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Risk and Returns Return Basics – Holding-Period Returns – Return Statistics Risk Statistics Return and Risk for Individual Securities Return and Risk for Portfolios – Efficient Set for Two Assets – Efficient Set for Many Securities Riskless Borrowing and Lending Capital Asset Pricing Model Chapters 9 & 10 – MBA504 Returns Dollar Return = Dividend + Change in Market Value dollar return percentage return beginning market val ue dividend change in market val ue beginning market val ue dividend yield capital gains yield Chapters 9 & 10 – MBA504 Example Suppose you bought 100 shares of Wal-Mart (WMT) one year ago today at $25. Over the last year, you received $20 in dividends (= 20 cents per share × 100 shares). At the end of the year, the stock sells for $30. How did you do? Chapters 9 & 10 – MBA504 3 Holding-Period Returns • The holding period return is the return that an investor would get when holding an investment over a period of n years, when the return during year i is given as ri: holding period return (1 r1 ) (1 r2 ) (1 rn ) 1 Chapters 9 & 10 – MBA504 4 Example • Suppose your investment provides the following returns over a four-year period: Year Return 1 10% 2 -5% 3 20% 4 15% Chapters 9 & 10 – MBA504 5 Holding Period Returns • Year-by-year historical rates of return starting in 1926 for the following five important types of financial instruments in the United States: – – – – – Large-Company Common Stocks Small-company Common Stocks Long-Term Corporate Bonds Long-Term U.S. Government Bonds U.S. Treasury Bills Chapters 9 & 10 – MBA504 6 Future Value of an Investment of $1 in 1925 $1,775.34 1000 $59.70 $17.48 10 Common Stocks Long T-Bonds T-Bills 0.1 1930 1940 1950 1960 1970 1980 1990 2000 Source: © Stocks, Bonds, Bills, and Inflation 2003 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved. Chapters 9 & 10 – MBA504 7 Return Statistics • The history of capital market returns can be summarized by describing the – average return ( R1 RT ) R T – the standard deviation of those returns ( R1 R) 2 ( R2 R) 2 ( RT R) 2 SD VAR T 1 – the frequency distribution of the returns. Chapters 9 & 10 – MBA504 8 Historical Returns, 1926-2002 Average Annual Return Series Standard Deviation Large Company Stocks 12.2% 20.5% Small Company Stocks 16.9 33.2 Long-Term Corporate Bonds 6.2 8.7 Long-Term Government Bonds 5.8 9.4 U.S. Treasury Bills 3.8 3.2 Inflation 3.1 4.4 – 90% Distribution 0% + 90% Source: © Stocks, Bonds, Bills, and Inflation 2003 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved. Chapters 9 & 10 – MBA504 9 Average Stock Returns and Risk-Free Returns • Risk Premium is the additional return (in excess of riskfree rate) resulting from bearing risk. • One of the most significant observations of stock market data is this long-run excess of stock return over the riskfree return. – The average excess return from large company common stocks for the period 1926 through 1999 was 8.4% = 12.2% – 3.8% – The average excess return from small company common stocks for the period 1926 through 1999 was 13.2% = 16.9% – 3.8% – The average excess return from long-term corporate bonds for the period 1926 through 1999 was 2.4% = 6.2% – 3.8% Chapters 9 & 10 – MBA504 10 Risk Premia • Suppose that The Wall Street Journal announced that the current rate for 1-year Treasury bills is 5%. • What is the expected return on the market of small-company stocks? • Recall that the average excess return from small company common stocks for the period 1926 through 1999 was 13.2% • Given a risk-free rate of 5%, we have an expected return on the market of small-company stocks of 18.2% = 13.2% + 5% Chapters 9 & 10 – MBA504 11 The Risk-Return Tradeoff 18% Small-Company Stocks Annual Return Average 16% 14% Large-Company Stocks 12% 10% 8% 6% T-Bonds 4% T-Bills 2% 0% 5% 10% 15% 20% 25% 30% 35% Annual Return Standard Deviation Chapters 9 & 10 – MBA504 12 Risk Premiums • Rate of return on T-bills is essentially risk-free. • Investing in stocks is risky, but there are compensations. • The difference between the return on T-bills and stocks is the risk premium for investing in stocks. Chapters 9 & 10 – MBA504 13 You can either sleep well or eat well. -- An old saying on Wall Street Chapters 9 & 10 – MBA504 14 Risk Statistics • There is no universally agreed-upon definition of risk. • The measures of risk that we discuss are variance and standard deviation. – The standard deviation is the standard statistical measure of the spread of a sample – In terms of normal distribution Chapters 9 & 10 – MBA504 15 Normal Distribution • A large enough sample drawn from a normal distribution looks like a bell-shaped curve. Probability The probability that a yearly return will fall within 20.1 percent of the mean of 13.3 percent will be approximately 2/3. – 3s – 49.3% – 2s – 28.8% – 1s – 8.3% 0 12.2% + 1s 32.7% + 2s 53.2% 68.26% + 3s 73.7% Return on large company common stocks 95.44% 99.74% Chapters 9 & 10 – MBA504 16 Risk and Return: Individual Securities • The characteristics of individual securities that are of interest are the: – Expected Return – Variance and Standard Deviation – Covariance and Correlation Chapters 9 & 10 – MBA504 17 Expected Return and Variance Rate of Return Scenario Probability Stock fund Bond fund Recession 33.3% -7% 17% Normal 33.3% 12% 7% Boom 33.3% 28% -3% Consider the following two risky asset world. There is a 1/3 chance of each state of the economy and the only assets are a stock fund and a bond fund. Chapters 9 & 10 – MBA504 18 Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Chapters 9 & 10 – MBA504 Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% 19 Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% Consider a portfolio that is 50% invested in bonds and 50% invested in stocks. Chapters 9 & 10 – MBA504 20 Risk and Return for a Portfolio Scenario Recession Normal Boom Expected return Variance Standard Deviation Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.160% 0.003% 0.123% 9.0% 0.0010 3.08% The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio: rP wB rB wS rS Chapters 9 & 10 – MBA504 21 % in stocks Risk Return 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50.00% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 8.2% 7.0% 5.9% 4.8% 3.7% 2.6% 1.4% 0.4% 0.9% 2.0% 3.08% 4.2% 5.3% 6.4% 7.6% 8.7% 9.8% 10.9% 12.1% 13.2% 14.3% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.00% 9.2% 9.4% 9.6% 9.8% 10.0% 10.2% 10.4% 10.6% 10.8% 11.0% Portfolio Return The Efficient Set for Two Assets Portfolo Risk and Return Combinations 12.0% 11.0% 100% stocks 10.0% 9.0% 8.0% 7.0% 100% bonds 6.0% 5.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) We can consider other portfolio weights besides 50% in stocks and 50% in bonds … Chapters 9 & 10 – MBA504 22 Risk Return 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 8.2% 7.0% 5.9% 4.8% 3.7% 2.6% 1.4% 0.4% 0.9% 2.0% 3.1% 4.2% 5.3% 6.4% 7.6% 8.7% 9.8% 10.9% 12.1% 13.2% 14.3% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.0% 9.2% 9.4% 9.6% 9.8% 10.0% 10.2% 10.4% 10.6% 10.8% 11.0% Portfolio Return % in stocks Portfolo Risk and Return Combinations 12.0% 100% stocks 11.0% 10.0% 9.0% 8.0% 7.0% 6.0% 100% bonds 5.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less. These compromise the efficient frontier. Chapters 9 & 10 – MBA504 23 Portfolio Risk as a Function of the Number of Stocks in the Portfolio s In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not. Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk Portfolio risk Nondiversifiable risk; Systematic Risk; Market Risk n Thus diversification can eliminate some, but not all of the risk of individual securities. Chapters 9 & 10 – MBA504 24 return Efficient Set for Many Securities Individual Assets sP Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios. Chapters 9 & 10 – MBA504 25 return minimum variance portfolio Individual Assets sP The section of the opportunity set above the minimum variance portfolio is the efficient frontier. Chapters 9 & 10 – MBA504 26 return Optimal Risky Portfolio with a RiskFree Asset 100% stocks rf 100% bonds s In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills Chapters 9 & 10 – MBA504 27 return Riskless Borrowing and Lending 100% stocks Balanced fund rf 100% bonds s Now investors can allocate their money across the T-bills and a balanced mutual fund Investor risk aversion is revealed in their choice of where to stay along the capital allocation line. Chapters 9 & 10 – MBA504 28 return Market Equilibrium 100% stocks Optimal Risky Portfolio rf 100% bonds s All investors have the same CML because they all have the same optimal risky portfolio given the risk-free rate. Chapters 9 & 10 – MBA504 29 Definition of Risk When Investors Hold the Market Portfolio • Researchers have shown that the best measure of the risk of a security in a large portfolio is the beta (b)of the security. • Beta measures the responsiveness of a security to movements in the market portfolio. Cov( R R ) bi i, M s ( RM ) 2 Chapters 9 & 10 – MBA504 30 Security Returns Estimating b with regression Slope = bi Return on market % Ri = a i + biRm + ei Chapters 9 & 10 – MBA504 31 Relationship between Risk and Expected Return (CAPM) • Expected Return on the Market: R M RF Market Risk Premium • Expected return on an individual security: Ri RF βi ( R M RF ) Market Risk Premium This applies to individual securities held within welldiversified portfolios. Chapters 9 & 10 – MBA504 32 • This formula is called the Capital Asset Pricing Model (CAPM) Expected return on a security RiskBeta of the = + × free rate security Market risk premium • Assume bi = 0, then the expected return is RF. • Assume bi = 1, then Ri R M Chapters 9 & 10 – MBA504 33 Expected return Relationship Between Risk & Expected Return 13.5% 3% βi 1.5 RF 3% 1.5 b R M 10% R i 3% 1.5 (10% 3%) 13.5% Chapters 9 & 10 – MBA504 34 Empirical Approaches to Asset Pricing • Empirical methods are based less on theory and more on looking for some regularities in the historical record. – E(R) = Rf + k1(beta) + k2(B/M) + k3(size) – k1, k2>0, and k3<0 Chapters 9 & 10 – MBA504 35 Style Portfolios • Related to empirical methods is the practice of classifying portfolios by style e.g. – Value portfolio – Growth portfolio Chapters 9 & 10 – MBA504 36