证券投资学英文课件：Lecture_04

1、InvestmentLecture 4Review 22111yCyDyVV)1)(1 (1)1 (12, 1122fssImmunizationnExample: Suppose an insurance company must make payments to a customer of $10 million in 1 year and $4 million in 5 years nSuppose the yield curve is flat at 10%nIf the company wants to fully fund and immunize its obligation w

2、ith 1 zero, what should it buy? What will the zero cost?Immunizationn(Example) Duration of payments: nvalue = nweight of 10 = nweight of 4 = nduration = nSo we should buy zeros with a maturity of _ yearsImmunizationn(Example) So we will buy zeros of maturity _, but how many should we buy?nThe market

3、 value of our zeros must be set equal to the market value of the obligationnSo buy _ worth of zerosnThis works out to _ of face value in zerosnMaybe they should buy 1 and 4 year zerosEquity InvesmentnReturnnRisknPortfolioRisknRisky return:nThis notation means realized returns may take on different v

4、alues (ie. D and P are risky therefore r is risky).1PPDr01RisknEg. Suppose a stock pays no dividends and tomorrows price is determined on the basis of the flip of two coins. Todays price is 900 and tomorrows price is 1000 x (# heads).nThis stock is risky.nWe dont know what tomorrows price will be bu

5、t we do know what it can be.Risk1PrProbability0-100.25100011.502000122.25nPrice can take on three possible values with different probabilities for each value.RisknWe also know the distribution of the stocks returns:neg. What is the probability that the above stocks return is negative?RisknIn reality

6、 a stocks return may take on any value and its distribution of returns is subjective (ie. Its based on beliefs).nEg. What does the probability distribution of Baidu returns look like?nRisk will have to be related to the distribution of returns.RisknMore risky returns demand higher expected returns:n

7、Expected Stock Return = riskless rate + risk premiumnBUT - how do we measure risk?Some StatisticsnWe need to calculate expected return and variance of stock returns:2/ 1r all2r all) rVar() rSDev() rp( )rE(-r() rVar() rp( r) rE(RisknFrom previous example:Properties of E() and Var()rVar()rVar(ara)rVar

8、(a)rE(a)rE(a)raE()rE(areturn) (eg. random - rrandom)(not constant - a2Scenario Analysis:Peace & WarnSuppose you are a shareholder of Tiffany (TIF) and of Lockheed Martin (LKM)nWhat is the return and the risk of the stocks?StateProbabilityReturn TIFReturn LKMPeace0.7530%0%War0.25-40%20%Distributi

9、on of ReturnsMeanVarianceStd. Dev.CovarianceCorrelation SssRpREssSsssRERRERpRRCOV, 2, 2, 1, 121,)(RVAR SsssRERpRVAR221212, 1),(RRCOVHistorical Returns (1999-2004)Distribution of Historical ReturnsArithmetic MeanGeometric MeanVarianceStd. Dev.CovarianceCorrelationttRRTRVAR211)()(RVAR21212, 1),(RRCOVt

10、tRTR122,11 ,2111),(RRRRTRRCOVtttTTTTVVRRRGR/10/121111Summary Statistics of Monthly Returns (1999-2004)TiffanyLockheed MartinStandard & Poors 500Arith. Mean2.12%1.23%0.03%Geom. Mean1.20%0.78%-0.08%Variance1.86%20.86%20.21%213.65%9.26%4.60%(Ri,RTIF)1(Ri,RLKM)0.031(Ri,RSPY)0.74-0.111A Simple Portfo

11、lionEg. Suppose we combine the stock above with a t-bill one-year t-bill which is yielding 5%. What is the mean, variance and standard deviation of a portfolio where 70% of the investment is in the stock and 30% is in the t-bill?A More Complicated PortfolionEg. Suppose we combine the S&P 500 ind

12、ex with a t-bill one-year t-bill which is yielding 5%. What is the mean, variance and standard deviation of a portfolio where 70% of the investment is in the index and 30% is in the t-bill? The market risk premium is 5% and the annual standard deviation is 20%.Portfolios of Risky Assets)r, rCov(xx2)

13、rVar( x )rVar(x)r x rVar(x)rE( x )rE( x)r x rE(xiasset on return random - riin investment of proportion - x2121222121221122112211iiProperties of Covariance)raVar()r, r Cov(a)r, rCov( b a)rb, r Cov(a)r, rCov()r-r )(rr(E)r, rCov(constant - b a,iasset on return random - r1112121211221221121iPortfolios

14、of Risky AssetsnSuppose we invest $120 in IBM stock and $180 in Bristol-Myers stock.nDetermine the portfolio mean. (BM=21%, IBM=17%)nDetermine the portfolio standard deviation. (BM=21%, IBM=31%, =0.135)A Simple Method for Calculating Portfolio CovarianceStock 1Stock 2Stock 3Stock 1Stock 2Stock 3)Var

15、(r121X)r ,Cov(r1212XX)r ,Cov(r1313XX)r ,Cov(r2121XX)Var(r222X)r ,Cov(r3131XX)r ,Cov(r3232XX)Var(r323X)r ,Cov(r2323XXPortfolio VariancenSuppose portfolio P is composed of N assets, let w denote portfolio weights, r denote components returnnThen 1 12 23 311122331.( )( )( )( ) .( )( )Npn ni iiNpnniiirw

16、rwrwrw rwrE rwE rw E rwE rw E rwE rPortfolio VariancenThe variance of P is:222112111,11() ( ) ( ) ( )( ) pppNNi iiiiiNiiiiNNijiijjijNNiji jijE rE rEwrwE rEw rE rEwwrE rrE rwwExamplenCalculate the expected return and variance of a portfolio with $1000 invested in each of XOM, WMT and AMZN.Annualized

17、Covariance Matrix (Based on last 14 returns)XOMWMTAMZNXOM0.015683 -0.0010510.002407WMT -0.0010510.0476630.023407AMZN0.0024070.0234070.130953Expected Returns0.063580.2358250.429066Std Dev0.164595Exp Ret0.242823Asset AllocationnAsset Allocation is the portfolio choice among broad investment classes:nO

18、ne risky asset and one risk-free assetnTwo risky assetsnTwo risky assets and one risk-free assetnMany risky assets and one risk-free assetAsset ClassesnRisky AssetsnStocks (S&P Comp. Index)nMean Real Return: 10%nStandard Deviation: 20%nBondsnMean Real Return: 4%nStandard Deviation: 10%nCorrelati

19、on with Real Stock Return: 0.2nT-Bills (Risk-free asset)nReal Return: 1%Portfolio of Risk-Free Asset and Risky AssetnWhat is the expected return and the standard deviation of a portfolio that invests:nw in stocksn1-w in the risk-free asset?Capital Allocation LinenExpected Return of Portfolio:nStanda

20、rd Deviation of Portfolio:nSubstituting for w, gives the Capital Allocation Line (CAL):FrwrwErEp1)()(pwpFSFrrErrE)()(Capital Allocation LineCapital Allocation LineCapital Allocation Line00.020.040.060.080.10.120.1400.050.10.150.20.250.30.35Standard DeviationExpected Return100% Stocks100% T-Bills50%

21、Stocks50% T-BillsCAL125% Stocks-25% T-BillsCapital Allocation LinenThe Capital Allocation Line shows the risk-return combinations available by changing the proportion invested in a risk-free asset and a risky assetnThe slope of the CAL is the reward-to-variability ratioRisk AversionnNow the question

22、 is, which risk-return combination along the CAL do you want?nTo answer this we need to bring your preferences for risk into the picturenWe will use indifference curves to represent risk aversionnIndifference curves represent utility functionsnInvestors view of risknRisk AversenRisk NeutralnRisk See

23、kingnExample: Risk Aversion & UtilityUtilitynUtilitynUtility FunctionU = E ( r ) .005 A 2nA measures the degree of risk aversionRisk Aversion and Value: The Sample Investment U = E ( r ) - .005 A 2 = 22% - .005 A (34%) 2Risk AversionAUtilityHigh5-6.903 4.66 Low 116.22T-bill = 5%Dominance Princip

24、le1234Expected ReturnVariance or Standard Deviation 2 dominates 1; has a higher return 2 dominates 3; has a lower risk 4 dominates 3; has a higher returnDominance PrincipleA dominates B if andand at least one inequality is strict)()(BArErEBAIndifference CurvesAsset AllocationnNow we can combine the

25、indifference curves with the capital allocation linenIf investors are maximizing their utility, they will choose the highest possible indifference curvenThe highest curve is tangent to the CALAsset AllocationUtility FunctionWe have:So:fprwrwErE1)()(222005. 0)1 ()( 005. 0)(pfpAwrwrwEArEUOptimal Holdi

26、ngFirst-order condition to maximize U:Solving for w:An example: if 2 ()0.010pfpdUE rrAydw2 ()/0.01pfpwE rrA%22%15)(%7, 4ppfrErA2(157)/(0.01 4 22 )0.41w Capital Allocation LinenIn practice, investors cannot borrow at the T-Bill ratenHow does the CAL change if the borrowing rate is 4%?Capital Allocati

27、on Line with Higher Borrowing Rates00.020.040.060.080.10.120.1400.050.10.150.20.250.30.35Standard DeviationExpected Return100% Stocks100% T-Bills50% Stocks50% T-BillsCAL125% Stocks-25% T-BillsPortfolio ChoicenThe investor chooses a point on the CAL that maximizes the utilitynThe choice is determined

28、 by the risk aversion of investorsnRisk-averse investors will invest more in the risk-free assetnRisk-tolerant investors will invest more in the risky assetAsset Allocation with Two Risky AssetsnWhat is the expected return and the standard deviation of a portfolio that invests:nw in Stocksn1-w in Bo

29、nds?Risk-Return TradeoffnExpected Return of Portfolio:nVariance of PortfolionSubstituting for w, gives the Capital Investment Opportunity Set)(1)()(BSPrEwrwErE )()(,121,1212222BSSBSSBSSSSBSSBSSSPRSDRSDRRCORRwwRVARwRVARwRRCOVwwRVARwRVARwRVARPortfolio Frontier with Stocks and Bonds00.020.040.060.080.1

30、0.120.1400.050.10.150.20.250.30.35Standard DeviationExpected Return100% Bonds / 0% Stocks0% Bonds / 100% Stocks86% Bonds / 14% Stocks 150% Bonds / -50% Stocks-50% Bonds / 150% StocksPortfolio FrontiernThe portfolio frontier depicts the feasible portfolio choices for investors holding stocks and bond

31、snThe minimum variance portfolio includes 86% bonds and 14% stocksnPortfolios below the minimum variance portfolio are inefficientnThe portfolio frontier above the minimum variance portfolio is called efficient frontierCorrelation: Two Risky AssetsnTo see the importance of correlation, we will look

32、at the set of feasible portfolios under three different assumptions:n1) AB = 1n2) AB = -1n3) AB = 0nThen we will discuss the intermediate casesPerfect CorrelationnCase 1: AB = 1nWhen AB = 1 we can simplify the variance:nP2 = (xA + (1-x) B)2nThe two relevant equations are therefore:nE (rP) = xE(rA) +

33、 (1-x) E(rB)nP = xA + (1-x)BnThese two equations give us the set of feasible portfoliosPerfect CorrelationPerfect CorrelationPerfect Negative CorrelationnCase 2: AB = -1nWhen AB = -1 we can simplify variance: P2 = (xA - (1-x) B)2nThe two relevant equations are therefore:n E(rP) = xE(rA) + (1-x) E(rB

34、)nP = |xA - (1-x) B|nThese two equations give us the set of feasible portfoliosPerfect Negative CorrelationPerfect Negative CorrelationNo CorrelationnCase 3: AB = 0nWhen AB = 0 we cannot simplify the variance equationnHowever, we can graph the set of feasible portfoliosNo CorrelationNo CorrelationAs

35、signmentnList two close-end funds in China and draw the graphic of their historical performancesnChoose two stocks in any fund above, and present their average return and risknSuppose you equally invested in those two stocks, present your portfolios average return and riskThe Benefits of Diversifica

36、tionnHow many stocks do you need to hold a well-diversified portfolio?nAssume that all the assets have the same standard deviation of 60% per year.nAssume that the correlation of the returns between two assets is 30%.nAssume that you invest the same amount in the different assets.The Benefits of Div

37、ersificationnThe variance of the return of a portfolio that includes N different assets depends on the weight w and on the covariances :),(),(),(),(),(),(),(),(),(),()(22112222221212112121111111NNNNNNNNNNNNNiNjjijiPrrCovwwrrCovwwrrCovwwrrCovwwrrCovwwrrCovwwrrCovwwrrCovwwrrCovwwrrCovwwrVarThe Benefits of DiversificationnIf you invest equally in N different assets, then the weights