Questions were written in the word file named questions, and the question related powerpoint slides were named as slides. Thanks in advance.

1) The common stock of Marielle Machinery will generate the following payoffs to investors next year: Boom Normal Recession Dividend $8 4 0 Stock Price $240 90 0 The common stock of Escapist Films will generate the following payoffs to investors next year: Boom Normal Recession Dividend $0 1 3 Stock Price $18 26 34 Marielle Machinery and Escapist Films stocks are selling today for $90 and $30 a share, respectively. Calculate the expected rate of return and standard deviation of returns for both companies. Then calculate the expected return and standard deviation of a portfolio half invested in Marielle Machinery and half invested in Escapist Films. 2) A share of stock with a beta of 0.80 now sells for $42. Investors expect the stock to pay a yearend dividend of $1.5. The T-bill rate is 3% and the market risk premium is 8%. Suppose investors actually believe that the stock will sell for $45 at year-end. Is the stock a good or bad buy? What will investors do? 3) According to the CAPM, would the expected rate of return on a security with a beta less than zero be more or less than the risk-free interest rate? Why would investors be willing to invest in such a security? (Hint: Look back to the auto and gold example in the lecture slides) 4) Stock A has a beta of 0.4 and investors expect it to return 5%. Stock B has a beta of 1.6 and investors expect it to return 14%. Use the CAPM to find the market risk premium and the expected rate of return on the market. Lecture 10 Risk, Return and CAPM The Historical Record The historical returns of stock or bond market indices can give an idea of the typical performance of different investments. We can focus on three distinct portfolios of securities since 1900: A portfolio of 3-month Treasury bills issued by US government A portfolio of 10-year Treasury bonds issued by US government A diversified portfolio of common stocks These portfolios are not equally risky. T-bills are about as safe as an investment as you can make. They are guaranteed by the government and their short maturity means that prices are relatively stable. Long term Treasury bonds are also certain to be repaid but their prices fluctuate more when interest rates vary. Common stocks are the riskiest because there is no promise that you will get your money back. The Historical Record The figure shows the performance of each group of securities assuming that all dividends and interest income had been reinvested in the portfolio. The performance of the portfolios fits our intuitive risk ranking. The Historical Record Long term government bonds gave a slightly higher return than T-bills. This difference of 1.2% is called the maturity premium. Investors who accepted the risk of common stocks received on average an extra return of 7.4% a year over the return on T-bills. This extra compensation is known as the market risk premium. Rate of return on common stocks = Interest rate on T-bills + Market risk premium The Historical Record Why do we look back over such a long period to measure average rates of return? The reason is that annual rates of return on common stocks fluctuate so much that averages taken over short periods are unreliable. By averaging the returns across both the rough years and the smooth, we should get a fair idea of the typical return that investors may justifiably expect. Using Historical Evidence to Estimate Today's Cost of Capital Earlier we have defined the opportunity cost of capital as the return that the firm's shareholders are giving up by investing in the project rather than in comparable risk alternatives. Measuring the cost of capital is easy if the project is a sure thing. Since T-bills promise sure payoffs, a firm should invest in a risk-free project if it can at least match the interest rate on a T-bill. If the project is risky - and most projects are - then the firm needs to at least match the return that shareholders could earn if they invested in securities of similar risk. It is not an easy task to put a precise figure on this. Using Historical Evidence to Estimate Today's Cost of Capital Suppose there is a project that has the same risk as an investment in a diversified portfolio of US common stocks. We will call this portfolio the market portfolio. The problem of estimating the cost of capital for the project boils down to estimating the expected rate of return on the market portfolio. One way to do this would be assuming that the future will be like the past and use the average return of 11.4% from the last table. However, investors are not likely to demand the same return each year. Even the interest rate on T-bills varies over time. They offered a return of 14% in 1981. Would you invest in the stock market in 1981 if the expected return was 11.4%? Using Historical Evidence to Estimate Today's Cost of Capital A better procedure is to take the current interest rate on T-bills plus 7.4%, the average risk premium shown earlier. In 1981, Exp. market return1981 = Interest rate on T-bills1981 + Risk premium = 14% + 7.4% = 21.4% Exp. market return2010 = Interest rate on T-bills2010 + Risk premium = 2.1% + 7.4% = 9.5% The expected return on an investment provides compensation to investors both for waiting (time value of money) and for worrying (the risk of the particular asset). Measuring Risk Now, we know that the opportunity cost of capital for safe projects must be the rate of return offered by T-bills and that the opportunity cost of capital for \"average-risk\" projects must be the expected return on the market portfolio. But how about projects that do not fit into these two simple cases? To answer this, we have to learn more about investment risk. We need some measure of how far the returns may differ from the average. One way to present this is by using histograms. The bars in each histogram show the number of years between 1900 and 2010 that the investment's return fell within a specific range. Common Stocks The risk reveals itself in widespread outcomes. Treasury Bonds and Treasury Bills Unusually high or low returns are much less common. Variance and Standard Deviation In general, figures are not enough and we need a numerical measure of dispersion. Suppose that you are offered a chance to play the following game. You start by investing $100. Then two coins are flipped. For each head that comes up your starting balance will be increased by 20% and for each tail that comes up your starting balance will be reduced by 10%. Head + Head = You make 40% with probability Head + Tail = You make 10% with probability Tail + Head = You make 10% with probability Tail + Tail = You make -20% with probability Expected return = (0.25 40) + (0.5 10) + (0.25 -20) = +10% Variance and Standard Deviation The table below shows how to calculate the variance and standard deviation of the game. Variance = Average of squared deviations around the average = 1800 (1/4) = 450 Standard deviation = Square root of variance = (450) 1/2 = 21.21% Measuring Variation in Investment Returns Many analysts estimate the spread of possible outcomes from investing in the stock market by calculating the standard deviation of past returns. Variance = 2,587.17 / (6-1) = 517.43 Standard Deviation = 517.431/2 = 22.75% Measuring Variation in Investment Returns It is, of course, difficult to measure the risk of securities on the basis of just six outcomes. The table below lists the annual standard deviations for three portfolios of securities over the period 1900-2007. As expected, T-bills were the least variable security and common stocks were the most variable. Diversification We can calculate the measures of variability for individual stocks as well. As can be seen, most stocks are much more variable than the market portfolio. The market portfolio is made up of individual stocks, so why isn't its variability equal to the average variability of its components? These standard deviation estimates are calculated using monthly returns between 2005 and 2010. Diversification The answer is diversification. Selling umbrellas only or selling ice cream only are risky businesses, but if you invest in both businesses, you will make an average profit come rain or shine. Portfolio diversification works because prices of different stocks do not move exactly together. In statistical jargon, this means that stock price changes are less than perfectly correlated. Diversification works best when the returns are negatively correlated (move in opposite directions), as is the case for the umbrella and ice cream businesses. Asset versus Portfolio Risk Volatility of returns can be a misleading measure of risk for an individual asset held as part of a portfolio. Consider the following example: There are three equally likely scenarios for the economy, a recession, normal growth and a boom. Auto firms are cyclical. They do well when the economy does well. Gold firms are countercyclical. They do well when other firms do poorly. Expected Return and Volatility for Auto Firm Expected Return and Volatility for Gold Firm Asset versus Portfolio Risk It appears that gold is the more volatile stock based on standard deviations. However, it also has a lower expected rate of return. In other words, it is a loser on both counts. Would anyone be willing to hold gold mining stocks in an investment portfolio? The answer is yes. Now suppose that you invest 75% in auto and 25% in gold firms. Portfolio return in recession = (0.75 -8) + (0.25 20) = -1% Portfolio return in normal growth = (o.75 5) + (0.25 3) = 4.5% Portfolio return in boom = (0.75 18) + (0.25 -20) = 8.5% Asset versus Portfolio Risk The volatility of the auto-plus-gold stock portfolio is considerably less than the volatility of either stock separately. This is the most important benefit of diversification. Asset versus Portfolio Risk In the boom, when auto stocks do best, the poor return on gold reduces the performance of the overall portfolio. However, when auto stocks are stalling in a recession, gold does well and boosts portfolio performance. The gold stock reduces the best-case return but improves the worst-case return and therefore provides a stabilizing effect. A gold stock can be considered to be a negative-risk asset since the incremental risk of the gold stock is negative. A security that is risky held in isolation may nevertheless serve to reduce the variability of the portfolio if its returns do not move in lockstep with the rest of the portfolio. Market Risk versus Unique Risk No matter how many securities you hold, you cannot eliminate all risk. The risk that can be eliminated is called unique or specific risk. Unique risk arises because of the perils that are peculiar to a particular company. The risk that cannot be eliminated is called market or systematic risk. Market risk stems from economywide perils that threaten all businesses. Events such as changes in interest rates, foreign exchange rates and energy prices are macroeconomic events that affect almost all companies and the returns on almost all stocks. We can therefore assess the impact of \"macro\" news by tracking the rate of return on a market portfolio of all securities. The performance of the market should reflect only macro news because firm-specific events (unique risks) average out for a large number of companies. Market Risk versus Unique Risk For a reasonably well-diversified portfolio, only market risk matters. You can get most of the diversification benefits with relatively few stocks: The improvement is little when the number of stocks is increased beyond 2530. Measuring Market Risk Market risk depends on exposure to macroeconomic events and can be measured as the sensitivity of a stock's returns to fluctuations in the returns of the market portfolio. This sensitivity is called beta. In principle, the market portfolio should contain all assets in the world economy (stocks, bonds, real estate, etc.). However, analysts generally focus on stock market indices such as S&P 500. Some stocks are less affected by market fluctuations than others. Defensive stocks are not very sensitive towards market fluctuations and thus have low betas. In contrast, aggressive stocks amplify any market movements and thus have high betas. Measuring Market Risk Aggressive stocks have high betas, betas greater than 1.0, meaning that their returns tend to respond more than one for one to changes in the return of the overall market. The betas of defensive stocks are less than 1.0. The returns of these stocks vary less than one for one with market return. The average beta of all stocks is 1.0 exactly since the market portfolio consists of all stocks. If the market goes up, it is better to be in aggressive stocks. If the market goes down, it is better to be in defensive stocks. Measuring Beta Suppose we look back at the trading history of a seafood company and pick out 6 months when the return on the market portfolio was plus or minus 1%. The plot on the next slide draws a line through the performance of the company when the market is up or down by 1%. The slope of this line is the company's beta. You can see right away that the beta is 0.8 because on average, the company gains or loses 0.8% when the market is up or down by 1%. Measuring Beta There are months when the company's returns lie above or below the line. This is due to news or events that affected the company but not the overall market. Thus, we can break down stock returns into two parts: the part explained by market returns and beta (market risk), and the part due to news that is specific to the firm (unique risk). Measuring Beta The numbers in the earlier example were very convenient but the procedure is the same for all stocks. Observe returns, usually monthly, for the stock and the market. Plot the observations and fit a line showing the average return to the stock at different market returns. Beta is the slope of the fitted line. Beta for Dow Chemical The circled point shows that during that month Dow Chemical's stock price rose by 19.8% whereas the market index rose by 9.7%. Notice that more often than not Dow Chemical outperformed the market when the index rose and underperformed the market when the index fell. Dow Chemical was a relatively aggressive high beta stock. The slope of the line is 2.04. The unique risk of the company shows up in the scatter of points around the line. Beta for Consolidated Edison In contrast, Consolidated Edison was a defensive, low- beta stock. It was not highly sensitive to market movements, usually lagging behind when the market rose and yet doing better (or less badly) when the market fell. The slope of the line of best fit below is 0.24. Portfolio Betas The beta of a portfolio is just an average of the betas of the securities in the portfolio, weighted by the investment in each security. Beta of portfolio = (fraction in first stock beta of first stock) + (fraction in second stock beta of second stock) A portfolio that is invested 50-50 in Dow Chemicals and Consolidated Edison would have a beta of (2.04 + 0.24)/2 = 1.14. Risk and Return Since the return on Treasury bills is risk-free and does not change, it is unaffected by what happens to the market and the beta of T-bills is zero. Wise investors require a higher return from the market portfolio than from T-bills. The difference between the return on the market and the interest rate on bills is termed the market risk premium. We have seen in the last lecture that over the past century the average market risk premium has been 7.4% a year. We will just assume that the normal risk premium is a round 7%. Risk and Return One can see that Treasury bills have a beta of zero and a risk-free return. We will assume that the risk-free return is 3%. The market portfolio has a beta of 1.0 and an assumed expected return of 3% + 7% = 10%. Risk and Return Given these benchmarks, what expected rate of return should an investor require from a portfolio that is equally divided between T-bills and the market? The answer is halfway between. Such a portfolio would have a beta of 0.5 and an expected return of 6.5% which includes a risk premium of 3.5% above T-bills. Capital Asset Pricing Model This calculation can be done in the following way: Market risk premium = rM - rF = 10% - 3% = 7% Since beta measures risk relative to the market: Risk premium = (rM - rF) = 0.5 7% = 3.5% 6.5% Expected return = rF + (rM - rF) = 3% + 3.5% = This relationship is known as the Capital Asset Pricing Model (CAPM) and holds for any asset. Capital Asset Pricing Model Suppose the only investment alternatives are Treasury bills and the market portfolio. If you are risk-averse, you may invest 20% of your money in the market portfolio and put the other 80% in T-bills. The beta of the portfolio would simply be Beta of portfolio = (market 0.2) + (bills 0.8) = (1 0.2) + (0 0.8) = 0.2 Expected portfolio return = rF + (rM - rF) = 3% + 0.2 7% = 4.4% The Security Market Line The last example illustrates the point that by investing some proportion of your money in the market portfolio and lending the balance, you can obtain any combination of risk and expected return along the sloping line below. This line is called the security market line. The Security Market Line The security market line describes the expected returns and risks from investing different fractions of your funds in the market. It also sets a standard for other investments. Investors would be willing to hold other investments only if they offer equally good prospects. The required risk premium for any investment is: Risk premium = Beta Expected market risk premium If X offered a lower return than 6.5%, then nobody would buy any of it, its price would drop and its return would increase. Similarly, if X offered a higher return than 6.5%, then everybody would want to buy more of it, its price would increase and its return would drop. How Well Does the CAPM Work? There is little doubt that the CAPM is too simple to capture everything. See below. The red line shows the cumulative difference between the returns of the smallfirm and large-firm stocks. The blue line shows the cumulative difference between the returns of the value (low MB ratio) and growth (high MB) stocks. The superior performance of small and value stocks contradicts the CAPM. How Well Does the CAPM Work? Defenders of the CAPM emphasize that these findings are concerned with the fact that the model is concerned with expected returns whereas all we can observe is the actual returns. Actual returns do reflect expectations but they also include lots of \"noise\" - the steady flow of surprises that conceal whether on average investors have received the returns they expected. In any case, CAPM captures two fundamental ideas in a simple way: Investors require some extra return for taking on risk. Investors appear to be concerned principally with the market risk that they cannot eliminate with diversification