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Chapter 10 Capital Markets and the Pricing of Risk

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Chapter 10 Capital Markets and the Pricing of Risk

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  1. Chapter 10 Capital Markets and the Pricing of Risk

  2. 10.1 Risk and Return • OVER the 5-year period between 2003 and 2007, investors in General Mills, Inc., earned an average return of 7% p.a. Within this period, the annual return ranged from -1.2% in 2003 to almost 20% in 2006. • Over the same period, investors in eBay, Inc., earned an average return of 25%. These investors, however, gained over 90% in 2003 and lost 30% in 2006. • Investors in 3-month U.S. T-bills earned an average return of 2.9% , with a low of 1.3% in 2004 and a high of 5% in 2007. • Clearly, these investments offered returns that were very different in terms of their average level and their variability. What accounts for these differences?

  3. 10.1 Risk and Return: Insights from 86 Years of Investor History • How would $100 have grown if it were placed in the following? • Standard & Poor’s 500: 90 U.S. stocks up to 1957 and 500 after that. Among the largest firms traded on U.S. Markets. • Small stocks: Securities traded on the NYSE with market capitalizations in the bottom 20%. • World Portfolio: International stocks from all the world’s major stock markets in North America, Europe, and Asia. • Corporate Bonds: Long-term, AAA-rated U.S. corporate bonds with maturities of approximately 20 years. • Treasury Bills: An investment in 3-month T-bills.

  4. Figure 10.1 Value of $100 Invested at the End of 1925 Source: Chicago Center for Research in Security Prices, Standard and Poor’s, MSCI, and Global Financial Data.

  5. 10.1 Risk and Return: Insights from 86 Years of Investor History • Small stocks had the highest long-term returns, while T-Bills had the lowest long-term returns. • Small stocks had the largest fluctuations in price, while T-Bills had the lowest • E.g. 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. • Even more importantly, your great-grandparents would have sustained losses at a time when they likely needed their savings the most––during the Great Depression. • Higher risk requires a higher return

  6. 10.1 Risk and Return: Insights from 86 Years of Investor History • Few people ever make an investment for 86 years. • More realistic investment horizons and different initial investment dates can greatly influence each investment's risk and return.

  7. Figure 10.2 Value of $100 Invested for Alternative Investment Horizons Source: Chicago Center for Research in Security Prices, Standard and Poor’s, MSCI, and Global Financial Data.

  8. Figure 10.2 Value of $100 Invested for Alternative Investment Horizons • Each panel shows the result of investing $100 at the end of the initial investment year, in each investment opportunity, for horizons of 1, 5, 10, or 20 years. • That is, each point on the plot is the result of an investment over the specified horizon, plotted as a function of the initial investment date. 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.

  9. 10.2 Common Measures of Risk and Return • Different securities have different prices, cash flows, etc. • To make them comparable, we express their performance in terms of their returns (i.e. the % increase in the value of an investment per dollar initially invested in the security). • Probability Distributions • When an investment is risky, there are different returns it may earn. • Each possible return has some likelihood of occurring. • This information is summarized with a probability distribution, which assigns a probability, PR , that each possible return, R , will occur.

  10. Table 10.1 Probability Distribution of Returns for BFI Assume BFI stock currently trades for $100 per share. 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.

  11. Figure 10.3 Probability Distribution of Returns for BFI

  12. Expected Return • Given the probability distribution of returns, we can compute the expected return. • Expected (Mean) Return • Calculated as a weighted average of the possible returns, where the weights correspond to the probabilities.

  13. Variance and Standard Deviation • Two common measures of the risk of a probability distribution • Variance • The expected squared deviation from the mean • Standard Deviation • The square root of the variance • Both are measures of the risk of a probability distribution

  14. Variance and Standard Deviation • For BFI, the variance and standard deviation are: • In finance, the standard deviation of a return is also referred to as its volatility. The standard deviation is easier to interpret because it is in the same units as the returns themselves.

  15. Textbook Example 10.1

  16. Textbook Example 10.1 (cont'd)

  17. Problem TXU stock is has the following probability distribution: What are its expected return and standard deviation? Alternative Example 10.1

  18. Alternative Example 10.1 (cont’d) • Solution • Expected Return • E[R] = (.25)(.08) + (.55)(.10) + (.20)(.12) • E[R] = 0.020 + 0.055 + 0.024 = 0.099 = 9.9% • Standard Deviation • SD(R) = [(.25)(.08 – .099)2 + (.55)(.10 – .099)2 + (.20)(.12 – .099)2]1/2 • SD(R) = [0.00009025 + 0.00000055 + 0.0000882]1/2 • SD(R) = 0.0001791/2 = .01338 = 1.338%

  19. Figure 10.4 Probability Distributions for BFI and AMC Returns

  20. 10.3 Historical Returns of Stocks and Bonds • Computing Historical Returns • Realized Return • The return that actually occurs over a particular time period.

  21. 10.3 Historical Returns of Stocks and Bonds • Computing Historical 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. • Let’s assume that all dividends are immediately reinvested and used to purchase additional shares of the same stock or security.

  22. 10.3 Historical Returns of Stocks and Bonds • Computing Historical Returns • If a stock pays dividends at the end of each quarter, with realized returns RQ1, . . . ,RQ4 each quarter, then its annual realized return, Rannual, is computed as:

  23. Textbook Example 10.2

  24. Table 10.2 Realized Return for the S&P 500, Microsoft, and Treasury Bills, 2001–2011

  25. 10.3 Historical Returns of Stocks and Bonds • Computing Historical Returns • By counting the number of times a realized return falls within a particular range, we can estimate the underlying probability distribution. • Empirical Distribution • When the probability distribution is plotted using historical data

  26. Figure 10.5 The Empirical Distribution of Annual Returns for U.S. Large & Small Stocks, Corporate Bonds, and T-Bills, 1926–2011

  27. Average Annual Return • Where Rt is the realized return of a security in year t, for the years 1 through T • Using the data from Table 10.2, the average annual return for the S&P 500 from 1999-2008 is:

  28. Table 10.3 Average Annual Returns for U.S. Small Stocks, Large Stocks, Corporate Bonds, and T-Bills, 1926–2011

  29. The Variance and Volatility of Returns • Variance Estimate Using Realized Returns • The estimate of the standard deviation is the square root of the variance.

  30. Textbook Example 10.3

  31. Excel calculations

  32. Textbook Example 10.3 (cont'd)

  33. Alternative Example 10.3 • Problem: • Using the data from Table 10.2, what are the variance and volatility of Microsoft’s returns from 2001 to 2011?

  34. Alternative Example 10.3 (cont’d) • Solution: • First, we need to calculate the average return for Microsoft’s over that time period, using equation 10.6:

  35. Alternative Example 10.3 (cont’d) Next, we calculate the variance using equation 10.7: The volatility or standard deviation is therefore

  36. Table 10.4 Volatility of U.S. Small Stocks, Large Stocks, Corporate Bonds, and Treasury Bills, 1926–2011

  37. Estimation Error: Using Past Returns to Predict the Future • To estimate the cost of capital for an investment, we need to determine the expected returnthat investors will require to compensate them for that investment’s risk. • If the distributionof past returns and the distribution of future returns are the same, we could look atthe return investors expected to earn in the past on the same or similar investments, andassume they will require the same return in the future. • However, there are two difficultieswith this approach. • First, we do not know what investors expected in the past; we can only observe the actual returnsthat were realized. • In 2008, for example, investors lost 37% investing in the S&P 500, which is surely notwhat they expected at the beginning of the year (or they would have invested inT-Bills instead!).

  38. Estimation Error: Using Past Returns to Predict the Future • If we believe that investors are neither overly optimistic nor pessimistic on average, thenover time, the average realized return should match investors’ expected return. • Armedwiththis assumption, we can use a security’s historical average return to infer (estimate) 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 estimationerror. • Given the volatility of stock returns, this estimation error can be large even with manyyears of data, as we will see next.

  39. Estimation Error: Using Past Returns to Predict the Future • We measure the estimation error of a statistical estimate by its standarderror. • This is the standard deviation of the estimated value of the mean ofthe actual distribution around its true value; that is, it is the standard deviation of the averagereturn. • The standard error provides an indication of how far the sample average mightdeviate from the expected return. • If the distribution of a stock’s return is identical eachyear, and each year’s return is independent of prior years’ returns, then we calculate thestandard error of the estimate of the expected return as follows:

  40. Estimation Error: Using Past Returns to Predict the Future • Standard Error of the Estimate of the Expected Return • 95% Confidence Interval • For the S&P 500 (1926–2004) • Or a range from 7.7% to 19.9%

  41. Normal Distribution

  42. Textbook Example 10.4

  43. Textbook Example 10.4 (cont'd)

  44. Limitations of Expected Return Estimates. • Individual stocks tend to be even morevolatile than large portfolios, and many have been in existence for only a few years, providinglittle data with which to estimate returns. • Because of the relatively large estimationerror in such cases, the average return investors earned in the past is not a reliable estimateof a security’s expected return. • Instead, we need to derive a different method to estimatethe expected return that relies on more reliable statistical estimates. • In the remainder, we will pursue the following alternative strategy: • First we will consider how tomeasure 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.

  45. 10.4 The Historical Tradeoff Between Risk and Return Excess Returns The difference between the average return for an investment and the average return for T-Bills Table 10.5 Volatility Versus Excess Return of U.S. Small Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury Bills, 1926–2011

  46. Figure 10.6 The Historical Tradeoff Between Risk and Return in Large Portfolios (Datafrom 1926–2011.) Source: CRSP, Morgan Stanley Capital International

  47. The Returns of Individual Stocks • Is there a positive relationship between volatility and average returns for individual stocks? • As shown on the next slide, there is no precise relationship between volatility and average return for individual stocks. • Larger stocks tend to have lower volatility than smaller stocks. • All stocks tend to have higher risk and lower returns than large portfolios.

  48. Figure 10.7 Historical Volatility and Return for 500 Individual Stocks, Ranked Annually by Size

  49. Volatility & Returns • We can make several important observations from these data. • First, there is a relationshipbetween size and risk: Larger stocks have lower volatility overall. • Second, even thelargest stocks are typically more volatile than a portfolio of large stocks, the S&P 500. • Finally, there is no clear relationship between volatility and return. While the smalleststocks have a slightly higher average return, many stocks have higher volatility andlower average returns than other stocks. • And all stocks seem to have higher risk and lowerreturns than we would have predicted from a simple extrapolation of our data from largeportfolios.