Capital Markets and the Pricing of Risk PDF

PA RT Risk and Return 4 THE LAW OF ONE PRICE CONNECTION. To apply the Law of One Price correctly requires comparing investment opportunities...

PA RT Risk and Return 4 THE LAW OF ONE PRICE CONNECTION. To apply the Law of One Price correctly requires comparing investment opportunities of equivalent risk. In this CHAPTER 10 part of the book, we explain how to measure and compare risks across invest- Capital Markets ment opportunities. Chapter 10 introduces the key insight that investors only de- and the Pricing mand a risk premium for risk they cannot costlessly eliminate by diversifying their of Risk portfolios. Hence, only non-diversifiable market risk will matter when comparing investment opportunities. Intuitively, this insight suggests that an investment’s risk CHAPTER 11 premium will depend on its sensitivity to market risk. In Chapter 11, we quantify Optimal these ideas and derive investors’ optimal investment portfolio choices. We then Portfolio consider the implications of assuming all investors choose their portfolio of risky investments optimally. This assumption leads to the Capital Asset Pricing Model Choice and (CAPM), the central model in financial economics that quantifies the notion of the Capital “equivalent risk” and thereby provides the relation between risk and return. In Asset Pricing Chapter 12, we apply these ideas and consider the practicalities of estimating Model the cost of capital for a firm and for an individual investment project. Chapter 13 takes a closer look at the behavior of individual, as well as professional, investors. CHAPTER 12 Doing so reveals some strengths and weaknesses of the CAPM, as well as ways Estimating the we can combine the CAPM with the principle of no arbitrage for a more general Cost of Capital model of risk and return. CHAPTER 13 Investor Behavior and Capital Market Efficiency 355 M10_BERK6318_06_GE_C10.indd 355 26/04/23 6:26 PM  C HAPT ER 10 Capital Markets and the Pricing of Risk NOTATION OVER THE TEN-YEAR PERIOD 2012 THROUGH 2021, INVESTORS IN p R probability of return R the consumer products firm Procter & Gamble (PG) earned an average return of 13% per year. Within this period, there was some variation, with the annual re- Var (R ) variance of return R turn ranging from -10% in 2015 to 40% in 2019. Over the same period, investors SD (R ) standard deviation of in the semi-conductor firm Advanced Micro Devices earned an average return of return R 65% per year. These investors, however, lost 56% in 2012 and gained over E [R ] expectation of return R 295% in 2016. Finally, investors in three-month U.S. Treasury bills earned an Div t dividend paid on date t average annual return of 0.6% during the period, with a high of 2.0% in 2019 Pt price on date t and a low of 0.02% in 2014. Clearly, these three investments offered returns that were very different in terms of their average level and their variability. What Rt  realized or total return accounts for these differences? of a security from date t − 1 to t In this chapter, we will consider why these differences exist. Our goal is to develop a theory that explains the relationship between the risk of an investment, R average return and the average (or expected) return that investors will require to hold that invest- β s beta of security s ment. We then use this theory to explain how to determine the cost of capital for r cost of capital of an an investment opportunity. investment opportunity We begin our investigation of the relationship between risk and return by examining historical data for publicly traded securities. We will see, for example, that while stocks are riskier investments than bonds, they have also earned higher average returns. We can interpret the higher average return on stocks as compen- sation to investors for the greater risk they are taking. But we will also find that not all risk needs to be compensated. By holding a portfolio containing many different investments, investors can eliminate risks that are specific to individual securities. Only those risks that cannot be eliminated by holding a large portfolio determine the risk premium required by investors. These observations will allow us to refine our definition of what risk is, how we can measure it, and thus, how to determine the cost of capital. 356 M10_BERK6318_06_GE_C10.indd 356 26/04/23 6:26 PM  10.1 Risk and Return: Insights from 96 Years of Investor History 357 10.1  Risk and Return: Insights from 96 Years of Investor History We begin our look at risk and return by illustrating how risk affects investor decisions and returns. Suppose your great-grandparents invested $100 on your behalf at the end of 1925. They instructed their broker to reinvest any dividends or interest earned in the account until the beginning of 2022. How would that $100 have grown if it were placed in one of the following investments? 1. Standard & Poor’s 500 (S&P 500): A portfolio, constructed by Standard and Poor’s, comprising 90 U.S. stocks up to 1957 and 500 U.S. stocks after that. The firms rep- resented are leaders in their respective industries and are among the largest firms, in terms of market value, traded on U.S. markets. 2. Small Stocks: A portfolio, updated quarterly, of U.S. stocks traded on the NYSE with market capitalizations in the bottom 20%. 3. World Portfolio: A portfolio of international stocks from all of the world’s major stock markets in North America, Europe, and Asia.1 4. Corporate Bonds: A portfolio of long-term, AAA-rated U.S. corporate bonds with maturities of approximately 20 years.2 5. Treasury Bills: An investment in one-month U.S. Treasury bills. Figure 10.1 shows the result, through the start of 2022, of investing $100 at the end of 1925 in each of these five investment portfolios, ignoring transactions costs. During this 96-year period in the United States, small stocks experienced the highest long-term return, followed by the large stocks in the S&P 500, the international stocks in the world portfolio, corporate bonds, and finally Treasury bills. All of the investments grew faster than infla- tion, as measured by the consumer price index (CPI). At first glance the graph is striking—had your great-grandparents invested $100 in the small stock portfolio, the investment would be worth over $10.3 million at the beginning of 2022! By contrast, if they had invested in Treasury bills, the investment would be worth only about $2,150. Given this wide difference, why invest in anything other than small stocks? But first impressions can be misleading. While over the full horizon stocks (especially small stocks) did outperform the other investments, they also endured periods of signifi- cant losses. Had your great-grandparents put the $100 in a small stock portfolio during the Depression era of the 1930s, it would have grown to $181 in 1928, but then fallen to only $15 by 1932. Indeed, it would take until World War II for stock investments to outperform corporate bonds. Even more importantly, your great-grandparents would have sustained losses at a time when they likely needed their savings the most––in the depths of the Great Depression. A similar story held during the 2008 financial crisis: All of the stock portfolios declined by more than 50%, with the small stock portfolio declining by almost 70% (over $1.5 million!) from its peak in 2007 to its lowest point in 2009. Again, many investors faced a double whammy: an increased risk of being unemployed (as firms started laying off employees) 1 Based on a World Market Index constructed by Global Financial Data, with approximate initial weights of 44% North America, 44% Europe, and 12% Asia, Africa, and Australia. 2 Based on Global Financial Data’s Corporate Bond Index. M10_BERK6318_06_GE_C10.indd 357 26/04/23 6:26 PM  358 Chapter 10 Capital Markets and the Pricing of Risk FIGURE 10.1 Value of $100 Invested in 1925 in Stocks, Bonds, or Bills $100,000,000 Small Stocks 2020: –50%, –33% S&P 500 2007–09: –67%, –51% $10,000,000 World Portfolio $10,323,401 Corporate Bonds 2000-02: –26%, –45% Treasury Bills 1987: –34%, –30% $1,273,189 $1,000,000 CPI Value of Investment $417,222 1968–74: –63%, –37% $100,000 $26,922 $10,000 $2,153 $1,558 $1,000 1979–81: Inflation, TBill yields peak at 15% $100 1937–38: –72%, –50% 1928–32: Small stocks –92%, S&P 500 –84% $10 1925 1935 1945 1955 1965 1975 1985 1995 2005 2015 Year The chart shows the growth in value of $100 invested in 1925 if it were invested in U.S. large stocks, small stocks, world stocks, corporate bonds, or Treasury bills, with the level of the consumer price index (CPI) shown as a refer- ence. Returns were calculated at year-end assuming all dividends and interest are reinvested and excluding transac- tions costs. Note that while stocks have generally outperformed bonds and bills, they have also endured periods of significant losses (numbers shown represent peak to trough decline, with the decline in small stocks in red and the S&P 500 in blue). Source: Chicago Center for Research in Security Prices, Standard and Poor’s, MSCI, and Global Financial Data precisely when the value of their savings eroded. Thus, while the stock portfolios had the best performance over this 96-year period, that performance came at a cost––the risk of large losses in a downturn. On the other hand, Treasury bills enjoyed steady––albeit modest––gains each year. Few people ever make an investment for 96 years, as depicted in Figure 10.1. To gain additional perspective on the risk and return of these investments, Figure 10.2 shows the results for more realistic investment horizons and different initial investment dates. Panel (a), for example, shows the value of each investment after one year and illustrates that if we rank the investments by the volatility of their annual increases and decreases in value, we obtain the same ranking we observed with regard to performance: Small stocks had the most variable returns, followed by the S&P 500, the world portfolio, corporate bonds, and finally Treasury bills. Panels (b), (c), and (d) of Figure 10.2 show the results for 5-, 10-, and 20-year invest- ment horizons, respectively. Note that as the horizon lengthens, the relative performance of the stock portfolios improves. That said, even with a 10-year horizon there were periods during which stocks underperformed Treasuries. And while investors in small stocks most often came out ahead, this was not assured even with a 20-year horizon: For investors in M10_BERK6318_06_GE_C10.indd 358 26/04/23 6:26 PM  10.1 Risk and Return: Insights from 96 Years of Investor History 359 FIGURE 10.2 Value of $100 Invested in Alternative Assets for Differing Horizons Small Stocks S&P 500 Corporate bonds Treasury Bills (a) Value after 1 Year (b) Value after 5 Years $300 $1,000 $200 $100 $100 $0 $10 1925 1935 1945 1955 1965 1975 1985 1995 2005 2015 1925 1935 1945 1955 1965 1975 1985 1995 2005 2015 (c) Value after 10 Years (d) Value after 20 Years $1,000 $10,000 $100 $1,000 $10 $100 1925 1935 1945 1955 1965 1975 1985 1995 2005 1925 1935 1945 1955 1965 1975 1985 1995 Initial Investment Year Initial Investment Year Each panel shows the result of investing $100, in each investment opportunity, for horizons of 1, 5, 10, or 20 years, plotted as a function of the year when the investment was initially made. Dividends and interest are reinvested and ­transaction costs are excluded. Note that small stocks show the greatest variation in performance at the one-year horizon, followed by large stocks and then corporate bonds. For longer horizons, the relative performance of stocks improved, but they remained riskier. Source Data: Chicago Center for Research in Security Prices, Standard and Poor’s, MSCI, and Global Financial Data the early 1980s, small stocks did worse than both the S&P 500 and corporate bonds over the subsequent 20 years. Finally, stock investors with long potential horizons might find themselves in need of cash in intervening years, and be forced to liquidate at a loss relative to safer alternatives. In Chapter 3, we explained why investors are averse to fluctuations in the value of their investments, and that investments that are more likely to suffer losses in downturns must compensate investors for this risk with higher expected returns. Figures 10.1 and 10.2 pro- vide compelling historical evidence of this relationship between risk and return, just as we should expect in an efficient market. Given this clear evidence that investors do not like risk and thus demand a risk premium to bear it, our goal in this chapter is to quantify this relationship. We want to explain how much investors demand (in terms of a higher expected return) to bear a given level of risk. To do so, we must first develop tools that will allow us to measure risk and return. That is our objective in the next section. M10_BERK6318_06_GE_C10.indd 359 26/04/23 6:26 PM  360 Chapter 10 Capital Markets and the Pricing of Risk CONCEPT CHECK 1. For an investment horizon from 1926 to 2021, which of the following investments had the highest return: the S&P 500, small stocks, world portfolio, corporate bonds, or Treasury bills? Which had the lowest return? 2. For an investment horizon of just one year, which of these investments was the most variable? Which was the least variable? 10.2 Common Measures of Risk and Return When a manager makes an investment decision or an investor purchases a security, they have some view as to the risk involved and the likely return the investment will earn. Thus, we begin our discussion by reviewing the standard ways to define and measure risks. Probability Distributions Different securities have different initial prices, pay different cash flows, and sell for dif- ferent future amounts. To make them comparable, we express their performance in terms of their returns. The return indicates the percentage increase in the value of an investment per dollar initially invested in the security. When an investment is risky, there are different returns it may earn. Each possible return has some likelihood of occurring. We summarize this information with a probability distribution, which assigns a probability, p R , that each possible return, R , will occur. Let’s consider a simple example. Suppose BFI stock currently trades for $100 per share. You believe that in one year there is a 25% chance the share price will be $140, a 50% chance it will be $110, and a 25% chance it will be $80. BFI pays no dividends, so these pay- offs correspond to returns of 40%, 10%, and −20%, respectively. Table 10.1 summarizes the probability distribution for BFI’s returns. We can also represent the probability distribution with a histogram, as shown in Figure 10.3. Expected Return Given the probability distribution of returns, we can compute the expected return. We cal- culate the expected (or mean) return as a weighted average of the possible returns, where the weights correspond to the probabilities.3 Expected (Mean) Return Expected Return = E [ R ] = ∑ p R × R (10.1) R TABLE 10.1     Probability Distribution of Returns for BFI Probability Distribution Current Stock Price ($) Stock Price in One Year ($) Return, R Probability, p R 140 0.40 25% 100 110 0.10 50% 80 −0.20 25% 3 The notation ∑ R means that we calculate the sum of the expression (in this case, p R × R ) over all ­possible returns R. M10_BERK6318_06_GE_C10.indd 360 26/04/23 6:26 PM  10.2 Common Measures of Risk and Return 361 FIGURE 10.3 50% Probability Distribution of Returns for BFI The height of a bar in the Probability histogram indicates the 25% 25% likelihood of the associated outcome. 2.25 2.20 2.15 2.10 2.05.00.05.10.15.20.25.30.35.40.45 Return Expected Return The expected return is the return we would earn on average if we could repeat the in- vestment many times, drawing the return from the same distribution each time. In terms of the histogram, the expected return is the “balancing point” of the distribution, if we think of the probabilities as weights. The expected return for BFI is E [ R BFI ] = 25% ( −0.20 ) + 50% ( 0.10 ) + 25% ( 0.40 ) = 10% This expected return corresponds to the balancing point in Figure 10.3. Variance and Standard Deviation Two common measures of the risk of a probability distribution are its variance and standard deviation. The variance is the expected squared deviation from the mean, and the standard deviation is the square root of the variance. Variance and Standard Deviation of the Return Distribution Var ( R ) = E  ( R − E [ R ] )  = ∑ p R × ( R − E [ R ] ) 2 2 R SD ( R ) = Var ( R ) (10.2) If the return is risk-free and never deviates from its mean, the variance is zero. Otherwise, the variance increases with the magnitude of the deviations from the mean. Therefore, the variance is a measure of how “spread out” the distribution of the return is. The variance of BFI’s return is Var ( R BFI ) = 25% × ( −0.20 − 0.10 ) 2 + 50% × ( 0.10 − 0.10 ) 2 + 25% × ( 0.40 − 0.10 ) 2 = 0.045 The standard deviation of the return is the square root of the variance, so for BFI, SD ( R ) = Var ( R ) = 0.045 = 21.2% (10.3) M10_BERK6318_06_GE_C10.indd 361 26/04/23 6:26 PM  362 Chapter 10 Capital Markets and the Pricing of Risk In finance, we refer to the standard deviation of a return as its volatility. While the vari- ance and the standard deviation both measure the variability of the returns, the standard deviation is easier to interpret because it is in the same units as the returns themselves.4 EXAMPLE 10.1 Calculating the Expected Return and Volatility Problem Suppose AMC stock is equally likely to have a 45% return or a −25% return. What are its expected return and volatility? Solution First, we calculate the expected return by taking the probability-weighted average of the possible returns: E [ R ] = ∑ p R × R = 50% × 0.45 + 50% × ( −0.25 ) = 10.0% R To compute the volatility, we first determine the variance: Var ( R ) = ∑ p R × ( R − E [ R ] ) = 50% × ( 0.45 − 0.10 ) 2 + 50% × ( −0.25 − 0.10 ) 2 2 R = 0.1225 Then, the volatility or standard deviation is the square root of the variance: SD ( R ) = Var ( R ) = 0.1225 = 35% Note that both AMC and BFI have the same expected return, 10%. However, the re- turns for AMC are more spread out than those for BFI—the high returns are higher and the low returns are lower, as shown by the histogram in Figure 10.4. As a result, AMC has a higher variance and volatility than BFI. FIGURE 10.4 50% 50% 50% Probability Distribution BFI AMC for BFI and AMC Returns While both stocks have the Probability same expected return, AMC’s 25% 25% return has a higher variance and standard deviation. 2.25 2.20 2.15 2.10 2.05.00.05.10.15.20.25.30.35.40.45 Return 4 While variance and standard deviation are the most common measures of risk, they do not differenti- ate upside and downside risk. Alternative measures that focus on downside risk include the semivariance (variance of the losses only) and the expected tail loss (the expected loss in the worst x% of outcomes). Because they often produce the same ranking (as in Example 10.1, or if returns are normally distributed) but are more complicated to apply, these alternative measures tend to be used only in special applications. M10_BERK6318_06_GE_C10.indd 362 26/04/23 6:26 PM  10.3 Historical Returns of Stocks and Bonds 363 If we could observe the probability distributions that investors anticipate for different securities, we could compute their expected returns and volatilities and explore the rela- tionship between them. Of course, in most situations we do not know the explicit prob- ability distribution, as we did for BFI. Without that information, how can we estimate and compare risk and return? A popular approach is to extrapolate from historical data, which is a sensible strategy if we are in a stable environment and believe that the distribution of future returns should mirror that of past returns. Let’s look at the historical returns of stocks and bonds, to see what they reveal about the relationship between risk and return. CONCEPT CHECK 1. How do we calculate the expected return of a stock? 2. What are the two most common measures of risk, and how are they related to each other? 10.3 Historical Returns of Stocks and Bonds In this section, we explain how to compute average returns and volatilities using historical stock market data. The distribution of past returns can be helpful when we seek to estimate the distribution of returns investors may expect in the future. We begin by first explaining how to compute historical returns. Computing Historical Returns Of all the possible returns, the realized return is the return that actually occurs over a par- ticular time period. How do we measure the realized return for a stock? Suppose you invest in a stock on date t for price Pt. If the stock pays a dividend, Div t +1 , on date t + 1, and you sell the stock at that time for price Pt +1 , then the realized return from your investment in the stock from t to t + 1 is Div t +1 + Pt +1 Div i +1 Pt +1 − Pt R t +1 = −1= + Pt Pt Pt = Dividend Yield + Capital Gain Rate (10.4) That is, as we discussed in Chapter 9, the realized return, Rt +1 , is the total return we earn from dividends and capital gains, expressed as a percentage of the initial stock price.5 Calculating Realized Annual Returns. If you hold the stock beyond the date of the first dividend, then to compute your return you must specify how you invest any dividends you receive in the interim. To focus on the returns of a single security, let’s assume that you reinvest all dividends immediately and use them to purchase additional shares of the same stock or security. In this case, we can use Eq. 10.4 to compute the stock’s return between dividend payments, and then compound the returns from each dividend interval to compute the return over a longer horizon. For example, if a stock pays dividends at the end of each quarter, with realized returns R Q 1 , … , R Q 4 each quarter, then its annual realized return, Rannual , is 1 + Rannual = ( 1 + R Q 1 )( 1 + R Q 2 )( 1 + R Q 3 )( 1 + R Q 4 ) (10.5) 5 We can compute the realized return for any security in the same way, by replacing the dividend payments with any cash flows paid by the security (e.g., with a bond, coupon payments would replace dividends). M10_BERK6318_06_GE_C10.indd 363 26/04/23 6:26 PM  364 Chapter 10 Capital Markets and the Pricing of Risk EXAMPLE 10.2 Realized Returns for Microsoft Stock Problem What were the realized annual returns for Microsoft ( MSFT ) stock in 2004 and 2008? Solution When we compute Microsoft’s annual return, we assume that the proceeds from the divi- dend payment were immediately reinvested in Microsoft stock. That way, the return corre- sponds to remaining fully invested in Microsoft over the entire period. To do that we look up Microsoft stock price data at the start and end of the year, as well as at any dividend dates (Yahoo! Finance is a good source for such data; see also capm.berkdemarzo.com for addi- tional sources). From these data, we can construct the following table (prices and dividends in $/share): Date Price Dividend Return Date Price Dividend Return 12/31/03 27.37 12/31/07 35.60 8/23/04 27.24 0.08 −0.18% 2/19/08 28.17 0.11 −20.56% 11/15/046 27.39 3.08 11.86% 5/13/08 29.78 0.11 6.11% 12/31/04 26.72 −2.45% 8/19/08 27.32 0.11 −7.89% 11/18/08 19.62 0.13 −27.71% 12/31/08 19.44 −0.92% The return from December 31, 2003, until August 23, 2004, is equal to 0.08 + 27.24 − 1 = −0.18% 27.37 The rest of the returns in the table are computed similarly. We then calculate the annual returns using Eq. 10.5: R2004 = ( 0.9982 )( 1.1186 )( 0.9755 ) − 1 = 8.92% R2008 = ( 0.7944 )( 1.0611 )( 0.9211 )( 0.7229 )( 0.9908 ) − 1 = −44.39% Example 10.2 illustrates two features of the returns from holding a stock like Microsoft. First, both dividends and capital gains contribute to the total realized return—ignoring ei- ther one would give a very misleading impression of Microsoft’s performance. Second, the returns are risky. In years like 2004 the returns are positive, but in other years like 2008 they are negative, meaning Microsoft’s shareholders lost money over the year. We can compute realized returns in this same way for any investment. We can also com- pute the realized returns for an entire portfolio, by keeping track of the interest and divi- dend payments paid by the portfolio during the year, as well as the change in the market value of the portfolio. For example, the realized returns for the S&P 500 index are shown in Table 10.2, which for comparison purposes also lists the returns for Microsoft and for three-month Treasury bills. Comparing Realized Annual Returns. Once we have calculated the realized annual returns, we can compare them to see which investments performed better in a given year. From Table 10.2, we can see that Microsoft stock outperformed the S&P 500 and 6 The large dividend in November 2004 included a $3 special dividend which Microsoft used to reduce its accumulating cash balance and disburse $32 billion in cash to its investors (at the time the largest aggregate dividend payment in history). M10_BERK6318_06_GE_C10.indd 364 26/04/23 6:26 PM  10.3 Historical Returns of Stocks and Bonds 365 TABLE 10.2       Realized Return for the S&P 500, Microsoft, and Treasury Bills, 2005–2021 Year End S&P 500 Dividends S&P 500 Realized Microsoft Realized 1-Month T-Bill Index Paid* Return Return Return 2004 1211.92 2005 1248.29 23.15 4.9% -0.9% 3.0% 2006 1418.30 27.16 15.8% 15.8% 4.8% 2007 1468.36 27.86 5.5% 20.8% 4.7% 2008 903.25 21.85 -37.0% -44.4% 1.5% 2009 1115.10 27.18 26.5% 60.5% 0.1% 2010 1257.64 25.44 15.1% -6.5% 0.1% 2011 1257.60 26.60 2.1% -4.5% 0.0% 2012 1426.19 32.67 16.0% 5.8% 0.1% 2013 1848.36 39.75 32.4% 44.3% 0.0% 2014 2058.90 42.47 13.7% 27.6% 0.0% 2015 2043.94 43.45 1.4% 22.7% 0.0% 2016 2238.83 49.56 12.0% 15.1% 0.2% 2017 2673.61 53.99 21.8% 40.7% 0.8% 2018 2506.85 49.54 -4.4% 20.8% 1.7% 2019 3230.78 65.39 31.5% 57.6% 2.1% 2020 3756.07 69.14 18.4% 42.5% 0.4% 2021 4766.18 68.08 28.7% 52.5% 0.0% * Total dividends paid by the 500 stocks in the portfolio, based on the number of shares of each stock in the index, adjusted until the end of the year, assuming they were reinvested when paid. Source: Standard & Poor’s, Microsoft, and U.S. Treasury Data Treasuries in 2007, 2009, and 2013–2021. On the other hand, Treasury bills performed better than Microsoft stock in 2005, 2008, and 2010–2011. Note also the overall tendency for Microsoft’s return to move in the same direction as the S&P 500, which it did in 13 out of the 17 years. Over any particular period we observe only one draw from the probability distribution of returns. However, if the probability distribution remains the same, we can observe mul- tiple draws by observing the realized return over multiple periods. By counting the number of times the realized return falls within a particular range, we can estimate the underlying probability distribution. Let’s illustrate this process with the data in Figure 10.1. Figure 10.5 plots the annual returns for each U.S. investment in Figure 10.1 in a histo- gram. The height of each bar represents the number of years that the annual returns were in each range indicated on the x-axis. When we plot the probability distribution in this way using historical data, we refer to it as the empirical distribution of the returns. Average Annual Returns The average annual return of an investment during some historical period is simply the average of the realized returns for each year. That is, if Rt is the realized return of a security in year t, then the average annual return for years 1 through T is Average Annual Return of a Security T 1 1 R= ( R1 + R2 +  + RT ) = ∑ Rt (10.6) T T t =1 Notice that the average annual return is the balancing point of the empirical distribution— in this case, the probability of a return occurring in a particular range is measured by the M10_BERK6318_06_GE_C10.indd 365 26/04/23 6:26 PM  366 Chapter 10 Capital Markets and the Pricing of Risk FIGURE 10.5 70 1-month Treasury Bills The Empirical 60 AAA Corporate Bonds 50 Distribution of Annual S&P 500 40 Returns for U.S. Large Small Stocks Frequency (number of years) 30 Stocks (S&P 500), Small 20 Stocks, Corporate 10 Bonds, and Treasury 0 Bills, 1926–2021 40 30 The height of each bar rep- 20 resents the number of years 10 that the annual returns were 0 in each 5% range. Note the 10 greater variability of stock 0 returns (especially small 10 stocks) compared to the re- 0 turns of corporate bonds or –60% –50% –40% –30% –20% –10% 0% 10% 20% 30% Annual Return 40% 50% 60% 70% 80% 90% >100% Treasury bills. number of times the realized return falls in that range. Therefore, if the probability distri- bution of the returns is the same over time, the average return provides an estimate of the expected return. Using the data from Table 10.2, the average return for the S&P 500 for the years 2005–2021 inclusive is 1 R= ( −0.049 + 0.158 + 0.055 + 0.37 + 0.265 + 0.151 + 0.21 + 0.160 + 0.324 17 + 0.137 + 0.014 + 0.120 + 0.218 − 0.044 + 0.315 + 0.184 + 0.287 ) = 12.0% The average Treasury bill return from 2005–2021 was 1.2%. Therefore, investors earned 10.8% more on average holding the S&P 500 rather than investing in Treasury bills over this period. Table 10.3 provides the average returns for different U.S. investments from 1926–2021. TABLE 10.3       Average Annual Returns for U.S. Small Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury Bills, 1926–2021 Investment Average Annual Return Small stocks 18.6% S&P 500 12.2% Corporate bonds 6.2% Treasury bills 3.3% M10_BERK6318_06_GE_C10.indd 366 26/04/23 6:26 PM  10.3 Historical Returns of Stocks and Bonds 367 The Variance and Volatility of Returns Looking at Figure 10.5, we can see that the variability of the returns is very different for each investment. The distribution of small stocks’ returns shows the widest spread. The large stocks of the S&P 500 have returns that vary less than those of small stocks, but much more than the returns of corporate bonds or Treasury bills. To quantify this difference in variability, we can estimate the standard deviation of the probability distribution. As before, we will use the empirical distribution to derive this esti- mate. Using the same logic as we did with the mean, we estimate the variance by comput- ing the average squared deviation from the mean. We do not actually know the mean, so instead we use the best estimate of the mean—the average realized return.7 Variance Estimate Using Realized Returns T 1 ∑ ( Rt − R ) 2 Var ( R ) = (10.7) T − 1 t =1 We estimate the standard deviation or volatility as the square root of the variance.8 EXAMPLE 10.3 Computing a Historical Volatility Problem Using the data from Table 10.2, what are the variance and volatility of the S&P 500’s returns for the years 2005–2021? Solution Earlier, we calculated the average annual return of the S&P 500 during this period to be 12.0%. Therefore, 1 ∑ ( Rt − R ) 2 Var ( R ) = T −1 t 1  = ( −0.049 − 0.120 ) 2 + ( 0.158 − 0.120 ) 2 +  + ( 0.287 − 0.120 ) 2  17 − 1  = 0.028 The volatility or standard deviation is therefore SD ( R ) = Var ( R ) = 0.028 = 16.7% We can compute the standard deviation of the returns to quantify the differences in the variability of the distributions that we observed in Figure 10.5. These results are shown in Table 10.4. Comparing the volatilities in Table 10.4 we see that, as expected, small stocks have had the most variable historical returns, followed by large stocks. The returns of corporate bonds and Treasury bills are much less variable than stocks, with Treasury bills being the least volatile investment category. 7 Why do we divide by T − 1 rather than by T here? It is because we do not know the true expected re- turn, and so must compute deviations from the estimated average return R. But in calculating the average return from the data, we lose a degree of freedom (in essence, we “use up” one of the data points), so that effectively we only have T − 1 remaining data points to estimate the variance. 8 If the returns used in Eq. 10.7 are not annual returns, the variance is typically converted to annual terms by multiplying the number of periods per year. For example, when using monthly returns, we multiply the variance by 12 and, equivalently, the standard deviation by 12. M10_BERK6318_06_GE_C10.indd 367 26/04/23 6:26 PM  368 Chapter 10 Capital Markets and the Pricing of Risk TABLE 10.4       Volatility of U.S. Small Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury Bills, 1926–2021 Investment Return Volatility (Standard Deviation) Small stocks 38.6% S&P 500 19.7% Corporate bonds 6.3% Treasury bills 3.1% Estimation Error: Using Past Returns to Predict the Future To estimate the cost of capital for an investment, we need to determine the expected return that investors will require to compensate them for that investment’s risk. If the distribu- tion of past returns and the distribution of future returns are the same, we could look at the return investors expected to earn in the past on the same or similar investments, and assume they will require the same return in the future. However, there are two difficulties with this approach. First, We do not know what investors expected in the past; we can only observe the actual returns that were realized. In 2008, for example, investors lost 37% investing in the S&P 500, which is surely not what they expected at the beginning of the year (or they would have invested in Treasury Bills instead). If we believe that investors are neither overly optimistic nor pessimistic on average, then over time, the average realized return should match investors’ expected return. Armed with this assumption, we can use a security’s historical average return to infer its expected return. But now we encounter the second difficulty: The average return is just an estimate of the true expected return, and is subject to estimation error. Given the volatility of stock returns, this estimation error can be large even with many years of data, as we will see next. Standard Error. We measure the estimation error of a statistical estimate by its standard error. The standard error is the standard deviation of the estimated value of the mean of the actual distribution around its true value; that is, it is the standard deviation of the aver- age return. The standard error provides an indication of how far the sample average might deviate from the expected return. If the distribution of a stock’s return is identical each year, and each year’s return is independent of prior years’ returns,9 then we calculate the standard error of the estimate of the expected return as follows: Standard Error of the Estimate of the Expected Return SD (Individual Risk ) SD ( Average of Independent, Identical Risks) = (10.8) Number of Observations 9 Saying that returns are independent and identically distributed (IID) means that the likelihood that the return has a given outcome is the same each year and does not depend on past returns (in the same way that the odds of a coin coming up heads do not depend on past flips). It turns out to be a reasonable first approximation for stock returns. M10_BERK6318_06_GE_C10.indd 368 26/04/23 6:26 PM  10.3 Historical Returns of Stocks and Bonds 369 Because the average return will be within two standard errors of the true expected return approximately 95% of the time,10 we can use the standard error to determine a reasonable range for the true expected value. The 95% confidence interval for the expected return is Historical Average Return ± ( 2 × Standard Error ) (10.9) For example, from 1926 to 2021 the average return of the S&P 500 was 12.2% with a volatility of 19.7%. Assuming its returns are drawn from an independent and identical dis- tribution (IID) each year, the 95% confidence interval for the expected return of the S&P 500 during this period is  19.7%  12.2% ± 2  = 12.2% ± 4.0%  96  or a range from 8.2% to 16.2%. Thus, even with 96 years of data, we cannot estimate the expected return of the S&P 500 very accurately. If we believe the distribution may have changed over time and we can use only more recent data to estimate the expected return, then the estimate will be even less accurate. Limitations of Expected Return Estimates. Individual stocks tend to be even more vol- atile than large portfolios, and many have been in existence for only a few years, providing little data with which to estimate returns. Because of the relatively large estimation error in such cases, the average return investors earned in the past is not a reliable estimate of a security’s expected return. Instead, we need to derive a different method to estimate the expected return that relies on more reliable statistical estimates. In the remainder of this chapter, we will pursue the following alternative strategy: First we will consider how to measure a security’s risk, and then we will use the relationship between risk and return— which we must still determine—to estimate its expected return. EXAMPLE 10.4 The Accuracy of Expected Return Estimates Problem Using the returns for the S&P 500 from 2005–2021 only (see Table 10.2), what is the 95% confidence interval you would estimate for the S&P 500’s expected return? Solution Earlier, we calculated the average return for the S&P 500 during this period to be 12.0%, with a volatility of 16.7% (see Example 10.3). The standard error of our estimate of the expected return is 16.7% ÷ 17 = 4.0%, and the 95% confidence interval is 12.0% ± ( 2 × 4.0% ), or from 4.0% to 20.0%. As this example shows, with only a few years of data, we cannot reliably estimate expected returns for stocks! CONCEPT CHECK 1. How do we calculate the average annual return of an investment? 2. We have 96 years of data on the S&P 500 returns, yet we cannot estimate the expected return of the S&P 500 very accurately. Why? 10 If returns are independent and from a normal distribution, then the estimated mean will be within two standard errors of the true mean 95.44% of the time. Even if returns are not normally distributed, this formula is approximately correct with a sufficient number of independent observations. M10_BERK6318_06_GE_C10.indd 369 26/04/23 6:26 PM  370 Chapter 10 Capital Markets and the Pricing of Risk Arithmetic Average Returns Versus Compound Annual Returns We compute average annual returns by calculating an This logic implies that the compound annual return will al- arithmetic average. An alternative is the compound annual ways be below the average return, and the difference grows return (also called the compound annual growth rate, or with the volatility of the annual returns. (Typically, the dif- CAGR), which is computed as the geometric average of the ference is about half of the variance of the returns.) annual returns R1 , … , RT : Which is a better description of an investment’s return? Compound Annual Return = The compound annual return is a better description of the long-run historical performance of an investment. It  ( 1 + R1 ) × ( 1 + R2 ) × … × ( 1 + RT )  − 1 1T describes the equivalent risk-free return that would be It is equivalent to the IRR of the investment over the period: required to duplicate the investment’s performance over the same time period. The ranking of the long-run performance ( Final Value Initial Investment ) 1T −1 of different investments coincides with the ranking of their For example, using the data in Figure 10.1, the com- compound annual returns. Thus, the compound annual pound annual return of the S&P 500 from 1926–2021 was return is the return that is most often used for comparison ( 1,273,189 100 )1 96 − 1 = 10.35% purposes. For example, mutual funds generally report their compound annual returns over the last five or ten years. That is, investing in the S&P 500 from 1926 to 2022 was Conversely, we should use the arithmetic average return equivalent to earning 10.35% per year over that time period. when we are trying to estimate an investment’s expected Similarly, the compound annual return for small stocks was return over a future horizon based on its past performance. 12.78%, for corporate bonds was 6.00%, and for Treasury If we view past returns as independent draws from the bills was 3.2%. same distribution, then the arithmetic average return pro- In each case, the compound annual return is below the vides an unbiased estimate of the true expected return.* average annual return shown in Table 10.3. This difference For example, if the investment mentioned above is reflects the fact that returns are volatile. To see the effect equally likely to have annual returns of +20% and −20% of volatility, suppose an investment has annual returns in the future, then if we observe many two-year periods, a of +20% one year and −20% the next year. The average $1 investment will be equally likely to grow to annual return is 21 ( 20% − 20% ) = 0%, but the value of $1 invested after two years is (1.20)(1.20) = $1.44, $1 × ( 1.20 ) × ( 0.80 ) = $0.96 (1.20)( 0.80) = $0.96, That is, an investor would have lost money. Why? (0.80)(1.20) = $0.96, Because the 20% gain happens on a $1 investment, whereas or (0.80)( 0.80) = $0.64. the 20% loss happens on a larger investment of $1.20. In Thus, the average value in two years will be this case, the compound annual return is ( 1.44 + 0.96 + 0.96 + 0.64 ) 4 = $1, so that the expect­ ed ( 0.96 )1 2 − 1 = −2.02% annual and two-year returns will both be 0%. * Note that we can estimate the future annual return using the average of past annual returns, or by compounding the average of past monthly re- turns. Because of estimation error these estimates will generally differ, but as long as the monthly returns are independent the results will converge with sufficient data. 10.4  The Historical Tradeoff Between Risk and Return In Chapter 3, we discussed the idea that investors are risk averse: The benefit they receive from an increase in income is smaller than the personal cost of an equivalent decrease in income. This idea suggests that investors would not choose to hold a portfolio that is more volatile unless they expected to earn a higher return. In this section, we quantify the histori- cal relationship between volatility and average returns. M10_BERK6318_06_GE_C10.indd 370 26/04/23 6:26 PM  10.6 Diversification in Stock Portfolios 375 EXAMPLE 10.5 Diversification and Gambling Problem Roulette wheels are typically marked with the numbers 1 through 36 plus 0 and 00. Each of these outcomes is equally likely every time the wheel is spun. If you place a bet on any one number and are correct, the payoff is 35:1; that is, if you bet $1, you will receive $36 if you win ($35 plus your original $1) and nothing if you lose. Suppose you place a $1 bet on your favorite number. What is the casino’s expected profit? What is the standard deviation of this profit for a single bet? Suppose 9 million similar bets are placed throughout the casino in a typical month. What is the standard deviation of the casino’s average revenues per dollar bet each month? Solution Because there are 38 numbers on the wheel, the odds of winning are 1/38. The casino loses $35 if you win, and makes $1 if you lose. Therefore, using Eq. 10.1, the casino’s expected profit is E [ Payoff ] = ( 1 38 ) × ( −$35 ) + ( 37 38 ) × ( $1 ) = $0.0526 That is, for each dollar bet, the casino earns 5.26 cents on average. For a single bet, we calculate the standard deviation of this profit using Eq. 10.2 as SD(Payoff ) = ( 1 38 ) × ( −35 − 0.0526 ) 2 + ( 37 38 ) × ( 1 − 0.0526 ) 2 = $5.76 This standard deviation is quite large relative to the magnitude of the profits. But if many such bets are placed, the risk will be diversified. Using Eq. 10.8, the standard deviation of the casino’s average revenues per dollar bet (i.e., the standard error of their payoff) is only $5.76 SD( Average Payoff ) = = $0.0019 9, 000, 000 In other words, by the same logic as Eq. 10.9, there is roughly 95% chance the casino’s profit per dollar bet will be in the interval $0.0526 ± ( 2 × 0.0019 ) = $0.0488 to $0.0564. Given $9 million in bets placed, the casino’s monthly profits will almost always be between $439,000 and $508,000, which is very little risk. The key assumption, of course, is that each bet is separate so that their outcomes are independent of each other. If the $9 million were placed in a single bet, the casino’s risk would be large—sustaining a loss of 35 × $9 million = $315 million if the bet wins. For this reason, casinos often impose limits on the amount of any individual bet. CONCEPT CHECK 1. What is the difference between common risk and independent risk? 2. Under what circumstances will risk be diversified in a large portfolio of insurance contracts? 10.6 Diversification in Stock Portfolios As the insurance example indicates, the risk of a portfolio of insurance contracts depends on whether the individual risks within it are common or independent. Independent risks are diversified in a large portfolio, whereas common risks are not. Let’s consider the impli- cation of this distinction for the risk of stock portfolios.12 12 Harry Markowitz was the first to formalize the role of diversification in forming an optimal stock mar- ket portfolio. See H. Markowitz, “Portfolio Selection,” Journal of Finance 7 (1952): 77–91. M10_BERK6318_06_GE_C10.indd 375 26/04/23 6:26 PM  376 Chapter 10 Capital Markets and the Pricing of Risk Firm-Specific Versus Systematic Risk Over any given time period, the risk of holding a stock is that the dividends plus the final stock price will be higher or lower than expected, which makes the realized return risky. What causes dividends or stock prices, and therefore returns, to be higher or lower than we expect? Usually, stock prices and dividends fluctuate due to two types of news: 1. Firm-specific news is good or bad news about the company itself. For example, a firm might announce that it has been successful in gaining market share within its industry. 2. Market-wide news is news about the economy as a whole and therefore affects all stocks. For instance, the Federal Reserve might announce that it will lower interest rates to boost the economy. Fluctuations of a stock’s return that are due to firm-specific news are independent risks. Like theft across homes, these risks are unrelated across stocks. This type of risk is also referred to as firm-specific, idiosyncratic, unique, or diversifiable risk. Fluctuations of a stock’s return that are due to market-wide news represent common risk. As with earthquakes, all stocks are affected simultaneously by the news. This type of risk is also called systematic, undiversifiable, or market risk. When we combine many stocks in a large portfolio, the firm-specific risks for each stock will average out and be diversified. Good news will affect some stocks, and bad news will affect others, but the amount of good or bad news overall will be relatively constant. The systematic risk, however, will affect all firms—and therefore the entire portfolio—and will not be diversified. Let’s consider an example. Suppose type S firms are affected only by the strength of the economy, which has a 50–50 chance of being either strong or weak. If the economy is strong, type S stocks will earn a return of 40%; if the economy is weak, their return will be −20%. Because these firms face systematic risk (the strength of the economy), holding a large portfolio of type S firms will not diversify the risk. When the economy is strong, the portfolio will have the same return of 40% as each type S firm; when the economy is weak, the portfolio will also have a return of −20%. Now consider type I firms, which are affected only by idiosyncratic, firm-specific risks. Their returns are equally likely to be 35% or −25%, based on factors specific to each firm’s local market. Because these risks are firm specific, if we hold a portfolio of the stocks of many type I firms, the risk is diversified. About half of the firms will have returns of 35%, and half will have returns of −25%, so that the return of the portfolio will be close to the average return of 0.5 ( 35% ) + 0.5 ( −25% ) = 5%. Figure 10.8 illustrates how volatility declines with the size of the portfolio for type S and I firms. Type S firms have only systematic risk. As with earthquake insurance, the volatility of the portfolio does not change as the number of firms increases. Type I firms have only idiosyncratic risk. As with theft insurance, the risk is diversified as the number of firms increases, and volatility declines. As is evident from Figure 10.8, with a large number of firms, the risk is essentially eliminated. Of course, actual firms are not like type S or I firms. Firms are affected by both sys- tematic, market-wide risks and firm-specific risks. Figure 10.8 also shows how the volatility changes with the size of a portfolio containing the stocks of typical firms. When firms carry both types of risk, only the firm-specific risk will be diversified when we combine many firms’ stocks into a portfolio. The volatility will therefore decline until only the sys- tematic risk, which affects all firms, remains. M10_BERK6318_06_GE_C10.indd 376 26/04/23 6:26 PM  10.6 Diversification in Stock Portfolios 377 FIGURE 10.8 Volatility of Portfolio (standard deviation) 35% Volatility of Portfolios of Type Type S 30% S and I Stocks Because type S firms have only 25% systematic risk, the volatility of the portfolio does not change. Type I 20% firms have only idiosyncratic risk, Typical Firms 15% which is diversified and eliminated as the number of firms in the portfolio 10% increases. Typical stocks carry a mix of both types of risk, so that the risk 5% of the portfolio declines as idiosyn- Type I cratic risk is diversified away, but 0% systematic risk still remains. 1 10 100 1000 Number of Stocks This example explains one of the puzzles shown in Figure 10.7. There we saw that the S&P 500 had much lower volatility than any of the individual stocks. Now we can see why: The individual stocks each contain firm-specific risk, which is eliminated when we com- bine them into a large portfolio. Thus, the portfolio as a whole can have lower volatility than each of the stocks within it. EXAMPLE 10.6 Portfolio Volatility Problem What is the volatility of the average return of 10 type S firms? What is the volatility of the aver- age return of 10 type I firms? Solution Type S firms have equally likely returns of 40% or −20%. Their expected return is 2 ( 40% ) + 2 ( −20% ) = 10%, so 1 1 SD ( RS ) = 2 ( 0.40 1 − 0.10 ) 2 + 21 ( −0.20 − 0.10 ) 2 = 30% Because all type S firms have high or low returns at the same time, the average return of 10 type S firms is also 40% or −20%. Thus, it has the same volatility of 30%, as shown in Figure 10.8. Type I firms have equally likely returns of 35% or −25%. Their expected return is 2 ( 35% ) + 2 ( −25% ) = 5%, so 1 1 SD ( R I ) = 2 ( 0.35 1 − 0.05 ) 2 + 21 ( −0.25 − 0.05 ) 2 = 30% Because the returns of type I firms are independent, using Eq. 10.8, the average return of 10 type I firms has volatility of 30% ÷ 10 = 9.5%, as shown in Figure 10.8. No Arbitrage and the Risk Premium Consider again type I firms, which are affected only by firm-specific risk. Because each in- dividual type I firm is risky, should investors expect to earn a risk premium when investing in type I firms? M10_BERK6318_06_GE_C10.indd 377 26/04/23 6:26 PM  378 Chapter 10 Capital Markets and the Pricing of Risk In a competitive market, the answer is no. To see why, suppose the expected return of type I firms exceeds the risk-free interest rate. Then, by holding a large portfolio of many type I firms, investors could diversify the firm-specific risk of these firms and earn a return above the risk-free interest rate without taking on any significant risk. The situation just described is very close to an arbitrage opportunity, which investors would find very attractive. They would borrow money at the risk-free interest rate and invest it in a large portfolio of type I firms, which offers a higher return with only a tiny amount of risk.13 As more investors take advantage of this situation and purchase shares of type I firms, the current share prices for type I firms would rise, lowering their expected return—recall that the current share price Pt is the denominator when computing the stock’s return as in Eq. 10.4. This trading would stop only after the return of type I firms equaled the risk-free interest rate. Competition between investors drives the return of type I firms down to the risk-free return. The preceding argument is essentially an application of the Law of One Price: Because a large portfolio of type I firms has no risk, it must earn the risk-free interest rate. This no- arbitrage argument suggests the following more general principle: The risk premium for diversifiable risk is zero, so investors are not compensated for holding firm-specific risk. We can apply this principle to all stocks and securities. It implies that the risk premium of a stock is not affected by its diversifiable, firm-specific risk. If the diversifiable risk of stocks were compensated with an additional risk premium, then investors could buy the stocks, earn the additional premium, and simultaneously diversify and eliminate the risk. By doing so, investors could earn an additional premium without taking on additional risk. This op- portunity to earn something for nothing would quickly be exploited and eliminated.14 Because investors ca