Chapter 11 Risk ad Return in Capital markets

Chapter Outline 11.1 A First Look at Risk and Return 11.2 Historical Risks and Returns of Stocks 11.3 The Historical Tradeoff Between Risk and Return 11.4 Common Versus Independent Risk 11.5 Diversification in Stock Portfolios

Learning Objectives Identify which types of securities have historically had the highest returns and which have been the most volatile Compute the average return and volatility of returns from a set of historical asset prices Understand the tradeoff between risk and return for large portfolios versus individual stocks Describe the difference between common and independent risk Explain how diversified portfolios remove independent risk, leaving common risk as the only risk requiring a risk premium 

11.1 A First Look at Risk and Return Consider how an investment would have grown if it were invested in each of the following from the end of 1929 until the beginning of 2012: Standard & Poor’s 500 (S&P 500) Small Stocks World Portfolio Corporate Bonds Treasury Bills

Figure 11. 1 Value of $100 Invested at the End of 1925 in U. S Figure 11.1 Value of $100 Invested at the End of 1925 in U.S. Large Stocks (S&P 500), Small Stocks, World Stocks, Corporate Bonds, and Treasury Bills

Table 11.1 Realized Returns, in Percent (%) for Small Stocks, the S&P 500, Corporate Bonds, and Treasury Bills, Year-End 1925–1935

11.2 Historical Risks and Returns of Stocks Computing Historical Returns Realized Returns Individual Investment Realized Returns The realized return from your investment in the stock from t to t+1 is: (Eq. 11.1)

Example 11.1a Realized Return Problem: Health Management Associates (HMA) paid a one-time special dividend of $10.00 on March 2, 2007. Suppose you bought HMA stock for $20.33 on February 15, 2007 and sold it immediately after the dividend was paid for $10.29. What was your realized return from holding the stock?

Example 11.1a Realized Return Using Eq. 11.1, the return from February 15, 2007 until March 2, 2007 is equal to This -0.2% can be broken down into the dividend yield and the capital gain yield:

11.2 Historical Risks and Returns of Stocks Computing Historical Returns Individual Investment Realized Returns For quarterly returns (or any four compounding periods that make up an entire year) the annual realized return, Rannual, is found by compounding: (Eq. 11.2)

Example 11.2 Compounding Realized Returns Problem: Suppose you purchased Microsoft stock (MSFT) on Nov 1, 2004 and held it for one year, selling on Oct 31, 2005. What was your annual realized return?

Example 11.2 Compounding Realized Returns The table below includes the realized return at each period.

Example 11.2 Compounding Realized Returns We then determine the one-year return by compounding.

11.2 Historical Risks and Returns of Stocks Average Annual Returns Average Annual Return of a Security (Eq. 11.3)

Figure 11. 2 The Distribution of Annual Returns for U. S Figure 11.2 The Distribution of Annual Returns for U.S. Large Company Stocks (S&P 500), Small Stocks, Corporate Bonds, and Treasury Bills, 1926–2012

Figure 11. 3 Average Annual Returns in the U. S Figure 11.3 Average Annual Returns in the U.S. for Small Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury Bills, 1926–2012

11.2 Historical Risks and Returns of Stocks The Variance and Volatility of Returns: Variance Standard Deviation (Eq. 11.4) (Eq. 11.5)

Example 11.3 Computing Historical Volatility Problem: Using the data below, what is the standard deviation of the S&P 500’s returns for the years 2005-2009?

Example 11.3 Computing Historical Volatility Evaluate: Our best estimate of the expected return for the S&P 500 is its average return, 3.1%, but it is risky, with a standard deviation of 24.1%.

Example 11.3a Computing Historical Volatility Problem: Using the data below, what is the standard deviation of small stocks’ returns for the years 2005-2009? Year 2005 2006 2007 2008 2009 Small Stocks’ Return 5.69% 16.17% -5.22% -36.72% 28.09%

Example 11.3a Computing Historical Volatility Solution: Plan: With the five returns, compute the average return using Eq. 11.3 because it is an input to the variance equation. Next, compute the variance using Eq. 11.4 and then take its square root to determine the standard deviation, as shown in Eq. 11.5. Year 2005 2006 2007 2008 2009 Small Stocks’ Return 5.69% 16.17% -5.22% -36.72% 28.09%

Example 11.3a Computing Historical Volatility Execute: Using Eq. 11.3, the average annual return for small stocks during this period is: We now have all of the necessary inputs for the variance calculation:  Applying Eq. 11.4, we have:

Example 11.3a Computing Historical Volatility Alternatively, we can break the calculation of this equation out as follows: Summing the squared differences in the last row, we get 0.2462. Finally, dividing by (5-1=4) gives us 0.2462/4 =0.0615 The standard deviation is therefore:

Example 11.3a Computing Historical Volatility Evaluate: Our best estimate of the expected return for small stocks is its average return, 1.71%, and they are risky, with a standard deviation of 24.80%.

Example 11.3b Computing Historical Volatility Problem: Using the data below, what is the standard deviation of the Large Stock returns for the years 2008-2012?

Example 11.3b Computing Historical Volatility Evaluate: Our best estimate of the expected return for Large Stocks is the average return, 4.5%, but it is risky, with a standard deviation of 24.8%.

Figure 11. 4 Volatility (Standard Deviation) of U. S Figure 11.4 Volatility (Standard Deviation) of U.S. Small Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury Bills, 1926–2012

11.2 Historical Risks and Returns of Stocks The Normal Distribution 95% Prediction Interval About two-thirds of all possible outcomes fall within one standard deviation above or below the average (Eq. 11.6)

Figure 11.5 Normal Distribution

Example 11.4 Confidence Intervals Problem: In Example 11.3 we found the average return for the S&P 500 from 2005-2009 to be 3.1% with a standard deviation of 24.1%. What is a 95% confidence interval for 2010’s return? We can use Eq. 11.6 to compute the confidence interval. Using Eq. 11.6, we have: Average ± (2  standard deviation) = 3.1% – (2  24.1%) to 3.1% + (2  24.1% ) = –45.1% to 51.3%.

Example 11.4 Confidence Intervals Evaluate: Even though the average return from 2005 to 2009 was 3.1%, the S&P 500 was volatile, so if we want to be 95% confident of 2010’s return, the best we can say is that it will lie between –45.1% and +51.3%.

Example 11.4a Confidence Intervals Problem: The average return for small stocks from 2005-2009 was 1.71% with a standard deviation of 24.8%. What is a 95% confidence interval for 2010’s return?

Example 11.4a Confidence Intervals Evaluate: Even though the average return from 2005-2009 was 1.71%, small stocks were volatile, so if we want to be 95% confident of 2010’s return, the best we can say is that it will lie between -47.89% and +50.77%.

Example 11.4b Confidence Intervals Problem: The average return for corporate bonds from 2005-2009 was 6.49% with a standard deviation of 7.04%. What is a 95% confidence interval for 2010’s return? Even though the average return from 2005-2009 was 6.49%, corporate bonds were volatile, so if we want to be 95% confident of 2010’s return, the best we can say is that it will lie between -7.59% and +20.57%.

Table 11.2 Summary of Tools for Working with Historical Returns

11.3 Historical Tradeoff between Risk and Return The Returns of Large Portfolios Investments with higher volatility, as measured by standard deviation, tend to have higher average returns

Figure 11.6 The Historical Tradeoff Between Risk and Return in Large Portfolios, 1926–2011

11.3 Historical Tradeoff between Risk and Return The Returns of Individual Stocks Larger stocks have lower volatility overall Even the largest stocks are typically more volatile than a portfolio of large stocks The standard deviation of an individual security doesn’t explain the size of its average return All individual stocks have lower returns and/or higher risk than the portfolios in Figure 11.6

11.4 Common Versus Independent Risk Theft Versus Earthquake Insurance Types of Risk Common Risk Independent Risk Diversification

Table 11.3 Summary of Types of Risk

Example 11.5 Diversification Problem: You are playing a very simple gambling game with your friend: a $1 bet based on a coin flip. That is, you each bet $1 and flip a coin: heads you win your friend’s $1, tails you lose and your friend takes your dollar. How is your risk different if you play this game 100 times in a row versus just betting $100 (instead of $1) on a single coin flip?

Example 11.5 Diversification Solution: Plan: The risk of losing one coin flip is independent of the risk of losing the next one: each time you have a 50% chance of losing, and one coin flip does not affect any other coin flip. We can compute the expected outcome of any flip as a weighted average by weighting your possible winnings (+$1) by 50% and your possible losses (-$1) by 50%. We can then compute the probability of losing all $100 under either scenario.

Example 11.5 Diversification Execute: If you play the game 100 times, you should lose about 50 times and win 50 times, so your expected outcome is 50  (+$1) + 50  (-$1) = $0. You should break-even. Even if you don’t win exactly half of the time, the probability that you would lose all 100 coin flips (and thus lose $100) is exceedingly small (in fact, it is 0.50100, which is far less than even 0.0001%). If it happens, you should take a very careful look at the coin!

Example 11.5 Diversification Execute (cont’d): If instead, you make a single $100 bet on the outcome of one coin flip, you have a 50% chance of winning $100 and a 50% chance of losing $100, so your expected outcome will be the same: break-even. However, there is a 50% chance you will lose $100, so your risk is far greater than it would be for 100 one dollar bets.

Example 11.5 Diversification Evaluate: In each case, you put $100 at risk, but by spreading-out that risk across 100 different bets, you have diversified much of your risk away compared to placing a single $100 bet.

Example 11.5a Diversification Problem: You are playing a very simple gambling game with your friend: a $1 bet based on a roll of two six-sided dice. That is, you each bet $1 and roll the dice: if the outcome is even you win your friend’s $1, if the outcome is odd you lose and your friend takes your dollar. How is your risk different if you play this game 100 times in a row versus just betting $100 (instead of $1) on a roll of the dice?

Example 11.5a Diversification Solution: Plan: The risk of losing one dice roll is independent of the risk of losing the next one: each time you have a 50% chance of losing, and one roll of the dice does not affect any other roll. We can compute the expected outcome of any roll as a weighted average by weighting your possible winnings (+$1) by 50% and your possible losses (-$1) by 50%. We can then compute the probability of losing all $100 under either scenario.

Example 11.5a Diversification Execute: If you play the game 100 times, you should lose about 50% of the time and win 50% of the time, so your expected outcome is 50  (+$1) + 50  (-$1) = $0. You should break-even. Even if you don’t win exactly half of the time, the probability that you would lose all 100 dice rolls (and thus lose $100) is exceedingly small (in fact, it is 0.50100, which is far less than even 0.0001%). If it happens, you should take a very careful look at the dice!

Example 11.5a Diversification Execute (cont’d): If instead, you make a single $100 bet on the outcome of one roll of the dice, you have a 50% chance of winning $100 and a 50% chance of losing $100, so your expected outcome will be the same: break-even. However, there is a 50% chance you will lose $100, so your risk is far greater than it would be for 100 one dollar bets.

Example 11.5a Diversification Evaluate: In each case, you put $100 at risk, but by spreading-out that risk across 100 different bets, you have diversified much of your risk away compared to placing a single $100 bet.

11.5 Diversification in Stock Portfolios Unsystematic Versus Systematic Risk Stock prices are impacted by two types of news: Company or Industry-Specific News Market-Wide News Unsystematic Risk Systematic Risk

Figure 11.7 Volatility of Portfolios of Type S and U Stocks

Figure 11.8 The Effect of Diversification on Portfolio Volatility

11.5 Diversification in Stock Portfolios Diversifiable Risk and the Risk Premium The risk premium for diversifiable risk is zero Investors are not compensated for holding unsystematic risk

Table 11.4 Systematic Risk Versus Unsystematic Risk

11.5 Diversification in Stock Portfolios The Importance of Systematic Risk The risk premium of a security is determined by its systematic risk and does not depend on its diversifiable risk

11.5 Diversification in Stock Portfolios The Importance of Systematic Risk There is no relationship between volatility and average returns for individual securities

Table 11.5 The Expected Return of Type S and Type U Firms, Assuming the Risk-Free Rate Is 5%

Chapter 11 Quiz Historically, which types of investments have had the highest average returns and which have been the most volatile from year to year? Is there a relation? For what purpose do we use the average and standard deviation of historical stock returns? What is the relation between risk and return for large portfolios? How are individual stocks different? What is the difference between common and independent risk? Does systematic or unsystematic risk require a risk premium? Why?