up with a weighted average sigma equal to 27.84%. The fact that the risk of our portfolio is about 5.36% less than that number indicates that we have achieved a diversification benefit by combining these two assets into a single portfolio. In summary, the expected return for a portfolio is simply the weighted average of the expected returns of the assets in the portfolio. The risk of a portfolio is determined by three factors. Firstly, the proportion of our wealth we invest in each asset. Secondly, the risk of each individual asset. And then thirdly, and perhaps most importantly as you'll see, the relationship between the returns of each asset which we measure using the correlation coefficient rho. Finally, we demonstrated that as long as the returns of the assets are less than perfectly correlated, the risk of the portfolio would be the less than the weighted average risk of the individual assets. This is what we call the diversification benefit.