Return and Risk - Salisbury University



Return and RiskHistorical ReturnsHow do we calculate returns?Rt+1=Divt+1+Pt+1Pt-1=Divt+1Pt+ Pt+1-PtPt= Dividend Yield + Capital Gain YieldCalculate the return for the following stock:DatePriceDividendReturn12/31/9871.562/2/9989.440.500.25685/11/9985.750.50-0.03575/28/9969.0013.72-0.03538/10/9960.810.50-0.111411/8/9969.060.500.143912/31/9972.690.052612/31/0353.402/11/0449.800.50-0.05815/12/0444.480.50-0.09688/11/0441.740.50-0.050411/4/0439.500.50-0.041712/31/0440.060.0142What is the annual return?We should use the geometric mean formula:1+RAnnual=1+RQ11+RQ21+RQ31+RQ4R1999 = (1 + 0.2568)(1 + (-0.0357))(1 – 0.0353)(1 – 0.1114)(1 + 0.1439)(1 + 0.0526) – 1R1999 = 0.2509 → 25.09%R2004 = (0.9419)(0.9032)(0.9496)(0.9583)(1.0142) – 1R2004 = -0.2148 → -21.48%Let’s calculate the annual returns for the S&P 500 Index, GM, and the 3-Month T-Bill:Year EndS&P 500 IndexDividends PaidS&P 500 Realized ReturnsGM Realized Return3-Month T-Bill Return1995615.931996740.7416.6123.0%8.6%5.1%1997970.4317.2033.4%19.6%5.2%19981,229.2318.5028.6%21.3%4.9%19991,469.2518.1021.0%25.1%4.8%20001,320.2815.70-9.1%-27.8%6.0%20011,148.0815.20-11.9%-1.0%3.3%2002879.8214.53-22.1%-20.8%1.6%20031,111.9220.8028.7%52.9%1.0%20041,211.9220.9810.9%-21.5%1.4%Annual Average11.39%6.27%3.70%Standard Deviation20.59%26.59%1.91%The standard error is the measure of the degree of estimation error of the estimate of the returns. The standard error is:Standard Error=σNExample: From 1926 through 2004 the average annual return for the S&P 500 was 12.3% and the standard deviation was 20.36%. The standard error is:Standard Error=0.203679=0.0229We can develop confidence intervals using the Standard Error. The 95% confidence interval is:Historical Average Return ± (2)(Standard Error)0.123 ± (2)(0.0229) Or a range of 7.72% to 16.88%.Expected Returns and Standard DeviationsHow do we calculate expected returns and the risk associated with the returns?ERA=i=1NPiRAiVariance of asset A returns=σA2=i=1NPiRAi-E(RA)2Standard Deveation of asset A returns=σA=σA2Covariance between asset A & B=σA,B=i=1NPiRAi-E(RA)RBi-E(RB)Correlation Coefficient between asset A & B=ρA,B=σA,BσAσBCoefficient of Variation for asset A=CVA=σAE(RA)StateProbCTExpectedCT10.300.150.25(0.3)(0.15) = 0.04500.075020.500.100.20(0.5)(0.10) = 0.05000.100030.200.020.01(0.2)(0.02) = 0.00400.0020Expected Return =0.09900.1770sigma Csigma T(0.3)(0.15 - 0.0990)^2 =0.0008(0.3)(0.25 - 0.1770)^2 =0.0016(0.5)(0.10 - 0.0990)^2 =0.000001(0.5)(0.20 - 0.1770)^2 =0.0003(0.2)(0.02 - 0.0990)^2 =0.0012(0.2)(0.01 - 0.1770^2 =0.0056Var (C) = 0.0020Var (T) = 0.0074Std. Dev (C) = 0.0450Std. Dev (T) = 0.0863Covariance(0.3)(0.15 - 0.0990)(0.25 - 0.1770) =0.0011(0.5)(0.10 - 0.0990)(0.20 - 0.1770) =0.000012(0.2)(0.02 - 0.0990)(0.01 - 0.1770) =0.0026Covariance (C,T) = 0.0038Correlation CoefficientRho (C,T) = ρC,T = 0.9695Coefficient of VariationCV (C) = 0.4550CV (T) = 0.4874Another exampleState of the EconomyProbabilitiesReturns for LRecession0.20-200.20(-20) = -4.00Normal0.50250.50(25) = 12.50Boom0.30700.30(70) = 21.00Expected Return29.50State of the EconomyProbabilitiesReturns for URecession0.20300.20(30) = 6.00Normal0.50220.50(22) = 11.00Boom0.30100.30(10) = 3.00Expected Return20.00Variance CalculationsState of the EconomyProbLRecession0.20-200.20(-20 - 29.50)2 =490.05Normal0.50250.50(25 - 29.50)2 =10.125Boom0.30700.30(70 - 29.50)2 =492.075Expected Varianceσ2L = 992.25Standard Dev.σL = 31.50%State of the EconomyProbURecession0.20300.20(30 - 20.0)2 =20.0Normal0.50220.50(22 - 20.0)2 =2.0Boom0.30100.30(10 - 20.0)2 =30.00Expected Varianceσ2U = 52.0Standard Dev.σU = 7.21%Which asset would you buy? PortfoliosMost investors hold a portfolio of assets or they own more than a single stock, bond, or other asset. Given this portfolio risk and return is an important subject to study.The way a portfolio is described is by the weights of the assets in the port. The weights are the percentages of each asset held in the port. If we have $250 in one asset and $750 in another asset, then our total portfolio is $1000 and we have 25% in the first asset and 75% in the second asset.Portfolio Expected ReturnWe go back to stocks L & U. Remember E(RL) = 29.5% and E(RU) = 20% and we invest $400 in L and $800 in U. What is the expected return for our portfolio?Remember:ERP=i=1NwiRiRP = (wL)E(RL) + (wU)E(RU) RP = (.33)(29.5) + (.67)(20) RP = 23.17%How do we calculate the risk in a portfolio?Is it:σP = (wL)σL + (wU)σU σP = (.33)31.50% + (.67)7.21% = 15.12%No.33(-20) + .67(30) = 13.33.33(25) + .67(22) = 23.00.33(70) + .67(10) = 30.00State of the EconomyProbPort ReturnRecession0.2013.330.20(13.33 - 23.17)2 =19.3651Normal0.5023.000.50(23 - 23.17)2 =0.0145Boom0.3030.000.30(30.0 - 23.17)2 =13.9947E(RP) = 23.17%Expected Variance: σ2P = 33.3742Standard Dev.: σP = 5.777%Here are the equations to calculate the return and variance of a portfolio.RP=i=1NwiRiσP2=i=1Nwi2σi2+i=1Nj=1i≠jN2wiwjσiσjρi,jRemember:ρi,j=σi,jσiσj Therefore I can rewrite the variance equation like this:σP2=i=1Nwi2σi2+i=1Nj=1i≠jN2wiwjσi,jHere are the equations for a two asset portfolio:RP = w1R1 + w2R2σP2=w12σ12+w22σ22+2w1w2σ1σ2ρ1,2Here are the equations for a three asset portfolio:RP = w1R1 + w2R2 + w3R3σP2=w12σ12+w22σ22+w32σ32+2w1w2σ1σ2ρ1,2+2w1w3σ1σ3ρ1,3+2w2w3σ2σ3ρ2,3The variance equation gets long quickly. As a result of this growth, we will look at the variance-covariance matrix:12345?NIn a portfolio with N assets there are N variances and N2 – N covariances. However, the top half is equal to the bottom half of the matrix. This means there are N2-N2 unique covariances. This is the reason for the 2 in the last term of the variance equation.1σ12σ1,2σ1,3σ1,4σ1,5?σ1,N2σ22σ2,3σ2,4σ2,5?σ2,N3σ32σ3,4σ3,5?σ3,N4σ42σ4,5?σ4,N5σ52?σ5,N???N?σN2Here is an example. We have five stocks with the following characteristics:Expected Return:15%Standard Deviation:5%We will form a portfolio with an equal investment in each. Calculate the portfolio return and standard deviation under the following assumptions:Correlation coefficient is equal to 1 and 0.First examine the portfolio when the correlation coefficient is 1.RP = 5(.20)(.15) = 0.15σP2=50.2020.052+1020.200.200.050.051σP2=0.0005+0.0020=0.0025σP = 0.05 → 5.0 %Now examine the portfolio when the correlation coefficient is 0.RP = 5(.20)(.15) = 0.15σP2=50.2020.052+1020.200.200.050.050σP2=0.0005σP = 0.022361 → 2.2361 %Why does diversification work? To answer this question we divided total risk into market (systematic) and firm-specific (unsystematic) risk. The risk for any asset can be divided into these two categories of risk. Market risk is risk that affects a large number of stocks, whereas firm-specific risk affects a single firm or a very small number of firms. The firm-specific risks between companies will have a correlation coefficient of less than one and may approach negative one. The firm-specific risk is diversified away as assets are added to a portfolio. This leaves market risk as the important risk in a portfolio. This allows us to say a great deal about the risk and return of individual assets. It also leads to the SML and beta. The SML and beta begin to answer questions about the required return for individual stocks. Capital Asset Pricing ModelThe Capital Asset Pricing Model (CAPM) is a Quantify relationship between an asset’s risk and return.The CAPM model is:Ri=Rf+βiRM-RfThis model specifies a relationship between the required return for an asset and the market risk of the asset. This model can be applied to individual assets or a portfolio of assets. Use the following equation to calculate the beta of a portfolio:βP=i=1NwiβiHere is an example calculation of the portfolio beta:InvestmentWeightsBetaDCLK$2,0000.13334.030.5373KO$3,0000.20000.840.1680INTC$4,0000.26671.050.2800KEI$6,0000.40000.590.2360Total = $15,0001βP=1.2213What is the required return for this portfolio if the risk-free rate is 4 percent and the market return is 12 percent?RP = .04 + 1.2213(0.12 – 0.04)RP = 0.1377 → 13.77%An alternative calculation is:InvestmentWeightsBetaRequired ReturnDCLK$2,0000.13334.03.04 + 4.03(0.12 – 0.04) = 0.36240.1333(0.3624) = 0.0483KO$3,0000.20000.84.04 + 0.84(0.12 – 0.04) = 0.10720.2000(0.1072) = 0.0214INTC$4,0000.26671.05.04 + 1.05(0.12 – 0.04) = 0.12400.2667(0.1240) = 0.0331KEI$6,0000.40000.59.04 + 0.59(0.12 – 0.04) = 0.08720.4000(0.0872) = 0.0349Total = $15,00010.1377How is beta calculated?βi=σi,MσM2Linear RegressionRi = a + bRM + εExampleS&P 500ArcadisBakerEnGlobalEssexKeithLayneMTCShawTetraURSVersarVSEWash GroupPort Avg.Month RetMonth RetMonth RetMonth RetMonth RetMonth RetMonth RetMonth RetMonth RetMonth RetMonth RetMonth RetMonth RetMonth Ret10.0092930.0033020.021566-0.0281520.002716-0.007296-0.0458970.0049770.0426540.0729230.0164540.066098-0.036646-0.024224070.0068058272-0.0112220.1477890.1416850.2805640.0544150.0605370.1005990.0913740.1035560.0492350.0061420.073227-0.130734-0.020393030.07369202530.035968-0.0050270.2961930.696809-0.0843530.0064130.175642-0.150149-0.1111110.1108650.0026770.3656250.2171220.055164320.121220634-0.0001430.0656250.0359630.0358130.2235290.0297170.1485550.0984190.0674940.1228220.104050.032258-0.1356570.064778170.06872043250.0299520.137633-0.1798290.6575340.2174480.0104860.1161290.1058710.1151080.1378660.100163-0.1711230.2919990.158822110.1306236026-0.0201090.014426-0.046712-0.060086-0.05940.212717-0.10249-0.067077-0.171101-0.1608560.0695650.0080860.130826-0.0791287-0.0239407437-0.019118-0.053171-0.02649-0.033195-0.1472580.002899-0.0259450.0159420.048077-0.238842-0.003121-0.055980.1241740.02834286-0.02804354980.0189030.1484590.163926-0.0360.0776590.017099-0.041622-0.0150860.2373590.1180040.022333-0.096552-0.025740.110970040.0523699349-0.02529-0.00335-0.007143-0.193548-0.122469-0.0247270.019284-0.032469-0.058263-0.114098-0.1211840-0.069089-0.04533333-0.059414685100.0324580.180620.0425531.0945950.148610.035736-0.0644330.0920620.2118130.0984250.0685750.018735-0.0071370.057692310.152142024Regression StatisticsRegression StatisticsMultiple R0.416647074Multiple R0.867775569R Square0.173594784R Square0.753034437Adjusted R Square0.070294132Adjusted R Square0.722163742Standard Error0.12665944Standard Error0.038620118Observations10Observations10?CoefficientsStandard Errort Stat?CoefficientsStandard Errort StatIntercept0.0367477680.0410763870.894620256Intercept0.0356970220.0125247272.850124X Variable 12.3298613161.7972697551.296333681X Variable 12.7065964740.5480110284.938945The BAII Plus has a linear regression function. The following steps illustrate the use of this function.First, enter the data in the data worksheet.S&P 500ShawMonth RetMonth RetXY10.0092930.0426542-0.0112220.10355630.035968-0.1111114-0.0001430.06749450.0299520.1151086-0.020109-0.1711017-0.0191180.04807780.0189030.2373599-0.02529-0.058263100.0324580.211813Second, use the Stat worksheet to examine the results of the regression analysis.Also, the following equation will calculate beta:βi=σi,MσM2CoVar Mark, Shaw =0.001157CoVar Mark, Port =0.001344Var Mark =0.000552Var Mark =0.000552Beta =2.096875Beta =2.435937Another example:MarketAsset A10.01220.0616(0.0122 - 0.0285)(0.0616 - 0.0665) = 0.0000804220.0173-0.0109(0.0173 - 0.0285)(-0.0109 - 0.0665) = 0.0008657830.05080.0914(0.0508 - 0.0285)(0.0914 - 0.0665) = 0.0005548840.00710.1239(0.0071 - 0.0285)(0.1239 - 0.0665) = -0.0012263650.05500.0667(0.0550 - 0.0285)(0.0667 - 0.0665) = 0.00000424Mean = 0.02850.0665Sum = 0.00027896Covariance = 0.00005579Var = 0.000409726MarketAsset AXYN = 510.01220.0616Xbar = __________Ybar = __________a = __________20.0173-0.0109Sx = __________Sy = __________b = __________30.05080.0914x = __________y = __________R = __________40.00710.123950.05500.0667 ................
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