ESTIMATING RISK AND RETURNCHAPTER 10

AFTER STUDYING CHAPTER 5, YOU SHOULD BE ABLE TO:Understand the relationship (or trade-off) between risk and return.Define risk and return and show how to measure them by calculating expected return, standard deviation, and coefficient of variation.Discuss the different types of investor attitudes toward risk. Explain risk and return in a portfolio context, and distinguish between individual security and portfolio risk.Distinguish between avoidable (unsystematic) risk and unavoidable (systematic) risk and explain how proper diversification can eliminate one of these risks. Define and explain the capital-asset pricing model (CAPM), beta, and the characteristic line.Calculate a required rate of return using the capital-asset pricing model (CAPM).Demonstrate how the Security Market Line (SML) can be used to describe this relationship between expected rate of return and systematic risk. Explain what is meant by an efficient financial market and describe the three levels (or forms) to market efficiency.

RISK AND RETURNDefining Risk and ReturnUsing Probability Distributions to Measure RiskAttitudes Toward RiskRisk and Return in a Portfolio ContextDiversificationThe Capital Asset Pricing Model (CAPM)Efficient Financial Markets

DEFINING RETURNIncome received on an investment plus any change in market price, usually expressed as a percent of the beginning market price of the investment.Dt + (Pt - Pt-1 )Pt-1R =

RETURN EXAMPLEThe stock price for Stock A was $10 per share 1 year ago. The stock is currently trading at $9.50 per share and shareholders just received a $1 dividend. What return was earned over the past year?

RETURN EXAMPLEThe stock price for Stock A was $10 per share 1 year ago. The stock is currently trading at $9.50 per share and shareholders just received a $1 dividend. What return was earned over the past year?$1.00 + ($9.50 - $10.00 )$10.00R == 5%

DEFINING RISKWhat rate of return do you expect on your investment (savings) this year?What rate will you actually earn?Does it matter if it is a bank CD or a share of stock?The variability of returns from those that are expected.

DETERMINING EXPECTED RETURN (DISCRETE DIST.) R = S ( Ri )( Pi )R is the expected return for the asset,Ri is the return for the ith possibility,Pi is the probability of that return occurring,n is the total number of possibilities.ni=1

HOW TO DETERMINE THE EXPECTED RETURN AND STANDARD DEVIATIONStock BW RiPi (Ri)(Pi) -.15 .10 -.015 -.03 .20 -.006 .09 .40 .036 .21 .20 .042 .33 .10 .033 Sum 1.00 .090The expected return, R, for Stock BW is .09 or 9%

DETERMINING STANDARD DEVIATION (RISK MEASURE)s = S ( Ri - R )2( Pi )

Standard Deviation, s, is a statistical measure of the variability of a distribution around its mean.It is the square root of variance.Note, this is for a discrete distribution.ni=1

HOW TO DETERMINE THE EXPECTED RETURN AND STANDARD DEVIATIONStock BW RiPi (Ri)(Pi) (Ri - R )2(Pi) -.15 .10 -.015 .00576 -.03 .20 -.006 .00288 .09 .40 .036 .00000 .21 .20 .042 .00288 .33 .10 .033 .00576 Sum 1.00 .090 .01728

DETERMINING STANDARD DEVIATION (RISK MEASURE)s = S ( Ri - R )2( Pi )s = .01728s = .1315 or 13.15%ni=1

COEFFICIENT OF VARIATIONThe ratio of the standard deviation of a distribution to the mean of that distribution.It is a measure of RELATIVE risk.CV = s / RCV of BW = .1315 / .09 = 1.46

DISCRETE VS. CONTINUOUS DISTRIBUTIONS Discrete Continuous

Chart2

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Normal Distribution

Sheet1

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Normal Distribution

Returns

Liklihood

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DETERMINING EXPECTED RETURN (CONTINUOUS DIST.) R = S ( Ri ) / ( n )R is the expected return for the asset,Ri is the return for the ith observation,n is the total number of observations.ni=1

DETERMINING STANDARD DEVIATION (RISK MEASURE)s = S ( Ri - R )2 ( n )Note, this is for a continuous distribution where the distribution is for a population. R represents the population mean in this example.ni=1

CONTINUOUS DISTRIBUTION PROBLEMAssume that the following list represents the continuous distribution of population returns for a particular investment (even though there are only 10 returns).9.6%, -15.4%, 26.7%, -0.2%, 20.9%, 28.3%, -5.9%, 3.3%, 12.2%, 10.5%Calculate the Expected Return and Standard Deviation for the population assuming a continuous distribution.

Enter Data first. Press:2nd Data 2nd CLR Work9.6 ENTER -15.4 ENTER 26.7 ENTER Note, we are inputting data only for the X variable and ignoring entries for the Y variable in this case.Lets Use the Calculator!

Enter Data first. Press: -0.2 ENTER 20.9 ENTER 28.3 ENTER -5.9 ENTER 3.3 ENTER 12.2 ENTER 10.5 ENTER Lets Use the Calculator!

Examine Results! Press:2nd Stat through the results.Expected return is 9% for the 10 observations. Population standard deviation is 13.32%.This can be much quicker than calculating by hand, but slower than using a spreadsheet.Lets Use the Calculator!

RISK ATTITUDESCertainty Equivalent (CE) is the amount of cash someone would require with certainty at a point in time to make the individual indifferent between that certain amount and an amount expected to be received with risk at the same point in time.

RISK ATTITUDESCertainty equivalent > Expected valueRisk PreferenceCertainty equivalent = Expected valueRisk IndifferenceCertainty equivalent < Expected valueRisk AversionMost individuals are Risk Averse.

RISK ATTITUDE EXAMPLEYou have the choice between (1) a guaranteed dollar reward or (2) a coin-flip gamble of $100,000 (50% chance) or $0 (50% chance). The expected value of the gamble is $50,000.Mary requires a guaranteed $25,000, or more, to call off the gamble.Raleigh is just as happy to take $50,000 or take the risky gamble.Shannon requires at least $52,000 to call off the gamble.

RISK ATTITUDE EXAMPLEWhat are the Risk Attitude tendencies of each?Mary shows risk aversion because her certainty equivalent < the expected value of the gamble.Raleigh exhibits risk indifference because her certainty equivalent equals the expected value of the gamble.Shannon reveals a risk preference because her certainty equivalent > the expected value of the gamble.

DETERMINING PORTFOLIOEXPECTED RETURN RP = S ( Wj )( Rj )RP is the expected return for the portfolio,Wj is the weight (investment proportion) for the jth asset in the portfolio,Rj is the expected return of the jth asset,m is the total number of assets in the portfolio.mj=1

DETERMINING PORTFOLIO STANDARD DEVIATIONsP = S S Wj Wk sjk Wj is the weight (investment proportion) for the jth asset in the portfolio,Wk is the weight (investment proportion) for the kth asset in the portfolio,sjk is the covariance between returns for the jth and kth assets in the portfolio.mj=1mk=1

TIP SLIDE: APPENDIX ASlides 5-28 through 5-30 and 5-33 through 5-36 assume that the student has read Appendix A in Chapter 5

WHAT IS COVARIANCE?s jk = s j s k r jk sj is the standard deviation of the jth asset in the portfolio,sk is the standard deviation of the kth asset in the portfolio,rjk is the correlation coefficient between the jth and kth assets in the portfolio.

CORRELATION COEFFICIENTA standardized statistical measure of the linear relationship between two variables.

Its range is from -1.0 (perfect negative correlation), through 0 (no correlation), to +1.0 (perfect positive correlation).

VARIANCE - COVARIANCE MATRIXA three asset portfolio: Col 1 Col 2 Col 3Row 1W1W1s1,1 W1W2s1,2 W1W3s1,3Row 2W2W1s2,1 W2W2s2,2 W2W3s2,3Row 3W3W1s3,1 W3W2s3,2 W3W3s3,3

sj,k = is the covariance between returns for the jth and kth assets in the portfolio.

PORTFOLIO RISK AND EXPECTED RETURN EXAMPLEYou are creating a portfolio of Stock D and Stock BW (from earlier). You are investing $2,000 in Stock BW and $3,000 in Stock D. Remember that the expected return and standard deviation of Stock BW is 9% and 13.15% respectively. The expected return and standard deviation of Stock D is 8% and 10.65% respectively. The correlation coefficient between BW and D is 0.75.What is the expected return and standard deviation of the portfolio?

DETERMINING PORTFOLIO EXPECTED RETURNWBW = $2,000 / $5,000 = .4WD = $3,000 / $5,000 = .6

RP = (WBW)(RBW) + (WD)(RD) RP = (.4)(9%) + (.6)(8%)RP = (3.6%) + (4.8%) = 8.4%

DETERMINING PORTFOLIO STANDARD DEVIATIONTwo-asset portfolio: Col 1 Col 2Row 1WBW WBW sBW,BW WBW WD sBW,DRow 2 WD WBW sD,BW WD WD sD,D

This represents the variance - covariance matrix for the two-asset portfolio.

DETERMINING PORTFOLIO STANDARD DEVIATIONTwo-asset portfolio: Col 1 Col 2Row 1 (.4)(.4)(.0173) (.4)(.6)(.0105)Row 2 (.6)(.4)(.0105) (.6)(.6)(.0113)

This represents substitution into the variance - covariance matrix.

DETERMINING PORTFOLIO STANDARD DEVIATIONTwo-asset portfolio: Col 1 Col 2Row 1 (.0028) (.0025)Row 2 (.0025) (.0041)

This represents the actual element values in the variance - covariance matrix.

DETERMINING PORTFOLIO STANDARD DEVIATIONsP = .0028 + (2)(.0025) + .0041sP = SQRT(.0119)sP = .1091 or 10.91%

A weighted average of the individual standard deviations is INCORRECT.

DETERMINING PORTFOLIO STANDARD DEVIATIONThe WRONG way to calculate is a weighted average like:sP = .4 (13.15%) + .6(10.65%)sP = 5.26 + 6.39 = 11.65%

10.91% = 11.65%

This is INCORRECT.

SUMMARY OF THE PORTFOLIO RETURN AND RISK CALCULATIONStock C Stock D PortfolioReturn 9.00% 8.00% 8.64%Stand.Dev.13.15% 10.65% 10.91%CV 1.46 1.33 1.26

The portfolio has the LOWEST coefficient of variation due to diversification.

DIVERSIFICATION AND THE CORRELATION COEFFICIENTCombining securities that are not perfectly, positively correlated reduces risk.INVESTMENT RETURNTIMETIMETIMESECURITY ESECURITY FCombinationE and F

TOTAL RISK = SYSTEMATIC RISK + UNSYSTEMATIC RISKSystematic Risk is the variability of return on stocks or portfolios associated with changes in return on the market as a whole.Unsystematic Risk is the variability of return on stocks or portfolios not explained by general market movements. It is avoidable through diversification.Total Risk = Systematic Risk + Unsystematic Risk

TOTAL RISK = SYSTEMATIC RISK + UNSYSTEMATIC RISKTotalRiskUnsystematic riskSystematic riskSTD DEV OF PORTFOLIO RETURNNUMBER OF SECURITIES IN THE PORTFOLIOFactors such as changes in nations economy, tax reform by the Congress,or a change in the world situation.

TOTAL RISK = SYSTEMATIC RISK + UNSYSTEMATIC RISKTotalRiskUnsystematic riskSystematic riskSTD DEV OF PORTFOLIO RETURNNUMBER OF SECURITIES IN THE PORTFOLIOFactors unique to a particular companyor industry. For example, the death of akey executive or loss of a governmentaldefense contract.

CAPITAL ASSET PRICING MODEL (CAPM)CAPM is a model that describes the relationship between risk and expected (required) return; in this model, a securitys expected (required) return is the risk-free rate plus a premium based on the systematic risk of the security.

CAPM ASSUMPTIONS1.Capital markets are efficient.2.Homogeneous investor expectations over a given period.3.Risk-free asset return is certain (use short- to intermediate-term Treasuries as a proxy).4.Market portfolio contains only systematic risk (use S&P 500 Indexor similar as a proxy).

CHARACTERISTIC LINEEXCESS RETURNON STOCKEXCESS RETURNON MARKET PORTFOLIOBeta =RiseRunNarrower spreadis higher correlationCharacteristic Line

CALCULATING BETA ON YOUR CALCULATORThe Market and My Stock returns are excess returns and have the riskless rate already subtracted.

Time Pd.MarketMy Stock19.6%12%2-15.4%-5%326.7%19%4-.2%3%520.9%13%628.3%14%7-5.9%-9%83.3%-1%912.2%12%1010.5%10%

CALCULATING BETA ON YOUR CALCULATORAssume that the previous continuous distribution problem represents the excess returns of the market portfolio (it may still be in your calculator data worksheet -- 2nd Data ).Enter the excess market returns as X observations of: 9.6%, -15.4%, 26.7%, -0.2%, 20.9%, 28.3%, -5.9%, 3.3%, 12.2%, and 10.5%.Enter the excess stock returns as Y observations of: 12%, -5%, 19%, 3%, 13%, 14%, -9%, -1%, 12%, and 10%.

CALCULATING BETA ON YOUR CALCULATORLet us examine again the statistical results (Press 2nd and then Stat )The market expected return and standard deviation is 9% and 13.32%. Your stock expected return and standard deviation is 6.8% and 8.76%.The regression equation is Y=a+bX. Thus, our characteristic line is Y = 1.4448 + 0.595 X and indicates that our stock has a beta of 0.595.

WHAT IS BETA?An index of systematic risk.It measures the sensitivity of a stocks returns to changes in returns on the market portfolio.The beta for a portfolio is simply a weighted average of the individual stock betas in the portfolio.

CHARACTERISTIC LINES AND DIFFERENT BETASEXCESS RETURNON STOCKEXCESS RETURNON MARKET PORTFOLIOBeta < 1(defensive)Beta = 1Beta > 1(aggressive)Each characteristic line has a different slope.

SECURITY MARKET LINERj is the required rate of return for stock j,Rf is the risk-free rate of return,bj is the beta of stock j (measures systematic risk of stock j),RM is the expected return for the market portfolio.Rj = Rf + bj(RM - Rf)

SECURITY MARKET LINERj = Rf + bj(RM - Rf)bM = 1.0Systematic Risk (Beta)RfRMRequired ReturnRiskPremiumRisk-freeReturn

SECURITY MARKET LINEObtaining BetasCan use historical data if past best represents the expectations of the futureCan also utilize services like Value Line, Ibbotson Associates, etc.Adjusted BetaBetas have a tendency to revert to the mean of 1.0Can utilize combination of recent beta and mean2.22 (.7) + 1.00 (.3) = 1.554 + 0.300 = 1.854 estimate

DETERMINATION OF THE REQUIRED RATE OF RETURNLisa Miller at Basket Wonders is attempting to determine the rate of return required by their stock investors. Lisa is using a 6% Rf and a long-term market expected rate of return of 10%. A stock analyst following the firm has calculated that the firm beta is 1.2. What is the required rate of return on the stock of Basket Wonders?

BWS REQUIRED RATE OF RETURNRBW = Rf + bj(RM - Rf)RBW = 6% + 1.2(10% - 6%)RBW = 10.8%The required rate of return exceeds the market rate of return as BWs beta exceeds the market beta (1.0).

DETERMINATION OF THE INTRINSIC VALUE OF BWLisa Miller at BW is also attempting to determine the intrinsic value of the stock. She is using the constant growth model. Lisa estimates that the dividend next period will be $0.50 and that BW will grow at a constant rate of 5.8%. The stock is currently selling for $15.What is the intrinsic value of the stock? Is the stock over or underpriced?

DETERMINATION OF THE INTRINSIC VALUE OF BWThe stock is OVERVALUED as the market price ($15) exceeds the intrinsic value ($10).$0.5010.8% - 5.8%IntrinsicValue==$10

SECURITY MARKET LINESystematic Risk (Beta)RfRequired ReturnDirection ofMovementDirection ofMovementStock Y (Overpriced)Stock X (Underpriced)

DETERMINATION OF THE REQUIRED RATE OF RETURNSmall-firm EffectPrice / Earnings EffectJanuary EffectThese anomalies have presented serious challenges to the CAPM theory.