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INVESTMENTS|BODIE,KANE,MARCUSCopyright 2011 by The McGraw-Hill Companies,Inc.All rights reserved.McGraw-Hill/IrwinCHAPTER 5Introduction to Risk,Return,and the Historical RecordINVESTMENTS|BODIE,KANE,MARCUSInterest Rate Determinants SupplyHouseholds DemandBusinesses Governments Net Supply and/or DemandFederal Reserve ActionsINVESTMENTS|BODIE,KANE,MARCUSReal and Nominal Rates of Interest Nominal interest rate:Growth rate of your money Real interest rate:Growth rate of your purchasing power Let R=nominal rate,r=real rate and I=inflation rate.Then:iRriiRr1INVESTMENTS|BODIE,KANE,MARCUSEquilibrium Real Rate of Interest Determined by:SupplyDemandGovernment actionsExpected rate of inflationINVESTMENTS|BODIE,KANE,MARCUSFigure 5.1 Determination of the Equilibrium Real Rate of InterestINVESTMENTS|BODIE,KANE,MARCUSEquilibrium Nominal Rate of Interest As the inflation rate increases,investors will demand higher nominal rates of return If E(i)denotes current expectations of inflation,then we get the Fisher Equation:Nominal rate=real rate+inflation forecast()RrE iINVESTMENTS|BODIE,KANE,MARCUSTaxes and the Real Rate of Interest Tax liabilities are based on nominal income Given a tax rate(t)and nominal interest rate(R),the Real after-tax rate is:The after-tax real rate of return falls as the inflation rate rises.(1)()(1)(1)Rtiritirtit INVESTMENTS|BODIE,KANE,MARCUSRates of Return for Different Holding Periods100()1()frTP TZero Coupon Bond,Par=$100,T=maturity,P=price,rf(T)=total risk free returnINVESTMENTS|BODIE,KANE,MARCUSExample 5.2 Annualized Rates of ReturnINVESTMENTS|BODIE,KANE,MARCUSEquation 5.7 EAR EAR definition:percentage increase in funds invested over a 1-year horizon TfTrEAR111INVESTMENTS|BODIE,KANE,MARCUSEquation 5.8 APR APR:annualizing using simple interestTEARAPRT11INVESTMENTS|BODIE,KANE,MARCUSTable 5.1 APR vs.EARINVESTMENTS|BODIE,KANE,MARCUSTable 5.2 Statistics for T-Bill Rates,Inflation Rates and Real Rates,1926-2009INVESTMENTS|BODIE,KANE,MARCUSBills and Inflation,1926-2009 Moderate inflation can offset most of the nominal gains on low-risk investments.A dollar invested in T-bills from19262009 grew to$20.52,but with a real value of only$1.69.Negative correlation between real rate and inflation rate means the nominal rate responds less than 1:1 to changes in expected inflation.INVESTMENTS|BODIE,KANE,MARCUSFigure 5.3 Interest Rates and Inflation,1926-2009INVESTMENTS|BODIE,KANE,MARCUSRisk and Risk PremiumsPDPPHPR0101HPR=Holding Period ReturnP0=Beginning priceP1=Ending priceD1=Dividend during period oneRates of Return:Single PeriodINVESTMENTS|BODIE,KANE,MARCUSEnding Price=110Beginning Price=100Dividend=4HPR=(110-100+4)/(100)=14%Rates of Return:Single Period ExampleINVESTMENTS|BODIE,KANE,MARCUSExpected returnsp(s)=probability of a stater(s)=return if a state occurss =stateExpected Return and Standard Deviation()()()sE rp s r sINVESTMENTS|BODIE,KANE,MARCUSStateProb.of Stater in State Excellent.250.3100Good.450.1400Poor.25-0.0675Crash.05-0.5200E(r)=(.25)(.31)+(.45)(.14)+(.25)(-.0675)+(0.05)(-0.52)E(r)=.0976 or 9.76%Scenario Returns:ExampleINVESTMENTS|BODIE,KANE,MARCUSVariance(VAR):Variance and Standard Deviation22()()()sp sr sE r2STDStandard Deviation(STD):INVESTMENTS|BODIE,KANE,MARCUSScenario VAR and STD Example VAR calculation:2=.25(.31-0.0976)2+.45(.14-.0976)2+.25(-0.0675-0.0976)2+.05(-.52-.0976)2=.038 Example STD calculation:1949.038.INVESTMENTS|BODIE,KANE,MARCUSTime Series Analysis of Past Rates of ReturnnsnssrnsrsprE11)(1)()()(The Arithmetic Average of rate of return:INVESTMENTS|BODIE,KANE,MARCUSGeometric Average Return1/1TVgnTV=Terminal Value of the Investmentg=geometric average rate of return)1).(1)(1(21nnrrrTVINVESTMENTS|BODIE,KANE,MARCUSGeometric Variance and Standard Deviation Formulas Estimated Variance=expected value of squared deviations 21_21nsrsrnINVESTMENTS|BODIE,KANE,MARCUSGeometric Variance and Standard Deviation Formulas When eliminating the bias,Variance and Standard Deviation become:21_11njrsrnINVESTMENTS|BODIE,KANE,MARCUSThe Reward-to-Volatility(Sharpe)Ratio Sharpe Ratio for Portfolios:Returns Excess of SDPremiumRisk INVESTMENTS|BODIE,KANE,MARCUSThe Normal Distribution Investment management is easier when returns are normal.Standard deviation is a good measure of risk when returns are symmetric.If security returns are symmetric,portfolio returns will be,too.Future scenarios can be estimated using only the mean and the standard deviation.INVESTMENTS|BODIE,KANE,MARCUSFigure 5.4 The Normal DistributionINVESTMENTS|BODIE,KANE,MARCUSNormality and Risk Measures What if excess returns are not normally distributed?Standard deviation is no longer a complete measure of risk Sharpe ratio is not a complete measure of portfolio performance Need to consider skew and kurtosisINVESTMENTS|BODIE,KANE,MARCUSSkew and KurtosisSkewEquation 5.19Kurtosis Equation 5.2033_RRaverageskew344_RRaveragekurtosisINVESTMENTS|BODIE,KANE,MARCUSFigure 5.5A Normal and Skewed Distributions INVESTMENTS|BODIE,KANE,MARCUSFigure 5.5B Normal and Fat-Tailed Distributions(mean=.1,SD=.2)INVESTMENTS|BODIE,KANE,MARCUSValue at Risk(VaR)A measure of loss most frequently associated with extreme negative returns VaR is the quantile of a distribution below which lies q%of the possible values of that distribution The 5%VaR,commonly estimated in practice,is the return at the 5th percentile when returns are sorted from high to low.INVESTMENTS|BODIE,KANE,MARCUSExpected Shortfall(ES)Also called conditional tail expectation(CTE)More conservative measure of downside risk than VaR VaR takes the highest return from the worst cases ES takes an average return of the worst casesINVESTMENTS|BODIE,KANE,MARCUSLower Partial Standard Deviation(LPSD)and the Sortino Ratio Issues:Need to consider negative deviations separately Need to consider deviations of returns from the risk-free rate.LPSD:similar to usual standard deviation,but uses only negative deviations from rf Sortino Ratio replaces Sharpe RatioINVESTMENTS|BODIE,KANE,MARCUSHistoric Returns on Risky Portfolios Returns appear normally distributed Returns are lower over the most recent half of the period(1986-2009)SD for small stocks became smaller;SD for long-term bonds got biggerINVESTMENTS|BODIE,KANE,MARCUSHistoric Returns on Risky Portfolios Better diversified portfolios have higher Sharpe Ratios Negative skewINVESTMENTS|BODIE,KANE,MARCUSFigure 5.7 Nominal and Real Equity Returns Around the World,1900-2000INVESTMENTS|BODIE,KANE,MARCUSFigure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World,1900-2000INVESTMENTS|BODIE,KANE,MARCUSFigure 5.9 Probability of Investment Outcomes After 25 Years with a Lognormal DistributionINVESTMENTS|BODIE,KANE,MARCUSTerminal Value with Continuous Compounding 221120201()TggTTTeeE rWhen the continuously compounded rate of return on an asset is normally distributed,the effective rate of return will be lognormally distributed.The Terminal Value will then be:INVESTMENTS|BODIE,KANE,MARCUSFigure 5.10 Annually Compounded,25-Year HPRsINVESTMENTS|BODIE,KANE,MARCUSFigure 5.11 Annually Compounded,25-Year HPRsINVESTMENTS|BODIE,KANE,MARCUSFigure 5.12 Wealth Indexes of Selected Outcomes of Large Stock Portfolios and the Average T-bill Portfolio
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