Monetary Economics Risk and Return, Part 2 Gerald P. Dwyer Fall 2015
Reading Malkiel, Part 2, Part 3 Malkiel, Part 3
Outline Returns and risk Overall market risk reduced over longer periods Individual stocks are riskier than the overall market Capital Asset Pricing Model Market risk and idiosyncratic risk Other theories of risk and expected return
Overall Market Dividends Reinvested December 31, 1984 to December 31, 2014 25 vwcrspd_84 20 15 10 5 0 1984 1989 1994 1999 2004 2009 2014
Overall Market Price since 1983 1000 CRSP Value weighted index with dividends 100 10 1 1983 1988 1993 1998 2003 2008 2013
How Reduce Risk? Reduce risk of portfolio What is risk of portfolio? Variability of return on portfolio
Variability of Return CRSP Value weighted return with dividends 0.15 dlvwcrspd_85 0.1 0.05 0.05 0 1985 1990 1995 2000 2005 2010 0.1 0.15 0.2 0.25
Summary Statistics on Return 1984 through 2014 Mean 0.000323 Variance 0.000124 Standard Deviation 0.011113
What Do The Numbers Indicate? The average is the arithmetic average of all the returns ( R R R... R ) / T 1 2 3 where R t is a day s return and T is the number of days A return of 0.00032 per day 0.032 percent per day T Mean 0.000323 Variance 0.000124 Standard Deviation 0.011113
What Do The Numbers Mean? The variance is the average squared deviations of returns around the mean ( R R) ( R R) ( R R)... ( R R) T 2 2 2 2 1 2 3 T where R is the average return The standard deviation is 0.011 per day, the square root of the variance The standard deviation is 1.1 percent per day Mean 0.000323 Variance 0.000124 Standard Deviation 0.011113
What Do The Numbers Mean? The average return is 0.032 percent per day Not a lot but it is daily The standard deviation is 1.1 percent per day Very common to find a standard deviation of about 1 percent per day Variability seems quite high Mean 0.000323 Variance 0.000124 Standard Deviation 0.011113
What Do The Numbers Mean? In what sense does a variability of 1 percent per day seem high? On a typical day, it is not surprising to see the overall market index go up or down 1 percent About 1/3 of the days, the market will go up or down more than 1 percent About 5 percent of the days, the market will go up or down more than 2 percent Mean 0.000323 Variance 0.000124 Standard Deviation 0.011113
This Risk Averages Out Over Time 0.15 dlvwretd 0.1 0.05 0.05 0 1984 1989 1994 1999 2004 2009 2014 0.1 0.15 0.2 0.25 0.3
Variability of Return CRSP Value weighted return with dividends 0.15 dlvwcrspd_85 0.1 0.05 0.05 0 1985 1990 1995 2000 2005 2010 0.1 0.15 0.2 0.25
Summary Statistics on Return Daily Mean 0.000323 Variance 0.000123 Standard Deviation 0.011094 Monthly Mean 0.008512 Variance 0.002056 Standard Deviation 0.045342
Summary Statistics on Return 1984 to 2014 Daily Mean 0.000480 Variance 0.000121 Standard Deviation 0.010896 Monthly Mean 0.009426 Variance 0.002921 Standard Deviation 0.054050
Summary Statistics on Returns Mean Daily 0.000480 Variance 0.000121 Standard Deviation 0.010986 Monthly Mean 0.009426 Variance 0.002921 Standard Deviation 0.054050 Monthly average return is 19.7 times bigger than daily average return About 0.9 percent per month
Summary Statistics on Returns Mean Daily 0.000480 Variance 0.000121 Standard Deviation 0.010986 Monthly Mean 0.009426 Variance 0.002921 Standard Deviation 0.054050 Monthly variance is 16.9 times bigger than daily variance About 0.2 squared percent per month
Summary Statistics on Returns Mean Daily 0.000480 Variance 0.000121 Standard Deviation 0.010986 Monthly Mean 0.009426 Variance 0.002921 Standard Deviation 0.054050 Monthly standard deviation is 4.1 times bigger than daily standard deviation About 5.4 percent per month About what would be expected based on formal statistics
Summary Statistics on Returns Mean Daily 0.000480 Variance 0.000121 Standard Deviation 0.010986 Monthly Mean 0.009426 Variance 0.002921 Standard Deviation 0.054050 Bottom lines Now have some idea of risk of stocks overall The volatility of returns increases over time but not as fast as the return
How Risky Are Individual Stocks? 0.4 Return on Amazon Stock 0.3 0.2 0.1 0 5/15/1997 5/15/2001 5/15/2005 5/15/2009 5/15/2013 0.1 0.2 0.3
Variability of Return 0.15 CRSP Value weighted return with dividends 0.1 0.05 0.05 0 1983 1988 1993 1998 2003 2008 2013 0.1 0.15 0.2
Volatility of Amazon s Return in Numbers Amazon Mean 0.002143 Variance 0.001782 Standard Deviation 0.042210 Overall Market (CRSP Index) Mean 0.000480 Variance 0.000121 Standard Deviation 0.010986 The volatility of an individual stock is more than the volatility of the overall market Not surprising
Overall Market Return and Amazon The overall market return has less volatility (a lower standard deviation) than Amazon Amazon is an atypical stock but this result holds in general
Portfolio is Less Risky The overall market return includes Amazon but it includes many other stocks The volatility of individual stocks is averaged out in the index Just like the volatility of the overall index is reduced by averaging over time from daily to monthly
Portfolio is Less Risky The overall market return includes Amazon but it includes many other stocks The volatility of individual stocks is averaged out in the index Just like the volatility of the overall index is reduced by averaging over time from daily to monthly Suppose found a stock that generally has the same return as the market The risk will be greater
Capital Asset Pricing Model The basic result in the Capital Asset Pricing Model Return on stock = riskfree rate + risk premium + unsystematic risk What is systematic risk? Systematic risk is risk that is related to overall stock market Unsystematic risk is risk that is not related to the overall stock market Idiosyncratic
Systematic Risk Return on stock = riskfree rate + risk premium + unsystematic risk R S is the return on the individual stock r is the riskfree rate R m is the return on the market portfolio R r is the risk premium (systematic risk) is the unsystematic risk S m R r R r S m S
Risk and Return reflects the extent to which the return on the individual stock moves with the market The risk premium reflects the comovement of the individual stock with the market Some stocks have betas greater than one Some stocks have betas less than one Some stocks may even have betas that are negative A diversified portfolio of typical stocks will have a beta of one Beta goes to one as the number of stocks increases
Beta on Amazon Beta for Amazon greater than one Common to estimate for last 60 months Lots of idiosyncratic movement 0.4 0.3 Return on Amazon Stock 0.15 0.1 CRSP Value weighted return with dividends 0.2 0.05 0.1 0 0 1983 0.05 1988 1993 1998 2003 2008 2013 5/15/1997 5/15/2001 5/15/2005 5/15/2009 5/15/2013 0.1 0.1 0.2 0.3 0.15 0.2
Systematic Versus Unsystematic Risk Systematic risk cannot be diversified away The covariation with the market is there For the overall market, beta is one Unsystematic (idiosyncratic) risk can be diversified away If you hold enough stocks, events affecting one firm but not others is unimportant for your portfolio Has virtually no effect on your return
Estimate Beta for Amazon Get excess returns I approximate this by daily return on 3 month Treasury bills
Daily Return on 3 month Treasury Bills
Return Daily Amazon
Excess Return Daily Amazon
Return CRSP
Excess Return CRSP
Excess Returns on Amazon and Overall Market
Estimate of Beta Estimated Beta is 1.47 over whole period More common for estimating future beta to estimate over last 60 months with monthly data Assume that relationship probably changes over time Just use most recent data
Estimated Regression Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 0.0016886 0.0005873 2.875 0.00405 ** er_crsp 1.4748463 0.0458128 32.193 < 2e 16 *** Signif. codes: *** 0.001 ** 0.01 * 0.05. 0.1 Residual standard error: 0.03783 on 4150 degrees of freedom (357 observations deleted due to missing values) Multiple R squared: 0.1998, Adjusted R squared: 0.1996 F statistic: 1036 on 1 and 4150 DF, p value: < 2.2e 16
Market Model CAPM R r R r Estimate market model instead S m S R c R S m m S S Estimated coefficient is 1.46 instead of CAPM s 1.47
Overall Market Return and Amazon The overall market return has less volatility (a lower standard deviation) than Amazon Amazon is an atypical stock but this result will hold in general
Portfolio is Less Risky The overall market return includes Amazon but it includes many other stocks The volatility of individual stocks is averaged out in the index Just like the volatility of the overall index is reduced by averaging over time from daily to monthly
Portfolio is Less Risky The overall market return includes Amazon but it includes many other stocks The volatility of individual stocks is averaged out in the index Just like the volatility of the overall index is reduced by averaging over time from daily to monthly Suppose found a stock that generally has the same return as the market The risk will be greater
Capital Asset Pricing Model The basic result in the Capital Asset Pricing Model Return on stock = riskfree rate + risk premium + unsystematic risk What is systematic risk? Systematic risk is risk that is related to overall stock market Unsystematic risk is risk that is not related to the overall stock market Idiosyncratic
Systematic Risk Return on stock = riskfree rate + risk premium + unsystematic risk R S is the return on the individual stock r is the riskfree rate R m is the return on the market portfolio R r is the risk premium (systematic risk) is the unsystematic risk S m R r R r S m S
Risk and Return reflects the extent to which the return on the individual stock moves with the market The risk premium reflects the comovement of the individual stock with the market Some stocks have betas greater than one Some stocks have betas less than one Some stocks may even have betas that are negative A diversified portfolio of typical stocks will have a beta of one Beta goes to one as the number of stocks increases if they are typical
Risk and Return reflects the extent to which the return on the individual stock moves with the market The risk premium reflects the comovement of the individual stock with the market Some stocks have betas greater than one Some stocks have betas less than one Some stocks may even have betas that are negative A diversified portfolio will have a beta of one
Beta on Amazon Beta for Amazon greater than one Common to estimate for last 60 months Lots of idiosyncratic movement 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 Return on Amazon Stock May 16, 1997 to December 31, 2012 0.15 0.1 0.05 0 0.05 0.1 0.15 Return on CRSP Index May 15, 1997 to December 31, 2012
Estimate Beta for Amazon Get excess returns I approximate this by daily return on 3 month Treasury bills
Daily Return on 3 month Treasury Bills
Return Daily Amazon
Excess Return Daily Amazon
Return CRSP
Excess Return CRSP
Excess Returns on Amazon and Overall Market
Estimate of Beta Estimated Beta is 1.47 over whole period More common to do over last 60 months Assume that relationship probably changes over time
Estimated Regression Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 0.0016886 0.0005873 2.875 0.00405 ** er_crsp 1.4748463 0.0458128 32.193 < 2e 16 *** Signif. codes: *** 0.001 ** 0.01 * 0.05. 0.1 Residual standard error: 0.03783 on 4150 degrees of freedom (357 observations deleted due to missing values) Multiple R squared: 0.1998, Adjusted R squared: 0.1996 F statistic: 1036 on 1 and 4150 DF, p value: < 2.2e 16
Market Model CAPM R r R r Estimate market model instead S m S R c R S m m S S Estimated coefficient is 1.46 instead of CAPM s 1.47
Systematic Versus Unsystematic Risk Systematic risk cannot be diversified away The covariation with the market is there For the overall market, beta is one Unsystematic (idiosyncratic) risk can be diversified away If you hold enough stocks, what happens to an individual firm is unimportant if it is not related to what happens to other firms Has virtually no effect on your portfolio return
Relationship Across Stocks between Beta and Return Riskfree rate of 3 percent, market return of 6 percent 10 9 8 7 Expected Stock Return and Beta Expected return 6 5 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Beta
Relationship Between Unsystematic Risk and Return Same data as previous slide Straight line at 6 percent with a beta of one is the underlying relationship Expected return is unaffected by unsystematic risk 10 8 6 4 2 Expected Stock Return and Unsystematic Risk 0 0 0.2 0.4 0.6 0.8 1
Beta as a Theory of Risk The Capital Asset Pricing Model has deficiencies It does not good a good job of predicting actual returns using the overall stock market May work better with an overall measure of wealth Still, it is a useful summary of the comovement of a stock return with the market
Other Analyses of Risk and Return Arbitrage Pricing Theory (APT) Systematic risk may well reflect other things than just the stock market For example Interest rates Inflation Business cycles (GDP) Fama French Three Factor Pricing Model Theories can consider skewness and other additional characteristics of stocks
Other Analyses of Risk and Return Fama French Three Factor Pricing Model Empirical generalization, not a theory Can think of as a particular set of variables in the APT Factors Beta (from the CAPM) Size (total equity market capitalization) Smaller firms are riskier and have higher expected returns Value (ratio of book to market value) Low market value relative to book value is associated with a higher expected return
Summary Virtually all of Finance is based on the proposition that people are risk averse Investors must be paid to bear risk Higher risk, higher expected return The volatility of an individual stock reflects Systematic risk Unsystematic risk Unsystematic risk can be diversified away
Summary CAPM implies that systematic risk is related to beta of individual stock Expected returns of stock are linearly related to their betas CAPM does not do a particularly good job of predicting average returns across stocks Alternative theories APT Fama French three factor model