2Risk and ReturnRisk and ReturnDefining Risk and ReturnUsing Probability Distributions toMeasure RiskAttitudes Toward RiskRisk and Return in a Portfolio ContextDiversificationThe Capital Asset Pricing Model (CAPM)
3Defining ReturnDefining ReturnIncome receivedIncome received on an investmentplus any change in market pricechange in market price,usually expressed as a percent ofthe beginning market pricebeginning market price of theinvestment.DDtt + (PPtt - P- Pt-1t-1 )PPt-1t-1R =
4Return ExampleReturn ExampleThe stock price for Stock A was $10$10 pershare 1 year ago. The stock is currentlytrading at $9.50$9.50 per share, andshareholders just received a $1 dividend$1 dividend.What return was earned over the past year?
5Return ExampleReturn ExampleThe stock price for Stock A was $10$10 pershare 1 year ago. The stock is currentlytrading at $9.50$9.50 per share, andshareholders just received a $1 dividend$1 dividend.What return was earned over the past year?$1.00$1.00 + ($9.50$9.50 - $10.00$10.00 )$10.00$10.00RR = = 5%5%
6Defining RiskDefining RiskWhat rate of return do you expect on yourWhat rate of return do you expect on yourinvestment (savings) this year?investment (savings) this year?What rate will you actually earn?What rate will you actually earn?Does it matter if it is a bank CD or a shareDoes it matter if it is a bank CD or a shareof stock?of stock?The variability of returns fromThe variability of returns fromthose that are expected.those that are expected.
7Determining ExpectedDetermining ExpectedReturn (Discrete Dist.)Return (Discrete Dist.)R = Σ ( Ri )( Pi )R is the expected return for the asset,Ri is the return for the ithpossibility,Pi is the probability of that returnoccurring,n is the total number of possibilities.ni=1
8How to Determine the ExpectedHow to Determine the ExpectedReturn and Standard DeviationReturn and Standard DeviationStock BWRi Pi (Ri)(Pi)-.15 .10 -.015-.03 .20 -.006.09 .40 .036.21 .20 .042.33 .10 .033Sum 1.00 .090.090Theexpectedreturn, R,for StockBW is .09or 9%
9Determining StandardDetermining StandardDeviation (Risk Measure)Deviation (Risk Measure)σσ = Σ ( Ri - R )2( Pi )Standard DeviationStandard Deviation, σσ, is a statisticalmeasure of the variability of a distributionaround its mean.It is the square root of variance.Note, this is for a discrete distribution.ni=1
10How to Determine the ExpectedHow to Determine the ExpectedReturn and Standard DeviationReturn and Standard DeviationStock BWRi Pi (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 .00576Sum 1.00 .090.090 .01728.01728
11Determining StandardDetermining StandardDeviation (Risk Measure)Deviation (Risk Measure)σσ = Σ ( Ri - R )2( Pi )σσ = .01728σσ = .1315.1315 or 13.15%13.15%ni=1
12Coefficient of VariationCoefficient of VariationThe ratio of the standard deviationstandard deviation ofa distribution to the meanmean of thatdistribution.It is a measure of RELATIVERELATIVE risk.CV = σσ / RRCV of BW = .1315.1315 / .09.09 = 1.46
13Discrete vs. ContinuousDiscrete vs. ContinuousDistributionsDistributions00.050.10.150.18.104.22.1680.4-15% -3% 9% 21% 33%Discrete Continuous00.0050.010.0150.020.0250.030.035-50%-41%-32%-23%-14%-5%4%13%22%31%40%49%58%67%
14Determining ExpectedDetermining ExpectedReturn (Continuous Dist.)Return (Continuous Dist.)R = Σ ( 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
15Determining StandardDetermining StandardDeviation (Risk Measure)Deviation (Risk Measure)ni=1σσ = Σ ( Ri - R )2( n )Note, this is for a continuousdistribution where the distribution isfor a population. R represents thepopulation mean in this example.
16ContinuousContinuousDistribution ProblemDistribution ProblemAssume that the following list represents thecontinuous distribution of population returnsfor a particular investment (even thoughthere 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 andStandard Deviation for the populationassuming a continuous distribution.
17Let’s Use the Calculator!Let’s Use the Calculator!Enter “Data” first. Press:2ndData2ndCLR Work9.6 ENTER ↓ ↓-15.4 ENTER ↓ ↓26.7 ENTER ↓ ↓Note, we are inputting dataonly for the “X” variable andignoring entries for the “Y”variable in this case.
18Let’s Use the Calculator!Let’s 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 ↓ ↓
19Let’s Use the Calculator!Let’s Use the Calculator!Examine Results! Press:2ndStat↓ through the results.Expected return is 9% forthe 10 observations.Population standarddeviation is 13.32%.This can be much quickerthan calculating by hand,but slower than using aspreadsheet.
20Certainty EquivalentCertainty Equivalent (CECE) is theamount of cash someone wouldrequire with certainty at a point intime to make the individualindifferent between that certainamount and an amount expected tobe received with risk at the samepoint in time.Risk AttitudesRisk Attitudes
22Risk Attitude ExampleRisk Attitude ExampleYou have the choice between (1) a guaranteeddollar 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, tocall off the gamble.Raleigh is just as happy to take $50,000 or takethe risky gamble.Shannon requires at least $52,000 to call off thegamble.
23What are the Risk Attitude tendencies of each?What are the Risk Attitude tendencies of each?Risk Attitude ExampleRisk Attitude ExampleMary shows “risk aversion”“risk aversion” because her“certainty equivalent” < the expected value ofthe gamble..Raleigh exhibits “risk indifference”“risk indifference” because her“certainty equivalent” equals the expected valueof the gamble..Shannon reveals a “risk preference”“risk preference” becauseher “certainty equivalent” > the expected valueof the gamble..
24RP = Σ ( Wj )( Rj )RP is the expected return for the portfolio,Wj is the weight (investment proportion)for the jthasset in the portfolio,Rj is the expected return of the jthasset,m is the total number of assets in theportfolio.Determining PortfolioDetermining PortfolioExpected ReturnExpected Returnmj=1
25Determining PortfolioDetermining PortfolioStandard DeviationStandard Deviationmj=1mk=1σσPP = Σ Σ Wj Wk σjkWj is the weight (investment proportion)for the jthasset in the portfolio,Wk is the weight (investment proportion) forthe kthasset in the portfolio,σjk is the covariance between returns forthe jthand kthassets in the portfolio.
26What is Covariance?What is Covariance?σσ jk = σ j σ k rr jkσj is the standard deviation of the jthassetin the portfolio,σkis the standard deviation of the kthassetin the portfolio,rjk is the correlation coefficient between thejthand kthassets in the portfolio.
27Correlation CoefficientCorrelation CoefficientA standardized statistical measureof the linear relationship betweentwo variables.Its range is from -1.0-1.0 (perfectnegative correlation), through 00(no correlation), to +1.0+1.0 (perfectpositive correlation).
36Stock C Stock D PortfolioReturnReturn 9.00% 8.00% 8.64%Stand.Stand.Dev.Dev. 13.15% 10.65% 10.91%CVCV 1.46 1.33 1.26The portfolio has the LOWEST coefficientof variation due to diversification.Summary of the PortfolioSummary of the PortfolioReturn and Risk CalculationReturn and Risk Calculation
37Combining securities that are not perfectly,positively correlated reduces risk.Diversification and theDiversification and theCorrelation CoefficientCorrelation CoefficientINVESTMENTRETURNTIME TIMETIMESECURITY ESECURITY E SECURITY FSECURITY FCombinationCombinationE and FE and F
38Systematic RiskSystematic Risk is the variability of return onstocks or portfolios associated with changesin return on the market as a whole.Unsystematic RiskUnsystematic Risk is the variability of returnon stocks or portfolios not explained bygeneral market movements. It is avoidablethrough diversification.Total Risk = SystematicTotal Risk = SystematicRisk + Unsystematic RiskRisk + Unsystematic RiskTotal RiskTotal Risk = SystematicSystematic RiskRisk +UnsystematicUnsystematic RiskRisk
39Total Risk = SystematicTotal Risk = SystematicRisk + Unsystematic RiskRisk + Unsystematic RiskTotalTotalRiskRiskUnsystematic riskUnsystematic riskSystematic riskSystematic riskSTDDEVOFPORTFOLIORETURNNUMBER OF SECURITIES IN THE PORTFOLIOFactors such as changes in nation’seconomy, tax reform by the Congress,or a change in the world situation.
40Total Risk = SystematicTotal Risk = SystematicRisk + Unsystematic RiskRisk + Unsystematic RiskTotalTotalRiskRiskUnsystematic riskUnsystematic riskSystematic riskSystematic riskSTDDEVOFPORTFOLIORETURNNUMBER OF SECURITIES IN THE PORTFOLIOFactors unique to a particular companyor industry. For example, the death of akey executive or loss of a governmentaldefense contract.
41CAPM is a model that describes therelationship between risk andexpected (required) return; in thismodel, a security’s expected(required) return is the risk-free raterisk-free rateplus a premiuma premium based on thesystematic risksystematic risk of the security.Capital AssetCapital AssetPricing Model (CAPM)Pricing Model (CAPM)
421. Capital markets are efficient.2. Homogeneous investor expectationsover a given period.3. Risk-freeRisk-free asset return is certain(use short- to intermediate-termTreasuries as a proxy).4. Market portfolio contains onlysystematic risksystematic risk (use S&P 500 Indexor similar as a proxy).CAPM AssumptionsCAPM Assumptions
45Calculating “Beta”Calculating “Beta”on Your Calculatoron Your CalculatorAssume that the previous continuousdistribution problem represents the “excessreturns” of the market portfolio (it may still bein your calculator data worksheet -- 2ndData ).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” observationsof: 12%, -5%, 19%, 3%, 13%, 14%, -9%, -1%,12%, and 10%.
46Calculating “Beta”Calculating “Beta”on Your Calculatoron Your CalculatorLet us examine again the statisticalresults (Press 2ndand then Stat )The market expected return and standarddeviation is 9% and 13.32%. Your stockexpected return and standard deviation is6.8% and 8.76%.The regression equation is Y=a+bX. Thus, ourcharacteristic line is Y = 1.4448 + 0.595 X andindicates that our stock has a beta of 0.595.
47An index of systematic risksystematic risk.It measures the sensitivity of astock’s returns to changes inreturns on the market portfolio.The betabeta for a portfolio is simply aweighted average of the individualstock betas in the portfolio.What is Beta?What is Beta?
48Characteristic LinesCharacteristic Linesand Different Betasand Different BetasEXCESS RETURNON STOCKEXCESS RETURNON MARKET PORTFOLIOBeta < 1Beta < 1(defensive)(defensive)Beta = 1Beta = 1Beta > 1Beta > 1(aggressive)(aggressive)Each characteristiccharacteristiclineline has adifferent slope.
49RRjj is the required rate of return for stock j,RRff is the risk-free rate of return,ββjj is the beta of stock j (measuressystematic risk of stock j),RRMM is the expected return for the marketportfolio.Security Market LineSecurity Market LineRRjj = RRff + ββj(RRMM - RRff)
51Lisa Miller at Basket Wonders isattempting to determine the rate of returnrequired by their stock investors. Lisa isusing a 6% R6% Rff and a long-term marketmarketexpected rate of returnexpected rate of return of 10%10%. A stockanalyst following the firm has calculatedthat the firm betabeta is 1.21.2. What is therequired rate of returnrequired rate of return on the stock ofBasket Wonders?Determination of theDetermination of theRequired Rate of ReturnRequired Rate of Return
52RRBWBW = RRff + ββj(RRMM - RRff)RRBWBW = 6%6% + 1.21.2(10%10% - 6%6%)RRBWBW = 10.8%10.8%The required rate of return exceedsthe market rate of return as BW’sbeta exceeds the market beta (1.0).BWs RequiredBWs RequiredRate of ReturnRate of Return
53Lisa Miller at BW is also attempting todetermine the intrinsic valueintrinsic value of the stock.She is using the constant growth model.Lisa estimates that the dividend next perioddividend next periodwill be $0.50$0.50 and that BW will growgrow at aconstant rate of 5.8%5.8%. The stock is currentlyselling for $15.What is the intrinsic valueintrinsic value of thestock? Is the stock overover orunderpricedunderpriced?Determination of theDetermination of theIntrinsic Value of BWIntrinsic Value of BW
54The stock is OVERVALUED asthe market price ($15) exceedsthe intrinsic valueintrinsic value ($10$10).Determination of theDetermination of theIntrinsic Value of BWIntrinsic Value of BW$0.50$0.5010.8%10.8% - 5.8%5.8%IntrinsicIntrinsicValueValue== $10$10
55Security Market LineSecurity Market LineSystematic Risk (Beta)RRffRequiredReturnRequiredReturnDirection ofMovementDirection ofMovementStock YStock Y (Overpriced)Stock X (Underpriced)
56Small-firm EffectSmall-firm EffectPrice / Earnings EffectPrice / Earnings EffectJanuary EffectJanuary EffectThese anomalies have presentedserious challenges to the CAPMtheory.Determination of theDetermination of theRequired Rate of ReturnRequired Rate of Return