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Chapter 8 Risk and Return © 2000 South-Western College Publishing PORTFOLIO THEORY The Relationship Between Risk and Return Inherent in Investing in Securities Especially Stocks WHY STUDY RISK AND RETURN? Over a Long Period of Time, Stocks returned about 9%, Debt returned about 3% But Returns on stocks could be very low (or high) over shorter periods. Look for a way to capture the high average returns of equity while avoiding as much risk as possible. In General, Investments With High Returns Also Have High risk. We need a way to measure risk and relate it to return so we can choose among investment opportunities. TM 8-1 THE RETURN ON ONE-YEAR INVESTMENTS DEBT Interest paid divided by the loan principal k int erest paid loan amount EQUITY What the investor receives divided by what was invested k D1 ( P1 P0 ) P0 TM 8-2 Slide 1 of 2 The Expected Return Based on general knowledge about the stock The Required Return Based on perceived risk Substantial investment in a stock will take place only if the generally expected return exceeds most people's required return. TM 8-2 Slide 2 of 2 RISK - A PRELIMINARY DEFINITION The chance (probability) that the return on an investment will turn out to be less than expected when the investment is made Note: Includes earning slightly less as well as losing money TM 8-3 Slide 1 of 2 RISK AVERSION People have negative feelings about bearing risk in investments. They prefer lower risk if the expected return is about the same. E.g., most prefer an 8% bank account to a stock with an expected return of 8%. However there's a trade-off. If the choice is between the 8% bank account and a 10% stock, many will choose the stock. Risk aversion doesn't mean risk is to be avoided at all cost. It is a negative that can be compensated with more expected return. TM 8-3 Slide 2 of 2 RANDOM VARIABLES A random variable is the outcome of a chance process. Discrete random variables take on only specific values. Continuous random variables take on values over a range. Example of a Discrete Random Variable Toss a coin four times, call the number of heads X. X is a random variable which can take on any of five discrete values. The probability distribution of X is: X P(X) 0 .0625 1 .2500 2 .3750 3 .2500 4 .0625 1.0000 TM 8-4 Slide 1 of 3 OR P(X) .3750 .2500 .0625 0 1 2 3 4 Figure 8-1 Discrete Probability Distribution TM 8-4 Slide 2 of 3 The most likely value is the mean or expected value The weighted average of all possible outcomes where each is weighted by its probability. X 0 1 2 3 4 P(X) .0625 .2500 .3750 .2500 .0625 1.0000 X * P(X) 0.00 0.25 0.75 0.75 0.25 X = 2.00 TM 8-4 Slide 3 of 3 CONTINUOUS RANDOM VARIABLES Can take on any numerical value over some range. E.g. people's heights P(H) 4’10” 5’8” 6’6” Figure 8-2 Probability Distribution for a Continuous Random Variable TM 8-5 Slide 1 of 2 CONTINUOUS RANDOM VARIABLES (continued) The probability of an actual outcome is expressed within a range rather than as an exact amount. For example, the probability of being exactly 5'2" isn't meaningful, but being between 5' 1 7/8" and 5' 2 1/8" is. Probability is represented by the area under the curve. When the distribution is symmetrical and has only one peak, the mean is found under that peak. TM 8-5 Slide 2 of 2 PORTFOLIO THEORY The Return on an Investment in Stock is Represented as a Continuous Random Variable D1 ( P1 P0 ) P0 D1 and P1 are subject to a large number of uncertain factors. Therefore, k has the characteristics of a random variable. k P(kX) Variance ( 2 ) kX Expected return 8.0 8.5 kX Return Figure 8-3 The Probability Distribution of the Return on an Investment in Stock X TM 8-6 Slide 1 of 2 PORTFOLIO THEORY Portfolio theory assumes the investment community's knowledge about a stock is reflected in the probability distribution of returns. The mean or expected value is the statistical representation of the average investor's expected return. The variance ( ) shows how likely a return is to be some distance away from the expected value. 2 The diagram shows the variance conceptually as the width of the distribution. TM 8-6 Slide 2of 2 VARIANCE Think of variance as variability in successive annual returns The bigger the variance, the more different successive returns are likely to be P(kX) Small Variance (low risk) Large Variance (high risk) kX kX Expected Return Return Figure 8-4 Probability Distributions with Large and Small Variances TM 8-7 Slide 1of 2 VARIANCE (continued) The large variance distribution has more area under the curve further away from the mean When the variance is large: More returns are likely to fall far away from the mean Returns will be more different or more variable from year to year TM 8-7 Slide 2of 2 RISK: REDEFINED AS VARIANCE In portfolio theory, risk is defined as variability. A risky stock's return is likely to be significantly different from one year to the next A risky stock has a large probability of producing a return that's substantially away from the mean of its distribution Hence a large probability of a big loss (or a big gain) But this is exactly the idea of variance so: In Portfolio Theory, a stock investment's risk is defined as the variance of the probability distribution of its return TM 8-8 Slide 1 of 2 Seems inconsistent with earlier definition that risk is the probability that return is less than expected - the left side of the distribution This definition includes better outcomes than expected Done for mathematical convenience understanding that most distributions are symmetrical Hence there are two definitions of risk that are both correct: In practical terms, risk is the probability that return will be less than expected. In financial theory, risk is the variance of the probability distribution of returns. TM 8-8 Slide 2 of 2 AN ALTERNATE VIEW Risk as Variability of Return Over Time Return kX A - High risk kX B - Low Risk Time Figure 8-5 Investment Risk Viewed as Variability of Return Over Time TM 8-9 RISK AVERSION A More Precise Definition People prefer investments with less risk to those with more risk if the expected returns are equal Otherwise, individual preferences vary P(k) P(k) Neither preferred Preferred with certainty k k k kA (a) kB (b) Figure 8-6 Risk Aversion TM 8-10 DECOMPOSING RISK INTO COMPONENTS SYSTEMATIC (MARKET) RISK AND UNSYSTEMATIC (BUSINESS-SPECIFIC) RISK Returns on stock investments move up and down in response to stimuli which may affect all stocks or only specific businesses News of politics and economics tends to affect all stocks A labor dispute affects only firms in one industry TM 8-11 Slide 1 of 2 Market Risk Movement in return in response to stimuli which affect all stocks is known as systematic risk or market risk. In general, most stock's returns move together. Hence, market risk is the degree to which a stock's return moves with the (average) return on the market. Business-specific Risk Whatever movement in a stock's return is left over after market risk is removed is known as unsystematic risk or business-specific risk. TM 8-11 Slide 2 of 2 PORTFOLIOS Investors generally hold the stocks of several companies. An investor's total stock holding is a portfolio. Risk and Return for a Portfolio The return on a portfolio is the average return of the stocks in it, weighted by the dollars invested in each stock. The portfolio's return has a probability distribution with a mean and variance. These are the portfolio's expected return and risk. The expected return is the weighted average of the stock's expected returns. The variance depends on the stock's variances, but in a complex way. TM 8-12 Slide 1 of 2 The Goal of the Investor/Portfolio Owner To capture the high average returns of equities while avoiding as much risk as possible Done by constructing diversified portfolios of securities with minimum variation in return Portfolio Theory Assumes: Investors care only about portfolio performance, not about individual stock performance. A stock's risk can appear different in and out of a portfolio. TM 8-12 Slide 2 of 2 DIVERSIFICATION HOW RISK IS AFFECTED WHEN STOCKS ARE ADDED TO A PORTFOLIO Business-specific (Unsystematic) Risk Stimuli are random events that push the returns on individual stocks up or down. Over a large number of different (diverse) stocks the pluses and minuses wash out and Business-specific Risk is Diversified Away TM 8-13 DIVERSIFICATION Systematic (Market) Risk A more difficult concept since returns on most stocks tend to move together The Portfolio Assume portfolio mirrors the makeup of the stock market so its risk is the market's risk Consider the impact on the portfolio's risk of adding a little of either of two new stocks TM 8-14 Slide 1 of 3 Return, k A C C A P kp k B B Time Figure 8-7 Risk In and Out of a Portfolio A adds risk to the portfolio (perfectly positively correlated with market) B reduces the portfolio's risk (perfectly negatively correlated with market) BUT outside the portfolio A and B are equally risky TM 8-14 Slide 2 of 3 Portfolio risk depends on the timing of the variation of return There are very few stocks like B (gold mines) Variation in portfolio return can be reduced, but not eliminated, with stocks like C (not perfectly positively correlated with market) They bring a little of the "personality" of B along TM 8-14 Slide 3 of 3 The Importance of Market Risk In Portfolio Theory The risk attributes of stocks change when we assume investors focus on portfolios Only market risk counts because business-specific risk is diversified away This is a dangerous result Not applicable to small investors with limited portfolios in which a business-specific event can cause a major loss Nevertheless The central risk concept in portfolio theory is market risk The variation in a stock's return that accompanies variation in the market's return Business-specific risk is ignored TM 8-15 MEASURING MARKET RISK THE CONCEPT OF BETA A stock's beta coefficient captures the historical variation in a stock's return that accompanies variation in the market's return. Developing Beta Plot historical values for kX against kM Fit a regression line to the data Known as the characteristic line for the stock TM 8-16 Slide 1 of 2 kX . . Characteristic . . . . Line . . . . . . . . . . . . . . . . . . . . . . . . . Values of . (kM, kX) . . kM { kX{ . kM Slope k X k M b X Beta Figure 8-8 The Determination of Beta Represents the average relationship between the stock's return and the market's return. The slope indicates on the average how much of a change in kX comes with a change in kM. This is exactly the notion of market risk. TM 8-16 Slide 2 of 2 UNDERSTANDING BETA Example 8-1 - Projecting Returns with Beta Conroy Corp. has a beta of 1.8 and returns 14%. The stock market is reacting negatively to a new Middle East crisis which threatens world oil supplies and limited war. Experts estimate the return on an average stock will drop from 12% to 8%. Estimate Conroy's new return. Solution: b Conroy 1.8 k Conroy k M k Conroy 4% k Conroy 7.2% k Conroy 14% 7.2% 6.8% TM 8-17 Slide 1of 2 Example 8-2 - Business-specific effects Would the estimate of return be valid if Conroy was a defense contractor? Solution: Probably not because of a positive business-specific effect from the threat of war Beta Over Time Use of a stock's beta implicitly assumes it will remain what it has been in the past Example 8-3 Consider Conroy as a defense contractor in the early 1990's with the Cold War ending and military spending declining. Would a projection using beta have been valid? Solution: The changing conditions make it unlikely that the historical beta would be good in the future. TM 8-17 Slide 2of 2 Beta Measures Volatility With Respect To Market Changes Beta = 1.0 The stock's return moves on average as much as the market's return. Beta > 1.0 Return moves more than the market's Beta < 1.0 Return moves with the market but less. Beta < 0 Return moves against the market (Stock B - gold mines) TM 8-18 Slide 1of 2 Beta in Practice Widely used to discuss risk However, Many people probably forget the definition as market risk only. Beta for a Portfolio Weighted average of betas of individual stocks Weights are dollars invested in each stock TM 8-18 Slide 2of 2 THE CAPITAL ASSET PRICING MODEL (CAPM) A theory explaining how the market sets the prices of financial (capital) assets Explains how required rates of return (k) are determined, which in turn implies price One year return D1 P1 P0 (1 k ) Gordon model D0 (1 g ) P0 kg TM 8-19 Slide 1of 3 The Security Market Line (SML) Required rates of return are determined by: Stock’s Risk Premium k X k RF ( k M k RF )b X Market Risk Premium where: kX is the required return on Stock X kRF is the risk free rate kM is the return on the market, and bX is Stock X's beta coefficient TM 8-19 Slide 2 of 3 The Market Risk Premium Reflects investors' tolerance for risk Indicative of the degree of risk aversion in the investing community. The Risk Premium for Stock X "Average" risk premium multiplied by stock X's beta, the measure of its market risk The only thing related specifically to company X is bX, the measure of X's market risk Implication: only market risk counts. Business-specific risk doesn't enter the equation Investors are rewarded with extra return only for bearing market risk, not for bearing business-specific risk which is diversified away for portfolio investors. TM 8-19 Slide 3 of 3 THE SML AS A PORTRAYAL OF THE SECURITIES MARKET kX Security Market Line k X k RF ( k M k RF )b X kA * B Disequilibrium ke<kB kRF bA bB bX Figure 8-9 The Security Market Line TM 8-20 VALUATION USING CAPM AND THE SML Two Step Process: Find required return with SML Use in Gordon model Example 8-4 The Kelvin's last annual dividend was $1.50. The firm is expected to grow at 7% indefinitely. Short term treasury bills yield 6%. An average stock yields 10%. Kelvin stock is relatively volatile; its return moves twice as much as the average in response to political and economic changes. For what should the Kelvins sell? TM 8-21 Slide 1 of 3 Solution: kRF = treasury bill rate = 6% kM = "average" return = 10% bKelvin = 2.0 SML: k Kelvin k RF ( k M k RF )bKelvin k Kelvin 6 (10 6) 2.0 14% Gordon model: P0 D0 (1 g ) kg $1.50(1.07) $22.93 .14 .07 TM 8-21 Slide 2 of 3 Making Decisions Based On Stock Price Example 8-5 A new venture at Kelvin will: Increase growth rate from 7% to 9% Raise beta from 2.0 to 2.3 Should Kelvin undertake the new project? Solution: Changes move stock price in opposite directions More growth - good, more risk - bad Evaluate and choose option with highest stock price k Kelvin 6 (10 6 ) 2.3 15.2% P0 D0 (1 g ) $1.50(1.09 ) $26.37 kg .152.09 The higher price implies the venture is a good idea. TM 8-21 Slide 3 of 3 ADJUSTMENTS TO CHANGING MARKET CONDITIONS Response to a Change in the Risk Free Rate The SML shifts up or down parallel to itself to a position determined by the new kRF (For the slope to remain unchanged kM must also change) kX kRF' kRF bX Figure 8-10 A Shift in the Security Market Line to Accommodate an Increase in the Risk Free Rate TM 8-22 Slide 1 of 3 The Response to a Change in Risk Aversion Changes the market risk premium, (kM - kRF) the slope of the SML A rotation around the intercept at KRF kX SML2 SML1 kRF bX Figure 8-11 A Rotation of the Security Market Line to Accommodate a Change in Risk Aversion TM 8-22 Slide 2 of 3 Example 8-6 Sidel Co: bS = 1.2, kRF = 6%, kM = 10% kS = kRF + (kM - kRF)bS = 6 + (10 - 6)1.25 = 11.0 Calculate new required rates if: a. kRF increases to 8% b. kM increases to 11% Solution: a. kS = kRF + (kM - kRF)bS = 8 + (12 - 8)1.25 = 13.0% b. kS = kRF + (kM - kRF)bS = 6 + (11 - 6)1.25 = 12.25% TM 8-22 Slide 3 of 3