1、CHAPTER 5Introduction to Risk, Return, and the Historical RecordInterest Rate DeterminantsSupplyHouseholdsDemandBusinessesGovernments Net Supply and/or DemandFederal Reserve ActionsReal and Nominal Rates of InterestNominal interest rate: Growth rate of your moneyReal interest rate: Growth rate of yo

2、ur purchasing powerLet R = nominal rate, r = real rate and I = inflation rate. Then:Equilibrium Real Rate of InterestDetermined by:SupplyDemandGovernment actionsExpected rate of inflationFigure 5.1 Determination of the Equilibrium Real Rate of InterestEquilibrium Nominal Rate of InterestAs the infla

3、tion rate increases, investors will demand higher nominal rates of returnIf E(i) denotes current expectations of inflation, then we get the Fisher Equation:Nominal rate = real rate + inflation forecastTaxes and the Real Rate of InterestTax liabilities are based on nominal incomeGiven a tax rate (t)

4、and nominal interest rate (R), the Real after-tax rate is:The after-tax real rate of return falls as the inflation rate rises.Rates of Return for Different Holding PeriodsZero Coupon Bond, Par = $100, T=maturity, P=price, rf(T)=total risk free returnExample 5.2 Annualized Rates of ReturnEquation 5.7

5、 EAREAR definition: percentage increase in funds invested over a 1-year horizonEquation 5.8 APRAPR: annualizing using simple interestTable 5.1 APR vs. EARTable 5.2 Statistics for T-Bill Rates, Inflation Rates and Real Rates, 1926-2009Bills and Inflation, 1926-2009Moderate inflation can offset most o

6、f 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.Figure 5.3 Interest Rates

7、 and Inflation, 1926-2009Risk and Risk PremiumsHPR = Holding Period ReturnP0 = Beginning priceP1 = Ending priceD1 = Dividend during period oneRates of Return: Single PeriodEnding Price =110Beginning Price = 100Dividend = 4HPR = (110 - 100 + 4 )/ (100) = 14%Rates of Return: Single Period ExampleExpec

8、ted returnsp(s) = probability of a stater(s) = return if a state occurss = stateExpected Return and Standard DeviationStateProb. of Stater in State Excellent.250.3100E(r) = (.25)(.31) + (.45)(.14) + (.25)(-.0675) + (0.05)(-0.52)E(r) = .0976 or 9.76%Scenario Returns: ExampleVariance (VAR):Variance an

9、d Standard DeviationStandard Deviation (STD):Scenario VAR and STDExample VAR calculation:2 = .25(.31 - 0.0976)2+.45(.14 - .0976)2 + .25(-0.0675 - 0.0976)2 + .05(-.52 - .0976)2 = .038Example STD calculation:Time Series Analysis of Past Rates of ReturnThe Arithmetic Average of rate of return:Geometric

10、 Average ReturnTV = Terminal Value of the Investmentg= geometric average rate of returnGeometric Variance and Standard Deviation FormulasEstimated Variance = expected value of squared deviationsGeometric Variance and Standard Deviation FormulasWhen eliminating the bias, Variance and Standard Deviati

11、on become:The Reward-to-Volatility (Sharpe) RatioSharpe Ratio for Portfolios:The Normal DistributionInvestment 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

12、 scenarios can be estimated using only the mean and the standard deviation.Figure 5.4 The Normal DistributionNormality and Risk MeasuresWhat if excess returns are not normally distributed?Standard deviation is no longer a complete measure of riskSharpe ratio is not a complete measure of portfolio pe

13、rformanceNeed to consider skew and kurtosisSkew and KurtosisSkewKurtosisFigure 5.5A Normal and Skewed Distributions Figure 5.5B Normal and Fat-Tailed Distributions (mean = .1, SD =.2)Value at Risk (VaR)A measure of loss most frequently associated with extreme negative returnsVaR is the quantile of a

14、 distribution below which lies q % of the possible values of that distributionThe 5% VaR , commonly estimated in practice, is the return at the 5th percentile when returns are sorted from high to low.Expected Shortfall (ES)Also called conditional tail expectation (CTE)More conservative measure of do

15、wnside risk than VaRVaR takes the highest return from the worst casesES takes an average return of the worst casesLower Partial Standard Deviation (LPSD)and the Sortino RatioIssues:Need to consider negative deviations separatelyNeed to consider deviations of returns from the risk-free rate.LPSD: sim

16、ilar to usual standard deviation, but uses only negative deviations from rfSortino Ratio replaces Sharpe RatioHistoric Returns on Risky PortfoliosReturns appear normally distributedReturns are lower over the most recent half of the period (1986-2009)SD for small stocks became smaller; SD for long-te

17、rm bonds got biggerHistoric Returns on Risky PortfoliosBetter diversified portfolios have higher Sharpe RatiosNegative skewFigure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000Figure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World, 1900-2000Figure 5.9 Probability of Investment Outcomes After 25 Years with a Lognormal DistributionTerminal Value with Continuous Compounding When the continuously compounded rate of return on an asset is normally distributed, the effect