Return, Risk, and the Security Market Line 4/21/2017 Chapter 12 Return, Risk, and the Security Market Line Chapter 12 continues our discussion of risk and reward. Define risk more precisely and how to measure it.

Announcements and News Firms make periodic announcements about events that may significantly impact the profits of the firm. Earnings Product development Personnel The impact of an announcement depends on how much of the announcement represents new information.

Announcements & Surprises 4/21/2017 Announcements & Surprises Total return–Expected return = Unexpected return R – E(R) = U Announcement = Expected + Surprise The total return on an asset has two components: an “expected” portion and an “unexpected” element. Think about news announcements as also having two components: Expected and Unexpected or the surprise. Stock prices react to unexpected announcements. Consistent with market efficiency, the expected portion has already been incorporated or impounded into the current price.

Announcements & Surprises 4/21/2017 Announcements & Surprises A company announces earnings $1.00 per share higher than the previous quarter. Let’s look at a very simple example

Systematic & Unsystematic Components of Return 4/21/2017 Systematic & Unsystematic Components of Return (12.3) R – E(R)=Systematic +Unsystematic (12.4) R – E(R) = U = m+ε where m = market risk ε = unsystematic risk (12.5) Total risk=Systematic risk + Unsystematic risk The unexpected portion of return also has two components: a systematic or market component and an unsystematic or firm-specific component. In our previous example we ignored the market component, assuming that any change in price was totally driven by the company’s announcement. In reality, this isn’t the case. Some news impacts the market as a whole, such as announcements concerning inflation, unemployment, interest rates or GDP. Other news may be industry or firm-specific, such as a company announcing higher earnings, a special dividend or a break-through in technology.

Systematic & Unsystematic Components of Return 4/21/2017 Systematic & Unsystematic Components of Return (12.5) Total risk = Systematic risk + Unsystematic risk = Market risk + firm-specific risk If the latest Consumer Price Index numbers indicate that inflation has risen substantially. Will this impact systematic or unsystematic risk? If a company announces that it will have to restate its financials for the last 3 years, which risk component will be affected? An increase in inflation will probably have market-wide impact, affecting the systematic risk component. On the other hand, if a company announces that it has found some accounting inconsistencies and will have to restate its financials for the last three years, this is clearly firm-specific and will affect the unsystematic portion of risk.

Diversification and Risk In a large portfolio: Some stocks will go up in value because of positive company-specific events, while Others will go down in value because of negative company-specific events. Unsystematic risk is essentially eliminated by diversification, so a portfolio with many assets has almost no unsystematic risk. Unsystematic risk = diversifiable risk. Systematic risk = non-diversifiable risk.

“The expected return on an asset depends only on its systematic risk.” 4/21/2017 “Unsystematic risk is essentially eliminated by diversification, so a portfolio with many assets has almost no unsystematic risk.” “The systematic risk principle states that the reward for bearing risk depends only on the systematic risk of an investment.” “The expected return on an asset depends only on its systematic risk.” These three points – taken directly from the textbook – are important enough to be reiterated: Why is this so important? We have been looking at standard deviation as a measure of the risk of an asset or a portfolio. Standard deviation measures TOTAL risk, which includes both the systematic and unsystematic components. We need a way to measure JUST the systematic risk of an asset or portfolio in order to determine its expected return.

The Systematic Risk Principle The systematic risk principle states: The expected return on an asset depends only on its systematic risk. No matter how much total risk an asset has, only the systematic portion is relevant in determining the expected return (and the risk premium) on that asset.

Measuring Systematic Risk The Beta coefficient (  ) measures the relative systematic risk of an asset. Beta > 1.0  more systematic risk than average. Beta < 1.0  less systematic risk than average. Assets with larger betas = greater systematic risk = greater expected returns. Note that not all Betas are created equally.

4/21/2017 Risk & Beta Asset 2 has more total risk because it has the greater standard deviation. Asset 1 has more systematic risk because it has the larger Beta. Asset 1 should have the higher expected return since it has the larger beta. Beta vs Standard Deviation: Since standard deviation measures total risk, Asset 2 has more total risk. Beta measures systematic risk, so Asset 1 has more market risk. With a higher Beta, we would expect Asset 1 to demand a higher return. Though Asset 2 has higher total risk, the market expects the firm-specific portion to be eliminated through diversification.

Portfolio Beta βi = Beta of asset i 4/21/2017 Portfolio Beta Xi = % of portfolio invested in asset i βi = Beta of asset i Beta can be used to measure the market risk of a single asset or a portfolio. The Beta of a portfolio = simple weighted average of the Betas of the stocks in the portfolio. The weights are the percentages invested in each asset. Given the beta and percent invested in each asset: Multiply each percentage by the beta and sum the results. In the table above, combining these two assets into a 40-60 portfolio results in a portfolio with a Beta equaling the market. We have offset the market risk of Asset 1 with Asset 2.

Portfolio Beta What is this portfolio’s expected return and Beta? 4/21/2017 Portfolio Beta Expanding beyond two assets: In the table above we have 4 stocks, the amount invested in each, its expected return and its beta. We need to calculate the portfolio’s expected return and Beta. What is this portfolio’s expected return and Beta?

Portfolio Return & Beta Example 4/21/2017 Portfolio Return & Beta Example Step 1: Calculate each stock’s % or weight in the portfolio The first step is to calculate the % of the portfolio invested in each asset.

Portfolio Return & Beta 4/21/2017 Portfolio Return & Beta Step 2: Apply the weights to each stock’s expected return to arrive at the portfolio’s expected return. Next, we apply the portfolio weights to each stock’s expected return to arrive a an expected return on the portfolio of 10.3%.

Portfolio Return & Beta 4/21/2017 Portfolio Return & Beta Step 3: Apply the weights to each stock’s beta to arrive at the portfolio’s beta. Finally, we apply the same weights to each stock’s beta to arrive at the portfolio beta of 1.066.

Beta and the Risk Premium, I. Consider a portfolio made up of asset A and a risk-free asset. For asset A: E(RA) = 16% and A = 1.6 The risk-free rate Rf = 4%. Note that for a risk-free asset,  = 0 by definition. Varying the % invested in each asset will change the possible portfolio expected returns and betas. Note: if the investor borrows at the risk-free rate and invests the proceeds in asset A, the investment in asset A will exceed 100%.

Beta and the Risk Premium 4/21/2017 In this example we have one risky asset – A and the risk free asset The table combines these two into portfolios ranging from all risk-free to 150% in the risky asset ..which requires borrowing at the risk free rate. 12-18 18

Portfolio Expected Returns and Betas for Asset A

Beta and the Risk Premium 4/21/2017 Beta and the Risk Premium In this example we have one risky asset – A and the risk free asset The table combines these two into portfolios ranging from all risk-free to 150% in the risky asset ..which requires borrowing at the risk free rate. 12-20 20

Portfolio Expected Returns and Betas for Asset B

Portfolio Expected Returns and Betas for Assets A & B 4/21/2017 Portfolio Expected Returns and Betas for Assets A & B Asset A has a Reward-to-risk ratio of 7.50% Asset B’s is 6.67% This situation cannot exist in an efficiently performing market. Market pressure will bring the prices in line

The Fundamental Relationship between Risk and Return 4/21/2017 The Fundamental Relationship between Risk and Return Reward-to-Risk (A) = 7.50% Reward-to-Risk (B) = 6.67% (12.6) The fundamental relationship between risk and return. Reward-to-risk ratio = the expected return on an asset minus the risk-free rate then divided by the asset’s beta. The fundamental principle is that the reward-to-risk ratio should be the same for all assets in an competitive market. “The reward-to-risk ratio must be the same for all assets in a competitive market.”

The Fundamental Result The situation we have described for assets A and B cannot persist in a well-organized, active market Investors will be attracted to asset A (and buy A shares) Investors will shy away from asset B (and sell B shares) This buying and selling will make The price of A shares increase The price of B shares decrease This price adjustment continues until the two assets plot on exactly the same line. That is, until:

The Fundamental Result

Over- and Under-Valued 4/21/2017 Over- and Under-Valued We can use this reward-to-risk ratio to determine if stocks are under- or over-valued relative to each other. Given the above data, we’ll compare the ratios for these stocks on the next slide. If the Risk-free rate is 5%, are these stocks fairly valued?

Over- and Under-Valued 4/21/2017 Over- and Under-Valued Stocks W and X offer insufficient returns for their level of risk compared to stocks Y and Z. Stock Y offers the highest return for its level of risk. On a comparative basis, stocks W and X are clearly not offering a sufficient return for their risk level compared to Stocks Y and Z Stocks Y offers the highest reward to risk. The thing we don’t know yet is, are any of these rewards high enough? We’ve only compared them to each other, not to any market benchmark.

The Security Market Line (SML) The Security market line (SML) = a graphical representation of the linear relationship between systematic risk and expected return in financial markets. For a market portfolio,

The Security Market Line The term E(RM) – Rf = market risk premium because it is the risk premium on a market portfolio. For any asset i in the market: Setting the reward-to-risk ratio for all assets equal to the market risk premium results in an equation known as the capital asset pricing model.

The Security Market Line The Capital Asset Pricing Model (CAPM) is a theory of risk and return for securities in a competitive capital market. The CAPM shows that E(Ri) depends on: Rf, the pure time value of money. E(RM) – Rf, the reward for bearing systematic risk. i, the amount of systematic risk. (12.7)

The Security Market Line 4/21/2017 The Security Market Line (12.7) Market Risk Premium Pure time value of money Reward for bearing systematic risk Security Market Line = equation 12.7 SML = Capital Asset Pricing Model (CAPM) simple and elegant explanation of the relationship between risk and reward. The expected return on an asset or portfolio has 3 components. Risk free rate = compensation for the pure time value of money. Market risk premium = market determined reward for bearing market risk as opposed to investing in a risk-free asset. Market risk premium =difference between the expected return on the market and the risk-free rate. Amount of systematic risk borne by this specific asset or portfolio which is measured by its beta. The SML is a straight line which implies that fairly-valued assets will plot on the line. Amount of systematic risk

The Security Market Line

Over- and Under-Valued 4/21/2017 Applying the CAPM equation to each stock in our table, we have a good comparison for each stock. The expected return reflects what WE expect. The column labeled “CAPM Return” shows what the market expects or requires according to CAPM. Stock W and X are clearly overvalued Not providing enough return for their level of risk. Stock Z is fairly valued Required return equal to its expected return. Stock Y is undervalued. We think it will return 12% but the market is requiring only 11%. Note that “market required” means the market is pricing the stock consistent with this level of return.

The Security Market Line 4/21/2017 The Security Market Line Y Z X W RF = 5% SML intercept = risk-free rate which is 5%. As we saw numerically on the previous slide: Stocks W and X are overvalued Return is not sufficient for their level of risk. Plot BELOW the Security Market Line Expected return < market requires for their level of risk. Stock Y plots above the SML, Undervalued Returning more than required. Stock Z plots ON the SML since its return exactly matches the market’s requirements. Remember that “return” is just a percentage change in price so if a stock’s return is too low, the market will bid the price DOWN which will drive the return UP until it graphs on the SML. Similarly, for Stock Y, the market will recognize that it is undervalued and apply buying pressure which will bid the price UP, driving the return down until Y plots on the line.

Risk and Return Summary, I.

Risk and Return Summary, II.

More on Beta (12.8 and 12.4) R - E(R)= m + ε 4/21/2017 More on Beta (12.8 and 12.4) R - E(R)= m + ε (12.9) m = [RM –E(RM)] x β (12.10) R-E(R)= [RM –E(RM)] x β + ε Systematic portion of the unexpected return on the market M We’ve talked a lot about what Beta is and how to use it, but where does it come from? Looking at equations 12.8 and 12.9 we can substitute 12.9 into 12.8 to arrive at 12.10 12.10 gives us more insight into Beta and confirms what we have seen in previous examples and graphs: A high beta stock is one that is relatively sensitive to overall market movements.

Decomposition of Total Return 4/21/2017 Decomposition of Total Return Suppose the expected return on the market is 10% and the risk-free rate is 5%. Asset A has a beta of 0.80. The expected return on Asset A is 9%. One year the return on the market is actually 8% and the return on Asset A in the same year is 6.5%. Decompose these returns. E(RM) = 10% rRF = 5%. Asset A beta = 0.80. First task = compute the expected return on Asset A. Then we’ll look at decomposing the returns.

Decomposition of Total Return 4/21/2017 Decomposition of Total Return E(RM) = 10% Actual RM = 8% RF = 5% βA = 0.80 E(RA) = 9% Actual RA =6.5% UNE(RM)= [RM-E(RM)] = 8% - 10% = -2% UNE(RA) = [RA-E(RA)] = 6.5% - 9% = -2.5% Systematic UNE = [RM-E(RM)] x β = -2% x .80 = -1.6% Unsystematic portion = [RA-E(RA)] - [RM-E(RM)] x β = (-2.5%) – (-1.6%) = -0.90% The unexpected return on the market = negative 2% The unexpected return on Asset A = negative 2.5%. Since Asset A’s beta is 0.80 we did not expect A to react more strongly than the market! Using equation 12.9 we find that the portion of Asset A’s unexpected return attributable to the market is equal to a negative 1.6%. That leaves a negative 0.9% attributable to Asset A specifically.

Decomposition of Total Returns

Unexpected Returns and Beta

Where Do Betas Come From? A security’s Beta depends on: How closely correlated the security’s return is with the overall market’s return, and How volatile the security is relative to the market. A security’s Beta is equal to the correlation multiplied by the ratio of the standard deviations.

Where Do Betas Come From?

Using a Spreadsheet to Calculate Beta

4/21/2017 Calculating Beta (12.11) The table above details the calculations involved in computing Beta for an asset. Beta = .0044/.00247=1.77

4/21/2017 Calculating Beta Step 1: Compute average returns for Asset H and the Market First: compute average returns for Asset H and the market. These are simple averages – just add up the returns for each of the five years and divide by 5.

4/21/2017 Calculating Beta Step 2: Compute variance and standard deviations for Asset H and the Market Step 2 = computing the variances and standard deviations for Asset H and the market. Using the average returns computed in Step 1, we find each year’s deviation from the average. These deviations for Asset H and the market are in columns 3 and 4. In columns 5 and 6 we square the deviations from columns 3 and 4 and sum them up. Because we are dealing with historical rather than expectational data, we divide these sums by the number of years minus one or 4 to arrive at the variances. Taking the square root of each variance gives us the standard deviations.

4/21/2017 Calculating Beta Step 3: Compute the covariance between the Market and Asset H. Next, we need to calculate the covariance between Asset H and the market. On the previous slide we squared each set of deviations To arrive at the covariance between two assets, we multiply each deviation for Asset A against the corresponding deviation for the market. That means multiplying column 3 times column 4 and is shown in column 7. Summing column 7 and dividing by 4 as we did previously, we arrive at a covariance figure of 0.0044 for Asset H and the market. The sign of the covariance (positive) tells us that Asset H and the market move somewhat together, but the value itself tells us very little about how MUCH together. For that we need the correlation coefficient which we compute on the next slide.

4/21/2017 Calculating Beta Step 4: Compute the correlation coefficient and the Beta for Asset H. The correlation coefficient scales covariance to a value between negative 1 and positive 1. The scaling is accomplished by multiplying the covariance by the result of dividing the standard deviation of the asset by the standard deviation of the market. As we’ve seen earlier, a positive one means the returns move exactly together; a negative one means they move exactly opposite of each other. We find a correlation of 0.9914, meaning Asset H moves almost perfectly with the market. Our final calculation uses equation 12.11 to find Asset H’s beta of 1.77. (12.11)

Why Do Betas Differ? Betas are estimated from actual data. Different sources estimate differently, possibly using different data. For data, the most common choices are three to five years of monthly data, or a single year of weekly data. The S&P 500 index is commonly used as a proxy for the market portfolio. Calculated betas may be adjusted for various statistical and fundamental reasons.

Beta - β 4/21/2017 Beta (β) measures a specific asset’s market risk relative to an average asset. Beta is the universally accepted measure for a specific asset’s (or a portfolio’s) market or systematic risk. Beta is the ratio of the asset’s risk relative to an average asset which is defined as having a Beta of 1.0. The table on the slide above is excerpted from one in the text and recaps the Betas for a small sample of well-known stocks. Betas are published by both Value Line and Standard and Poors. While the published Betas from the two sources are similar – they aren’t identical. Differences are driven by different techniques used by the companies to calculate Betas. But what does a Beta of 1.45 (GM) mean? Market Beta = 1.00 Risk-free asset beta = 0.00 51

Finding a Beta on the Web

4/21/2017 Beta and Return Using two stocks from the previous table, the graph simulates each stock’s reaction to market moves. The market is the solid red line. GM is the dotted blue line Wal-Mart is the dashed yellow line. Notice that all three lines follow the same general pattern but, GM, with a Beta of 1.45, reacts more violently to market ups and downs. Wal-Mart, with a Beta of 0.80, reacts in a more damped manner. This is what Beta measures – how volatile a stock’s returns are, relative to the market. An asset with a beta greater than 1.0 reacts more strongly to market swings, and is, therefore riskier. An asset with a beta less than 1.0 reacts mildly to market movements and is thus less risky. 53

Extending CAPM The CAPM has a stunning implication: What you earn on your portfolio depends only on the level of systematic risk that you bear As a diversified investor, you do not need to worry about total risk, only systematic risk. But, does expected return depend only on Beta? Or, do other factors come into play? The above bullet point is a hotly debated question.

Important General Risk-Return Principles Investing has two dimensions: risk and return. It is inappropriate to look at the total risk of an individual security. It is appropriate to look at how an individual security contributes to the risk of the overall portfolio Risk can be decomposed into nonsystematic and systematic risk. Investors will be compensated only for systematic risk.

The Fama-French Three-Factor Model FF propose two additional factors as useful in explaining the relationship between risk and return. Size, as measured by market capitalization The book value to market value ratio Whether these two additional factors are truly sources of systematic risk is still being debated.

Returns from 25 Portfolios Formed on Size and Book-to-Market Note that the portfolio containing the smallest cap and the highest book-to-market have had the highest returns. Also, in three cases, the highest return belongs to the smallest cap quintile. In each row, the highest return belongs to the highest B/M quintile.

Useful Internet Sites earnings.nasdaq.com (to see recent earnings surprises) www.portfolioscience.com (helps you analyze risk) money.cnn.com (a source for betas) finance.yahoo.com (a terrific source of financial information) www.smartmoney.com (another fine source of financial information) www.moneychimp.com (for a CAPM calculator) http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ (source for data behind the FAMA-French model)