CHAPTER 8Risk and Rates of Return • This chapter is relatively important.
Investment returns The rate of return on an investment can be calculated as follows: (Amount received – Amount invested) Return =________________________ Amount invested For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is: ($1,100 - $1,000) / $1,000 = 10%.
What is investment risk? • Investment risk is related to the probability of earning a low or negative actual return. • The greater the chance of lower than expected or negative returns, the riskier the investment. • The greater the range of possible events that can occur, the greater the risk • The Chinese definition • Two types of investment risk • Stand-alone risk (when the return is analyzed in isolation.) • Portfolio risk (when the return is analyzed in a portfolio.)
PART I: Standard alone risk • The risk an investor would face if s/he held only one asset.
Firm X Firm Y Rate of Return (%) -70 0 15 100 Expected Rate of Return Probability distributions • A listing of all possible outcomes, and the probability of each occurrence. • Can be shown graphically.
Which firm is more likely to have a return closer to its expected value? • Firm X? • Firm Y?
Investor attitude towards risk • Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities. • Who wants to be a millionaire? • Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.
Selected Realized Returns, 1926 – 2001 Average Standard ReturnDeviation Small-company stocks 17.3% 33.2% Large-company stocks 12.7 20.2 L-T corporate bonds 6.1 8.6 L-T government bonds 5.7 9.4 U.S. Treasury bills 3.9 3.2 Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2002 Yearbook (Chicago: Ibbotson Associates, 2002), 28.
The Value of an Investment of $1 in 1926 6402 2587 64.1 48.9 16.6 Index 1 Year End Source: Ibbotson Associates
Rates of Return 1926-2000 Percentage Return Year Source: Ibbotson Associates
Suppose there are 5 possible outcomes over the investment horizon for the following securities:
Why is the T-bill return independent of the economy? T-bills will return the promised 5.5%, regardless of the economy. T-bills are risk-free in the default sense of the word.
How do the returns of HT and Coll. behave in relation to the market? • HT – Moves with the economy, and has a positive correlation. This is typical. • Coll. – Is countercyclical with the economy, and has a negative correlation. This is unusual.
Summary of expected returns Expected return HT 12.4% Market 10.5% USR 9.8% T-bill 5.5% Coll. 1.0% HT has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?
Prob. T - bill USR HT 0 5.5 9.8 12.4 Rate of Return (%) Comparing standard deviations
Comments on standard deviation as a measure of risk • Standard deviation (σi) measures “total”, or stand-alone, risk. • The larger σi is, the lower the probability that actual returns will be closer to expected returns. • Larger σi is associated with a wider probability distribution of returns.
Comparing risk and return ^ * Seem out of place.
(Not required) Coefficient of Variation (CV) A standardized measure of dispersion about the expected value, that shows the risk per unit of return.
PART II: Risk in a portfolio context • Portfolio risk is more important because in reality no one holds just one single asset. • The risk & return of an individual security should be analyzed in terms of how this asset contributes the risk and return of the whole portfolio being held.
In a portfolio… • The expected return is the weighted average of each individual stock’s expected return. • However, the portfolio standard deviation is generally lower than the weighted average of each individual stock’s standard deviation.
Portfolio construction:Risk and return • Assume a two-stock portfolio is created with $50,000 invested equally in both HT and Collections. That is, you invest 50% in each. What are the expected returns and standard deviation for the portfolio? • A portfolio’s expected return is a weighted average of the returns of the portfolio’s component assets. • Standard deviation is a little more tricky and requires that a new probability distribution for the portfolio returns be devised.
Comments on portfolio risk measures • σp = 3.4% is much lower than the σi of either stock (σHT = 20.0%; σColl. = 13.2%). This is not generally true. • σp = 3.4% is lower than the weighted average of HT and Coll.’s σ (16.6%). This is usually true (so long as the two stocks’ returns are not perfectly positively correlated). • Perfect correlation means the returns of two stocks will move exactly in same rhythm. • Portfolio provides average return of component stocks, but lower than average risk!
General comments about risk • Most stocks are positively (though not perfectly) correlated with the market. • σ 35% for an average stock. • Combining stocks in a portfolio generally lowers risk.
Stock W Stock M Portfolio WM 25 15 0 0 0 -10 -10 Returns distribution for two perfectly negatively correlated stocks 25 25 15 15 -10
Stock M’ Portfolio MM’ Stock M 25 25 25 15 15 15 0 0 0 -10 -10 -10 Returns distribution for two perfectly positively correlated stocks
Returns • A stock’s realized return is often different from its expected return. • Total return= expected return + unexpected return • Unexpected return=systematic portion + unsystematic portion • Total risk (stand-alone risk)= systematic portion of risk + unsystematic portion of risk
Systematic Risk • The systematic portion will be affected by factors such as changes in GDP, inflation, interest rates, etc. • This portion is not diversifiable because the factor will affect all stocks in the market. • Such risk factors affect a large number of stocks. Also called Market risk, non-diversifiable risk, beta risk.
Unsystematic Risk • This unsystematic portion is affected by factors such as labor strikes, part shortages, etc, that will only affect a specific firm, or a small number of firms. • Also called diversifiable risk, firm specific risk.
Diversification • Portfolio diversification is the investment in several different classes or sectors of stocks. • Diversification is not just holding a lot of stocks. • For example, if you hold 50 internet stocks, you are not well diversified.
Creating a portfolio:Beginning with one stock and adding randomly selected stocks to portfolio • σp decreases as stocks added, because stocks usually would not be perfectly correlated with the existing portfolio. • Expected return of the portfolio would remain relatively constant. • Diversification can substantially reduce the variability of returns with out an equivalent reduction in expected returns. • Eventually the diversification benefits of adding more stocks dissipates after about 10 stocks, and for large stock portfolios, σp tends to converge to 20%.
sp (%) Company-Specific Risk 35 Stand-Alone Risk, sp 20 0 Market Risk 10 20 30 40 2,000+ # Stocks in Portfolio Illustrating diversification effects of a stock portfolio
Breaking down sources of total risk (stand-alone risk) Stand-alone risk = Market risk + Firm-specific risk • Market risk (systematic risk, non-diversifiable risk) – portion of a security’s stand-alone risk that cannot be eliminated through diversification. • Firm-specific risk (unsystematic risk, diversifiable risk) – portion of a security’s stand-alone risk that can be eliminated through proper diversification. • If a portfolio is well diversified, unsystematic is very small.
Failure to diversify • If an investor chooses to hold just one stock in her/his portfolio (thus exposed to more risk than a diversified investor), should the investor be compensated for the firm-specific risk (earn higher returns)? • No. • An analogy, food diversification • Firm-specific risk is not important to a well-diversified investor, who only cares about the systematic risk.
So, • Rational, risk-averse investors are concerned with σp, which is based upon market risk. • No compensation should be earned for holding unnecessary, diversifiable risk. • Only systematic risk will be compensated.
How do we measure systematic risk?Beta • Measures a stock’s market risk, and shows a stock’s volatility relative to the market (i.e., the degree of co-movement with the market return.) • Indicates how risky a stock is if the stock is held in a well-diversified portfolio. • Measure of a firm’s market risk or the risk that remains after diversification • Beta will decide a stock’s required rate of return.
Calculating betas • Run a regression of past returns of a security against past market returns. (Market return is the weighted average of all stocks’ returns at a certain time.) • The slope of the regression line (called the security’s characteristic line) is defined as the beta coefficient for the security.
_ ri . 20 15 10 5 . Year rM ri 1 15% 18% 2 -5 -10 3 12 16 _ -5 0 5 10 15 20 rM Regression line: ri = -2.59 + 1.44 rM . -5 -10 ^ ^ Illustrating the calculation of beta (security’s characteristic line)
Security Character Line • What does the slope of SCL mean? • Beta • What variable is in the horizontal line? • Market return. • The steeper the line, the more sensitive the stock’s return relative to the market return, that is, the greater the beta.
Comments on beta • A stock with a Beta of 0 has no systematic risk • A stock with a Beta of 1 has systematic risk equal to the “typical” stock in the marketplace • A stock with a Beta greater than 1 has systematic risk greater than the “typical” stock in the marketplace • A stock with a Beta less than 1 has systematic risk less than the “typical” stock in the marketplace • The market index has a beta=1. • Most stocks have betas in the range of 0.5 to 1.5.
Can the beta of a security be negative? • Yes, if the correlation between Stock i and the market is negative. • If the correlation is negative, the regression line would slope downward, and the beta would be negative. • However, a negative beta is highly unlikely. • A stock that delivers higher return in recession is generally more valuable to investors, thus required rate of return is lower.
_ ri HT: b = 1.30 40 20 T-bills: b = 0 _ kM -20 0 20 40 Coll: b = -0.87 -20 Beta coefficients for HT, Coll, and T-Bills
Comparing expected returns and beta coefficients SecurityExpected Return Beta HT 12.4% 1.32 Market 10.5 1.00 USR 9.8 0.88 T-Bills 5.5 0.00 Coll. 1.0 -0.87 Riskier securities have higher returns, so the rank order is OK.
Until now… • We have argued that well-diversified investors only cares about a stock’s systematic risk (measured by beta). • The higher the systematic risk (non-diversifiable risk), the higher the rate of return investors will require to compensate them for bearing the risk. • This extra return above risk free rate that investors require for bearing the non-diversifiable risk of a stock is called risk premium.