In this first session on linking risk with expected return, we're going to focus on the key factors that drive the benefits from diversifying a portfolio of investments, or from a company's point of view, a set of projects. Now you might recall from the last module, that when two assets are combined into a portfolio that provided the returns of the two assets, so less than perfectly positively correlated, the risk of the portfolio will be less than the weighted average risk of the individual assets. So in the example graphed here on the left, we have the risk return combinations generated as we shift from the bottom from 100% investment in Kellogg shares to 100% investment in American Airlines. The straight red line provides the combination that would be created if risk was merely averaged. That is, if the shares of the two were perfectly positively correlated. The curved blue line represents combinations of risk and return that reflect the benefits of diversification in that any given level of expected return, the portfolio has a level of risk below the weighted average of the assets in the portfolio. The lower the value of the correlation coefficient between the returns of the two assets, the lower the risk of the portfolio for any set of portfolio weights. Now, while on the topic of how asset returns move or co-vary together, you might also recall from our last session that as the number of assets in a portfolio increases, the number of covariance terms in the calculation of portfolio standard deviation also increases. Indeed, for a 50 asset portfolio, we saw that there were only 50 measures of standalone risk as measured by standard deviation of returns, yet there are 2,450 covariance measures. Now this suggests that, as we diversify a portfolio across a greater and greater number of assets, the contribution to the risk of the portfolio of any individual asset is driven mainly by the nature of that asset's relationship with other assets in the portfolio, rather than the asset's individual risk. So, this very point is highlighted in this graph, which demonstrates that as you increase the number of assets in a portfolio, the risk of the portfolio reduces as we diversify away variability that is linked to individual firm effects. Now, not surprisingly, this risk that we are able to eliminate is known as diversifiable risk. So, what happens if you fully diversify your portfolio? Does that mean you've eliminated all risk? Well, the answer is a resounding no. The value of your portfolio would still vary. It's just that the source of the variation would be market wide factors as opposed to firm specific influences. Now consistent with all of that, let's bear down some definitions. Total risk is simply the total variability in the returns from an asset. Our common measure of total risk is standard deviation or the Greek letter sigma. Diversifiable risk is the risk that can be eliminated by adding more assets to a portfolio. Other names for diversifiable risk include unsystematic or firm-specific or idiosyncratic risk. Finally, undiversifiable risk is the risk that, not surprisingly, cannot be eliminated via diversification. It is also referred to as systematic, market or covariance risk. So for which of these risks can we demand from the market compensation in the form of higher expected returns? Well it's not total risk because we know that some of the total risk can be eliminated via diversification. Now it's definitely not diversifiable risk because this is the risk that we can get rid of. So you're right, it's undiversifiable or systematic risk, because this is the risk that we can't eliminate through diversification. So to illustrate this point, let's start with the assumption that the only assets in existence are the shares of companies that are included in the S&P 500. Let's assume that you own a very well diversified portfolio that closely mimics the return to the S&P500 index. You are now considering adding another asset to your portfolio. As you will only be compensated for systematic or, undiversifiable risk, the only measure that you're really interested in is how the new assets returns covary with all of the other assets in the portfolio. When we group all those other assets into a portfolio we call that portfolio the market portfolio. Now this relationship between the new asset and the market portfolio can be captured by the covariance measure, sigma i,M which itself can be calculated or broken down to sigma i, times sigma M, times rho i,M. Now those measures of course are the standard deviation of the returns of asset i, the standard deviation of returns of asset M, the market portfolio and the correlation coefficient between the two. Now this leads us neatly to our measure of systematic risk, beta, as signified by the greek letter, beta. This measure of systematic risk is standardized, in the sense that it measures risk relative to the standalone risk of the market portfolio. Which you'll recall is the portfolio consisting of all risky assets in existence. Specifically, we calculate an asset's beta by dividing the covariance between the asset’s returns and the returns of the market portfolio by the variance, that is standard deviation squared of the market portfolios returns. And as we discussed on the previous slide, the covariance is simply the product of standard deviation of the asset returns, standard deviation of the market portfolios returns, and the correlation coefficient, rho, between the two. So beta is measuring systematic risk and it reflects how an asset's returns behave relative to the broad market portfolio. Well, what do we know about beta? Firstly, almost all assets have positive beta, in that there are general macroeconomic factors that tend to influence all assets in the same direction. For example, during a recession, the returns for most companies would decline, along with the market, as the real demand for goods and services also declines. These market wide influences invoke positive correlation coefficients amongst assets, which in turns invokes positive betas. Secondly, assets with betas greater than 1, have higher systematic risks than the market portfolio, which will tend to show up in asset returns that tend to rise by more than the market in a rising market. But fall by more than the market in a falling market. These are frequently referred to as aggressive assets or aggressive investments. Conversely, assets with betas less than 1 are less risky than the market. And will tend to not move up as much in a rising market, nor fall as much in a falling market. They are often called defensive assets. Now, while it's tempting to think that companies can't do much about their beta, as it will largely be driven by the industry in which the company operates, this is not the full story. Firms can impact upon their own beta through the choices that they make within the firm. For example, higher levels of leverage will increase the beta of a company's stock. As returns to shareholders become more sensitive to market-wide fluctuations as leverage increases. You'll recall that we saw this effect in our last course together, Corporate Financial Decision Making for Value Creation, when we discussed the concept of financial risk. To properly measure beta for an asset, we would need to observe the way in which the asset's returns covaried with a portfolio consisting of every single risky asset in existence. Now obviously, this is not all that practicable. So instead, what we do is use a proxy. Most commonly, a stock price index that reflects stock price movements in the country in which the company we're interested in is listed. So for Kellogg's we'll use the S&P500 Index. To measure beta we first gather stock price data for both the company and the market proxy. Let's say we use one year of daily price data. That is we did when we estimated standard deviations in the last module, we would convert the price series to return series for both the stock and the market proxy. Finally, we utilize the Excel function, STDEV, to estimate standard deviations for each of the return series, as well as the CORREL function, to estimate the correlation between each of the return series. So let's do this using data from the 2014 calendar year for a sample of companies, as well as for the S&P500 index as our proxy for the market portfolio. There are a number of interesting results in this table. Firstly, note that Kellogg's and Kraft, the two firms that are most closely aligned in terms of their operating activities, have very similar betas and that their betas are less than one. This is a common observation for companies that derive cashflows from consumer staples, like breakfast cereals and snack foods, in that no matter how bad things become in the broader economy, we still like our corn flakes in the morning and our macaroni and cheese at night. The other thing to note is that the order of riskiness as measured by standard deviation of returns is different to the order as indicated by beta. American Airlines is the riskiest of the companies in 2014. If we measure risk using standard deviation yet Facebook has a higher beta than American Airlines using 2014 data. Now this suggests that a lot of the volatility in returns from shares in American Airlines is due to firm-specific factors, which ultimately can be diversified away. Whereas more of Facebook's risk is due to market-wide risk, as indicated by the higher correlation between the returns on Facebook shares and the returns of the S&P500, our market proxy. So what happens to systematic risk as we increase the number of assets in our portfolio? Well, the simple answer is, it simply averages out. That's right diversification doesn't reduce systematic risk because systematic risk, by definition, is undiversifiable risk. It follows then, that the beta of the portfolio is simply the weighted average of the betas of the assets in the portfolio. Where the weight reflect the proportion of wealth invested in each asset. So for example, let's assume that we invest 40% of our wealth in Kellogg's shares, 30% in American Airlines, and 30% in Facebook. What is the beta of our portfolio? While we simply multiply each beta by the proportion of our portfolio that each asset represents and we end up with a beta of 1.311. So that's all pretty straightforward, right? Well, unfortunately not quite. Now recall that when we calculated Kellogg's beta, it worked out to be 0.773. When we used one year of daily data and the S&P500 Index is our proxy for the market portfolio. So what happens when we change our approach slightly? Well, if we were to use the NYSE 100 Index. as opposed to the S&P500, as our proxy for the market portfolio then Kellogg's would have ended up with a Beta of 0.330. If we had have stuck with the S&P500, but had have use 5 years of of daily returns instead of only a single year, then Beta would've been 0.469. Finally, if we had have use five years of weekly returns, then beta would have been calculated as 0.384. So what we see is that the beta estimated can be very sensitive to the choices made with respect to the estimation procedure followed. As a general point, it's best practice to use a broad based market proxy rather than a narrowly defined one and to also select the period of time that you think is long enough to yield sensible estimates of the parameters underlying beta, yet not so long that it encompasses a period of time when the underlying asset was significantly different to the one that you're dealing with today. So in summary, in this session, we have distinguished between diversifiable or firm-specific or idiosyncratic risk and undiversifiable, systematic, or covariance risk. We defined our measure of systematic risk as beta, which is calculated as the ratio of covariance between the returns of an asset and the market portfolio and the variance of returns for the market portfolio. Finally we highlighted some of the implementation issues that might arise when we attempt to estimate beta in practice. The next question, is how are we going to use beta to help assist in corporate financial decision making? That will be the subject of the next session.
