Okay then. So we have no worked out together how to quantify the risk of an asset and the expected return and risk of a portfolio of risky assets when combined together. This next session is going to be spent digging a little deeper into how we can demonstrate a diversification benefit. We will then turn our attention to how we can expand our analysis to allow for literally hundreds or even thousands of different risky assets being combined into a single portfolio. First, a quick recap. We saw in our last session that the expected return for a portfolio, was simply the weighted average of the expected returns of the assets that are in the portfolio. Then, the risk of a portfolio, as measured by standard deviation, or Sigma, has three distinct parts to it. The first, w 1 sigma 1, reflects the contribution of the risk of the first asset. The second part, w2 sigma 2 is the contribution of the risk of the second asset. And the third part reveals the impact of the way in which the returns of the two assets co-vary together, as represented by the correlation between the returns of the two assets. This final session of the module is going to spend digging into how we can demonstrate a diversification benefit for risk averse investors. We will then turn out attention to how we can expand our analysis to allow for many more assets when combined together into a single portfolio. So let's head back to our simple two asset portfolio, consisting of an investment in American Airlines and an investment in Kellogg's. This table demonstrates the impact on expected return and risk varying proportion in wealth invested into two assets. The first column reflects investment in American Airlines. The second column is the proportion of wealth invested in Kellogg's. The third column is the expected return of the portfolio. And the fourth column is the standard deviation of returns for the portfolio. The fifth and final column is an interesting one. In that, it records the weighted average risk of the individual assets. Now, you'll recall from our last session together, that this number is our benchmark against which we can measure the benefits of diversification. The first row and the last row of this table document risk and return from 100% investment in American Airlines and Kellogg's, respectively. As expected, if you invest in only one asset, there is no opportunity for a diversification benefit. And this is indeed so for these two portfolios, where the portfolio risk simply matches the individual risk of each of the two assets. But let's turn our attention to the more interesting intermediate portfolios. As we can see, in each and every case, the risk of the portfolio is less than the weighted average of the two assets. So, we have clearly documented diversification benefits here. One other important point that we should make here is that not all of the portfolios would be attractive to investors. No matter the level of risk aversion. Consider the two bottom portfolios and note their expected return of risk levels. Can you see how both of these portfolios offer a lower expected returns, yet a higher risk than the portfolio that consists of 10% investment in American Airways and 90% investment in Kellogg's? These two portfolios are what we call inefficient portfolios. In that, they offer a level of risk and return that is clearly inferior to an alternatively constructed portfolio. Now, let's have a look at graphing the risk return combinations provided in the final two columns of this table. We have expected return on the vertical axis here, plotted against risk on the horizontal axis. The straight red line we see here is the benchmark expected return in risk combination that reflects the weighted average risk of each portfolio combination. The blue line represents the actual risk return combination for each portfolio. So let's choose a level of expected return. Say at about 10%. In the absence of any diversification benefits, the combination of these two assets will yield a portfolio standard deviation of around 27.5%. Yet, owing to the fact that these two assets don't move in perfect lock step with each other. There's a chance for the variability in the returns of one asset to dampen down the impact of the variability of the other asset's returns. Thus, reducing the risk of the portfolio to around 22% per annum. This is the benefit of diversification. So what determines the extent of the diversification benefit? Well, we know that, the key element for a given pair of assets is the correlation coefficient rho. The lower the value of rho, as in the further it is from positive one, the smaller the portfolio's risk at any given level of expected return. But the question is, what sort of factors might impact upon the value of rho? Well, here we need to think about the nature of the cash flows being generated by the two assets. Obviously, industry is gong to be very important. With two assets that are both located in the same industry, aren't likely to offer the same diversification benefits as the combination of two assets that are in unrelated industries. Similarly, assets located in the same country may be expected to have higher correlation coefficients and hence, lower diversification benefits, than assets located in different countries. But there are also other, more nuanced influences upon correlation coefficients. And these have to do with the strategic decisions made by the company. For example, firms operating in the same industry, who've adopted similar leverage levels, would be likely to have higher correlation in returns, relative to firms with quite different leverage levels still operating in the same industry. Similarly, firms that adopt similar hedging policies, might also be expected to have similar exposures to risk. And hence have higher correlations and returns than firms that don't adopt the same type of hedging policy. Now, let's have a look at different correlation coefficients for the stock of firms operating in different industries. These correlation coefficients have been estimated over the 2014 calendar year. The diagonal of this matrix features values of 1, which of course makes perfect sense as a specific asset's returns are perfectly correlated with itself. Now, let's focus on the first column, which documents the correlation between returns on Kellogg's shares and shares in different companies as well as the S&P 500, which, as we know, is a broad-based index consisting of 500 of the largest and most liquid companies listed on US exchanges. Consider first the correlation between the returns of Kellogg's, and one of its biggest competitors, Kraft Foods. It is no surprise that out of all the companies that we've compared Kellogg's to, its returns are most highly correlated with the returns of Kraft. In contrast, the correlation between Kellogg's and the two internet-based firms Trip Advisor and Facebook is much, much lower. Yet, the correlation between these two internet-based firms is as expected much higher. So that's all well and good when we're dealing with just two assets. Question is though, what happens when we diversify our wealth over a portfolio consisting of many more assets? Now, recall that for a two asset portfolio, we have two stand alone risk measures, as well as two co-variance terms. Noting that the two at front of the third term in the first equation. For a three asset portfolio, we add another measure of stand alone risk. But we also now need to account for the relationship between the returns of the third asset, and the returns of the first two assets. This increases the number of covariance terms to six. So as you can see, all of a sudden the way in which asset returns are related to each other is starting to become almost more important than the stand-alone risk of individual assets. Let's demonstrate via example. Let's build a portfolio consisting 40% of Kellogg's shares, 30% Facebook and 30% Microsoft shares and let me pause for a second. The expected returns for the individual assets is simply given here. And you may be wondering how do we get these values? The answer is as fascinating as it is important. We're going to spend the whole of the next module on just this point. At the moment though, let's just take it as given. So the expected return for the portfolio is simply the weighted average of the individual assets. So in this case, that is that is equal to a 9.858% per annum. What about the risk of the portfolio? It looks a little daunting, doesn't it? But if we just take each term in turn, you can't go wrong. Also, remember that, the covariance between the returns of two assets is calculated as the standard deviation of one asset multiplied by the standard deviation of the other asset, multiplied by the correlation coefficient between the returns of the two assets. The standard deviation of the portfolio's returns, therefore, is calculated as 17.89% per annum. So, how do we know, if there was a diversification benefit or not? Well, as always, we simply compare this figure with the weighted average standard deviation of the assets in the portfolio. The weighted average risk is 24.8% which is almost 7% higher than the risk of the portfolio itself. Hence, there is a very clear diversification benefit. As you keep adding more and more assets to the portfolio, you will see that the number of covariance terms far outweighs the number of stand alone risk factors. A four asset portfolio has four stand alone risk terms, but 12 covariant risk terms. A 50 asset portfolio has 50 stand-alone risk terms, yet 2,450 covariance risk terms. It's starting to look like a pretty complicated set of calculations when you're dealing with a massive portfolio. Isn't it? Well, not necessarily. Imagine you have an investment portfolio consisting of 200 of the largest listed companies in the US, and you're considering adding another asset to that portfolio. Do you really need to estimate hundreds of stand alone risk terms and thousands of correlation coefficients? Well, the answer is, probably not. What you can do is treat the existing 200 asset portfolio as a single asset. And then simply estimate the stand alone risk of the existing portfolio, the risk of the new asset that you're considering for inclusion, and a single correlation coefficient that reflects how the returns between the two have historically varied. In this session, we began by demonstrating the diversification benefit graphically. We highlighted the factors that influenced the value of the correlation coefficient. And hence, impacted upon the diversification benefits of combining different assets together. We also worked out how to estimate the risk of a portfolio consisting of more than two assets. And in so doing, highlighted how standalone risk increasingly took a backseat to the ever more influential covariance risk. We began this module by identifying a relevant measure of risk, the standard deviation, which accounted for the dispersion in an asset's returns. We then documented alternative attitudes towards risk, differentiating between risk-averse, risk-neutral, and risk-seeking investors. We made a point in that session of stressing that, risk averse investors do not dislike risk. They simply demand compensation for being exposed to it. And the compensation demanded is dependent upon the level of risk aversion and investor displays. In the third module, we worked out how to calculate the expected return and risk of the two-asset portfolio. It was during this session that we first came across what has become an increasingly important influence on risk: the correlation coefficient. So, where to next? In our next module together, we're going to work out how to link together the concepts of risk and expected return. Whereas in this module we have simply assumed an expected return for different individual assets. In the next, we're going to discuss alternative models that explicitly link risk and return. I look forward to working with you on what promises to be a fascinating set of topics.