Chap風(fēng)險與收益實用

會計學(xué)1Chap風(fēng)險與收益實用5-2RealandNominalRatesofInterest

真實與名義利率Nominalinterestrate:Growthrateofyourmoney金錢增長Realinterestrate:Growthrateofyourpurchasingpower購買力增長LetR=nominalrate,r=realrateandI=inflationrate.Then:第1頁/共43頁5-3EquilibriumRealRateofInterest

真實利率的均衡如何達(dá)成Determinedby:Supply供給Demand需求Governmentactions政府行為Expectedrateofinflation通脹預(yù)期第2頁/共43頁5-4Figure5.1DeterminationoftheEquilibriumRealRateofInterest第3頁/共43頁5-5EquilibriumNominalRateofInterest

名義利率的均衡如何達(dá)成Astheinflationrateincreases,investorswilldemandhighernominalratesofreturn通脹上升時投資者要求更高的名義利率IfE(i)denotescurrentexpectationsofinflation,thenwegettheFisherEquation:費雪方程式Nominalrate=realrate+inflationforecast第4頁/共43頁5-6TaxesandtheRealRateofInterest

稅收與真實利率Taxliabilitiesarebasedonnominalincome稅賦是基于名義收入的Givenataxrate(t)andnominalinterestrate(R),theRealafter-taxrateis:i是通脹率Theafter-taxrealrateofreturnfallsastheinflationraterises稅后真實收益被通脹侵蝕第5頁/共43頁5-7RatesofReturnforDifferentHoldingPeriods比較不同持有期利率ZeroCouponBond,Par=$100,T=maturity,P=price,rf(T)=totalriskfreereturn第6頁/共43頁5-8Example5.2AnnualizedRatesofReturn

收益的年化率第7頁/共43頁5-9Equation5.7EAREARdefinition:percentageincreaseinfundsinvestedovera1-yearhorizon投資1年所獲得的收益第8頁/共43頁5-10Equation5.8APRAPR:annualizingusingsimpleinterest簡單年化率第9頁/共43頁5-11Table5.1APRvs.EAR兩者相隔不大

前者計算簡便，后者更為精確第10頁/共43頁5-12Table5.2StatisticsforT-BillRates,InflationRatesandRealRates,1926-2009短期國庫券、通脹以及真實利率的統(tǒng)計數(shù)據(jù)第11頁/共43頁5-13BillsandInflation,1926-2009通貨膨脹與短期國庫券Moderateinflationcanoffsetmostofthenominalgainsonlow-riskinvestments.低風(fēng)險投資的大多數(shù)收益都會被通脹抵消AdollarinvestedinT-billsfrom1926–2009grewto$20.52,butwitharealvalueofonly$1.69.投資短期國庫券名義收益1:20.52，真實收益只有1:1.69Negativecorrelationbetweenrealrateandinflationratemeansthenominalraterespondslessthan1:1tochangesinexpectedinflation.名義利率在實際上也沒有隨著預(yù)期通脹的上升完全上漲，這說明真實利率被侵蝕的更厲害第12頁/共43頁5-14Figure5.3InterestRatesandInflation,1926-2009利率與通脹第13頁/共43頁5-15RiskandRiskPremiums風(fēng)險與溢價HPR=HoldingPeriodReturn持有期P0=Beginningprice初始價格P1=Endingprice結(jié)束價格D1=Dividendduringperiodone期間紅利RatesofReturn:SinglePeriod單期收益第14頁/共43頁5-16EndingPrice= 110BeginningPrice= 100Dividend= 4HPR=(110-100+4)/(100)=14%RatesofReturn:SinglePeriodExample

單期收益的例子第15頁/共43頁5-17Expectedreturnsp(s)=probabilityofastate狀態(tài)概率r(s)=returnifastateoccurs狀態(tài)下收益s=state狀態(tài)ExpectedReturnandStandardDeviation期望收益第16頁/共43頁5-18State

Prob.ofState rinState

Excellent .25 0.3100 Good .45 0.1400Poor .25 -0.0675Crash .05 -0.5200E(r)=(.25)(.31)+(.45)(.14)+(.25)(-.0675) +(0.05)(-0.52)E(r)=.0976or9.76%ScenarioReturns:Example案例第17頁/共43頁5-19Variance(VAR):VarianceandStandardDeviation方差與標(biāo)準(zhǔn)差StandardDeviation(STD):第18頁/共43頁5-20ScenarioVARandSTD如何計算ExampleVARcalculation:σ2=.25(.31-0.0976)2+.45(.14-.0976)2+.25(-0.0675-0.0976)2+.05(-.52-.0976)2=.038ExampleSTDcalculation:第19頁/共43頁5-21TimeSeriesAnalysisofPastRatesofReturn歷史收益的時間序列分析TheArithmeticAverageofrateofreturn:收益率的算術(shù)平均值第20頁/共43頁5-22GeometricAverageReturn幾何平均值TV=TerminalValueoftheInvestment最終投資價值g=geometricaveragerateofreturn收益的幾何平均值第21頁/共43頁5-23GeometricVarianceandStandardDeviationFormulas幾何方差與標(biāo)準(zhǔn)差EstimatedVariance=expectedvalueofsquareddeviations誤差平方的期望第22頁/共43頁5-24GeometricVarianceandStandardDeviationFormulas幾何方差與標(biāo)準(zhǔn)差公式Wheneliminatingthebias,VarianceandStandardDeviationbecome:標(biāo)準(zhǔn)差公式第23頁/共43頁5-25TheReward-to-Volatility(Sharpe)Ratio風(fēng)險收益比：夏普比SharpeRatioforPortfolios:組合的夏普比第24頁/共43頁5-26TheNormalDistribution

正態(tài)分布Investmentmanagementiseasierwhenreturnsarenormal.如果收益正態(tài)分布投資管理就比較輕松Standarddeviationisagoodmeasureofriskwhenreturnsaresymmetric.如果收益對稱的話標(biāo)準(zhǔn)差是風(fēng)險的一個好的度量Ifsecurityreturnsaresymmetric,portfolioreturnswillbe,too.如果證券收益對稱那么組合的收益也是對稱的Futurescenarioscanbeestimatedusingonlythemeanandthestandarddeviation.未來的收益就可以只用均值和方差估計出來第25頁/共43頁5-27Figure5.4TheNormalDistribution

正態(tài)分布第26頁/共43頁5-28NormalityandRiskMeasures

正態(tài)性與風(fēng)險測度Whatifexcessreturnsarenotnormallydistributed?如果超額收益不是正態(tài)分布Standarddeviationisnolongeracompletemeasureofrisk標(biāo)準(zhǔn)差不再是風(fēng)險的完全測度Sharperatioisnotacompletemeasureofportfolioperformance夏普比也就不再是組合表現(xiàn)的完全測度Needtoconsiderskewandkurtosis需要考慮的偏度和峰度第27頁/共43頁5-29SkewandKurtosis

偏度與峰度SkewEquation5.19KurtosisEquation5.20第28頁/共43頁5-30Figure5.5ANormalandSkewedDistributions正態(tài)與有偏分布第29頁/共43頁5-31Figure5.5BNormalandFat-TailedDistributions(mean=.1,SD=.2)正態(tài)與厚尾分布第30頁/共43頁5-32ValueatRisk(VaR)在險價值A(chǔ)measureoflossmostfrequentlyassociatedwithextremenegativereturns度量一定概率下發(fā)生極端負(fù)值所造成的損失VaRisthequantileofadistributionbelowwhichliesq%ofthepossiblevaluesofthatdistribution在險價值的另一個名字是分布的分位數(shù)The5%VaR,commonlyestimatedinpractice,isthereturnatthe5thpercentilewhenreturnsaresortedfromhightolow.通常估計的5%的VaR是指收益從高到底排列后的，有95%的收益都會好過的那一個值。第31頁/共43頁5-33ExpectedShortfall(ES)預(yù)期損失Alsocalledconditionaltailexpectation(CTE)也叫做條件尾部期望MoreconservativemeasureofdownsideriskthanVaR比起在險價值是下側(cè)風(fēng)險更為保守的風(fēng)險測度VaRtakesthehighestreturnfromtheworstcases在險價值取所有最壞情況的最高值來表達(dá)風(fēng)險EStakesanaveragereturnoftheworstcases預(yù)期損失取所有最壞情況的平均值第32頁/共43頁5-34LowerPartialStandardDeviation(LPSD)

andtheSortinoRatio下標(biāo)準(zhǔn)偏差與Sortino比Issues:分布不對稱帶來的問題Needtoconsidernegativedeviationsseparately需要單獨考察負(fù)值偏差Needtoconsiderdeviationsofreturnsfromtherisk-freerate.需要考察收益偏離無風(fēng)險收益的水平LPSD:similartousualstandarddeviation,butusesonlynegativedeviationsfromrf下標(biāo)準(zhǔn)和標(biāo)準(zhǔn)差的計算一樣，不同的是它只用負(fù)收益為樣本SortinoRatioreplacesSharpeRatio在這種情況下我們用Sortino比率代替夏普比衡量風(fēng)險收益比第33頁/共43頁5-35HistoricReturnsonRiskyPortfolios

風(fēng)險組合的歷史收益Returnsappearnormallydistributed收益通常是正態(tài)分布的Returnsareloweroverthemostrecenthalfoftheperiod(1986-2009)近年來收益降低了SDforsmallstocksbecamesmaller;SDforlong-termbondsgotbigger小型股的標(biāo)準(zhǔn)差變小，但是長期債券的標(biāo)準(zhǔn)差變大第34頁/共43頁5-36HistoricReturnsonRiskyPortfolios

風(fēng)險組合的歷史收益BetterdiversifiedportfolioshavehigherSharpeRatios分散較好的組合有較高的夏普比Negativeskew右偏，或者說向負(fù)值偏斜第35頁/共43頁5-37Figure5.7NominalandRealEquityReturnsAroundtheWorld,1900-2000名義與真實收益第36頁/共43頁5-38Figure5.8StandardDeviationsofRealEquityandBondReturnsAroundtheWorld,1900-2000股票與債券真實收益的標(biāo)