risk return and cost of capital风险回报和资本成本.ppt

1、Return, Risk, and the Security Market Line,Types of Returns Expected Returns and Variances Portfolios Announcements, Surprises, and Expected Returns Risk: Systematic and Unsystematic Diversification and Portfolio Risk Systematic Risk and Beta The Security Market Line The SML and the Cost of Capital

2、Summary and Conclusions,Types of Returns,Total Monetary return = Dividend Income + Capital Gain Eg an investment of 1000 rises in value to 1500 providing a capital gain of 500. Over the same period the dividend income is 5% = 50. Total return is then 500 +50 = 550. Total monetary return is an absolu

3、te measure of returns. It tells you how much money you have made in s. It is often more useful to know the Percentage Return. The Percentage Return is the total monetary return divided by the amount of capital invested. Percentage Return = Dividends + Capital Gains amount invested OrRit = Dit + (Pit

4、 Pit-1) = Div. Yield + % capital gain Pit-1,Expected Returns and Variances: Basic Ideas,The quantification of risk and return is a crucial aspect of modern finance. It is not possible to make “good” (i.e., value-maximizing) financial decisions unless one understands the relationship between risk and

5、 return. Rational investors like returns and dislike risk. Consider the following proxies for return and risk: Expected return - weighted average of the distribution of possible returns in the future. Variance of returns - a measure of the dispersion of the distribution of possible returns in the fu

6、ture. How do we calculate these measures?.,Calculating the Expected Return.Example 1,s E(R) = (pi x Ri) i =1 pi Ri Probability Return in i pi x Ri State of Economy of state i state i +1% change in GNP .25 -5% i=1 -1.25% +2% change in GNP.50 15% i=2 7.5% +3% change in GNP .25 35% i=3 8.75% Expected r

7、eturn = (-1.25 + 7.50 + 8.75) = 15%,Calculating the Variance(Example 1 of Calculating the expected return),Var(R) i (Ri E(R)2 pi x (Ri E(R)2 i=1 (-0.05-0.15) 2 = 0.04 0.25*0.04 = 0.01 i=2 (0.15-0.15) 2 = 0 0.5*0 = 0 i=3 (0.35-0.15) 2 = 0.04 0.25*0.04 = 0.01 Var(R) = .02 What is the standard deviatio

8、n?,Expected Returns and VariancesExample 2,State of the ProbabilityReturn onReturn oneconomyof stateasset Aasset B Boom 0.4030%-5% Bust 0.60-10%25% 1.00 A.Expected returns E(RA) = 0.40 x (.30) + 0.60 x (-.10) = .06 = 6% E(RB) = 0.40 x (-.05) + 0.60 x (.25) = .13 = 13%,Example: Expected Returns and V

9、ariances (concluded),B.Variances Var(RA) = 0.40 x (.30 - .06)2 + 0.60 x (-.10 - .06)2 = .0384 Var(RB) = 0.40 x (-.05 - .13)2 + 0.60 x (.25 - .13)2 = .0216 C.Standard deviations SD(RA) = .0384 = .196 = 19.6% SD(RB) = .0216 = .147 = 14.7%,Calculating Expected Returns and Variance in practice,The most

10、common method is to use a time series of returns calculated from past prices and dividends.,Calculating Expected Returns and Variance in practice (2),E(Ri) is assumed to be equal to the sample average return = ( 0.0116+0.0046+0.0092-0.0136+0-0.0345 )/6 = -0.00378 To calculate the variance we calcula

11、te the deviation for each days return from the expected return, square to make it positive and then divide by n-1. In this case n=6.,Calculating Expected Returns and Variance in practice (3),Measuring risk,If we were to plot the daily returns on a security over a long period then it might look somet

12、hing like a normal distribution (picture next slide) What we want to do is to summarise this picture as simply as possible. The mean is the expected return, the spread or variation is the standard deviation or variance. We argue that this spread represents risk to investors and hence that the St. De

13、v. or variance is a measure of the risk of a share. In fact return distributions dont usually look exactly like this. They tend to have a truncated left tail and a longer right tail. Variance may not be the best measure of risk.,Describing a distribution,Portfolio Expected Returns and Variances,What

14、 we have done so far is describe the risk and return of individual securities. We also want to be able to describe the risk and return of portfolios of securities. We have two equivalent alternatives open to us. Component - We can determine the return and risk of the portfolio by combining the retur

15、ns and risks of the securities that make up the portfolio. Security - We can treat the portfolio as just another security and calculate its return and risk as we have been doing. Both of these approaches give the same answer but the first allows us to see how individual securities affect the return

16、and risk of a portfolio.,Portfolio Expected Returns and Variances(using returns from Example 2),Portfolio weights: put 50% in Asset A and 50% in Asset B: State of the ProbabilityReturnReturnReturn oneconomyof stateon Aon Bportfolio Boom0.4030%-5%12.5% Bust0.60-10%25%7.5% 1.00,Example: Portfolio Expe

17、cted Returns and Variances (continued),Calculate expected returns: Security approach E(RP) = 0.40 x (.125) + 0.60 x (.075) = .095 = 9.5% Component approach E(RP) = .50 x E(RA) + .50 x E(RB) = 9.5% Calculate variance of portfolio: Security approach Var(RP) = 0.40 x (.125 - .095)2 + 0.60 x (.075 - .09

18、5)2 = .0006 Portfolio approach The sum of the variances is not the variance of the portfolio Var (RP) .50 x Var(RA) + .50 x Var(RB),Ft this week,Olympus saga continues resignation of President, open letter by major shareholder, questions (at last!) by Japanese Press and Government. Eurozone the deal

19、 more of the same, bigger (voluntary) haircuts, more austerity but the debtor strikes back (Greek referendum). MF Global collapse broker-dealer suffering from eurozone ratings downgrades ($6.3bn exposure). Management greed huge increase in senior management pay over last year.,The Story so far,Our a

20、im is to relate return to risk. Basic principle is that investors require a reward for taking on risk. The larger the risk, the larger the reward. But how are we to measure risk and return? Many different types of risk. We concentrate on risk as perceived by the capital markets. The price of a share

21、 at any time reflects everything that is known about the company. Suggests that we can use price changes to provide information about the company. By examining the distribution of percentage price changes (returns) we can determine the likely or expected return, and the dispersion of returns that mi

22、ght occur.,The story so far (2),An obvious measure of expected return is the arithmetic mean. A measure of dispersion is the variance. This is used as a measure of the risk of a share. The variance is a reasonable measure if the distribution of returns is symmetric. Most companies are not held in is

23、olation but are held as part of a portfolio. We use two share portfolios to demonstrate how risk changes. The proportion of each company in the portfolio is known as the portfolio Weight. Our interest is in how one company relates to another. We are concerned about the joint distribution of returns.

24、,Joint Distribution of returns,probabilityReturn on Security X Return on Security Y,Covariance and Correlation,The Covariance is a measure of how the two securities are related. Similar to Variance but uses cross deviations. Variance = E (RAt E(RAt) (RAt E(RAt) Covariance = average (deviation of ret

25、urn on A from its mean) * (deviation of return on B from its mean) CAB = E (RAt E(RAt)(RBt E(RBt) Correlation is a standardised Covariance. Correlation between A and B is the Covariance between A and B divided by the standard deviation of A times the standard deviation of B. AB = CovAB/ A B,Covarian

26、ce and Correlation,The risk of a portfolio is comprised of the risk of the individual securities plus the correlation between them. If there are two securities then the risk of the portfolio can be calculated from the variance of each security plus the correlation between them. For two securities we

27、 have: p2 = X12Var1+X22Var2+2X1X2Cov12 Remember:Cov12 = 1 2 12 p2 = X12 12 + X22 22 + 2X1X2 1 2 12 Cov12 = E(R1t - E(R1t)(R2t - E(R2t) = E(R2t - E(R2t)(R1t - E(R1t) = Cov21,Two security Portfolio Selection Example,Rpt= X1R1t + X2R2t E(Rpt)= E(X1R1t + X2R2t) = X1E(R1t) + X2E(R2t) p2=E(Rpt - E(Rpt)2 p

28、2=EX1R1t + X2R2t - (X1E(R1t) + X2 E(R2t)2 p2=EX1(R1t - E(R1t) + X2(R2t - E(R2t)2 From algebra we know that (a+b)2 = a2 + b2 + 2ab p2 = X12E(R1t - E(R1t)2+X22E(R2t - E(R2t)2 +2X1X2E(R1t -E(R1t)(R2t - E(R2t) = X12 12 + X22 22 + 2X1X2Cov12,How Correlation affects risk (2 security example),How Correlati

29、on affects risk (2 security example),How Correlation affects risk (2 security example),The Effect of correlation on Portfolio Variance,Stock A returns,0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05,0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03,Stock B returns,0.04 0.03 0.02 0.01 0 -0.01 -0.

30、02 -0.03,Portfolio returns:50% A and 50% B,Covariance and Correlation: more than 2 securities,One way of thinking of the covariance of securities within a portfolio is to visualise a matrix of securities. Each security must pair with each other. If the numbers are the same it is a variance, otherwis

31、e a covariance. eg if there are five securities we can think of:,Components of Portfolio Risk,Variance Covariance Expression,Covariance and Correlation (cont.),Impact of correlation (covariance),Standard Deviations of Annual Portfolio Returns,( 3) (2)Ratio of Portfolio (1)Average StandardStandard De

32、viation to Number of StocksDeviation of AnnualStandard Deviation in PortfolioPortfolio Returns(%)of a Single Stock 1 49.24 1.00 1023.93 0.49 5020.20 0.41 10019.69 0.40 30019.34 0.39 50019.27 0.39 1,00019.21 0.39 Figures from Table 1 in Meir Statman, “How Many Stocks Make a Diversified Portfolio?” Jo

33、urnal of Financial and Quantitative Analysis 22 (September 1987), pp. 35364, and derived from E. J. Elton and M. J. Gruber, “Risk Reduction and Portfolio Size: An Analytic Solution,” Journal of Business 50 (October 1977), pp. 41537.,Portfolio Diversification,Average annualstandard deviation (%),Numb

34、er of stocksin portfolio,Diversifiable risk,Nondiversifiablerisk,49.2,23.9,19.2,1,10,20,30,40,1000,Diversification: analytical solution,Diversification: analytical solution (2),Diversification: analytical solution (3),If we were to look at the case where covariances are not equal to zero we would fi

35、nd that the risk of a large portfolio of stocks is approximately equal to the average covariance between all the stocks. P2CovAV,Peter Bernstein on Risk and Diversification,“Big risks are scary when you cannot diversify them, especially when they are expensive to unload; even the wealthiest families

36、 hesitate before deciding which house to buy. Big risks are not scary to investors who can diversify them; big risks are interesting. No single loss will make anyone go broke . . . by making diversification easy and inexpensive, financial markets enhance the level of risk-taking in society.” Peter B

37、ernstein, in his book, Capital Ideas,How correlation affects risk: The Efficient Frontier,FT this week,Olympus admits wrong doing. Eurozone many interesting articles highlighting the power of Greece, German role and interest, dangers to Italy and others. Focus on one article: Robert Jenkins, Insight

38、 (nov.8) - Greek restructuring exit from the eurozone Greek govt decides on exit. Greek citizens and companies withdraw euro deposits whilst they are still euros. Foreign lenders stop lending and recall loans as quickly as possible. Govt. announces a new drachma. Capital controls are introduced. Gov

39、t debt is redenominated in drachma.,Olympus share price,FT this week (cont) Greek restructuring,Value of the drachma plunges, Greek inflation soars. Disputes over private sector debt. Are they in drachma or euros? If drachma then foreign banks have a problem asset values have fallen. If in euros the

40、n Greek borrowers have a problem Contagion commences. Portugese citizens think it might happen to them and move out euros from the banks. Similar moves in several other countries. European banks in difficulties because of exposure to euro debt of various countries with likely difficulties. Counterpa

41、rty risk means market in bank loans dries up. Bank lending halts! Banks collapse unless Govt rescue them.,The Story to date,The risk of a portfolio depends on the Covariance or Correlation between assets. Variance is important for an individual asset but becomes less and less important as a portfoli

42、o includes more and more stocks. The risk of a portfolio depends on the average covariance between stocks. The relationship between risk and return can be represented graphically by a quadratic frontier. The best combinations of risk and return are on the Efficient Frontier. The shape of the frontie

43、r arises from the covariance between assets.,How Correlation affects risk: a risk free asset,How Correlation affects risk: a risk free asset (2),Tobins Separation Theorem,Simplifying our Risk Measure,Our message so far has been that when we add securities together risk is affected by the correlation

44、 (covariance) between them. Because securities are less than perfectly correlated, risk is reduced. Whilst this is useful as a concept it is operationally very difficult to use. The number of correlations that we need to consider to construct optimal portfolios using this sort of approach is very la

45、rge. We need to find some other measure of risk that will enable us to simplify the problem. One such measure is the beta of a security or portfolio. The beta of a security can be thought of as: the (standardised) sum of the securitys covariance with all securities Since all securities is just anoth

46、er way of saying the market, the beta of a security is: the (standardised) covariance of the security with the market,Estimating Beta,Beta is usually estimated using linear regression. Beta is an output from the Market Model. This assumes that there is a linear relationship between the return on the

47、 market and the return on a share. Returns on a share are regressed against returns on a market index. Rit = ai + bi Rmt cit ai is the alpha of share I bi is the beta of share I,Beta Coefficients for Selected Companies (Table 10.7),Beta Company Coefficient ( i) Alcatel-Lucent1.44 LOreal0.45 SAP0.56

48、Siemens1.51 Daimler1.25 Philips Electron0.92 Renault1.64 Volkswagen0.40,Source: Hillier, Ross, Westerfield, Jaffe, Jordan. Corporate Finance.,Portfolio Beta Calculations,Portfolio Beta has a very desirable characteristic. It is the (weighted) average of the individual betas. AmountPortfolioStockInve

49、stedWeightsBeta (1)(2)(3)(4)(3) x (4) Haskell Mfg. $ 6,00050%0.900.450 Cleaver, Inc.4,00033%1.100.367 Rutherford Co.2,00017%1.300.217 Portfolio$12,000100%1.034,Cash (riskless asset), Portfolio Expected Returns and Betas,Assume you wish to hold a portfolio consisting of a risky asset A and cash (a ri

50、skless asset). Given the following information, calculate portfolio expected returns and portfolio betas, letting the proportion of funds invested in asset A range from 0 to 125%. Asset A has a beta ( ) of 1.2 and an expected return of 18%. The return on cash at the Central Bank (risk-free rate) is

51、7%. Asset A weights: 0%, 25%, 50%, 75%, 100%, and 125%.,Cash (riskless asset), Portfolio Expected Returns and Betas,Proportion ProportionPortfolio Invested in Invested inExpected Portfolio Asset A (%) Risk-free Asset (%) Return (%) Beta 01007.000.00 25759.750.30 505012.500.60 752515.250.90 100018.00

52、1.20 125-2520.751.50 Plot this and measure the slope - (.18-.07)/1.2 = 0.092. This is the risk premium per unit of systematic risk.,Cash (riskless asset), Portfolio Expected Returns and Betas,Expected return 18% 7% 0 1.2 beta Slope = (.18-.07)/1.2 = .092,Return, Risk, and Equilibrium,Key issues: Wha

53、t is the relationship between risk and return? What does security market equilibrium look like? The fundamental conclusion is that the ratio of the risk premium to beta is the same for every asset. In other words, the reward-to-risk ratio is constant and equal to E(Ri ) - Rf slope = Reward/risk rati

54、o = i,Return, Risk, and Equilibrium (concluded),Example: Asset A has an expected return of 12% and a beta of 1.40. Asset B has an expected return of 8% and a beta of 0.80. Are these assets valued correctly relative to each other if the risk-free rate is 5%? a.For A, (.12 - .05)/1.40 = _ b.For B, (.0

55、8 - .05)/0.80 = _ What would the risk-free rate have to be for these assets to be correctly valued? (.12 - Rf)/1.40 = (.08 - Rf)/0.80 Rf = _,The Capital Asset Pricing Model,The Capital Asset Pricing Model (CAPM) - an equilibrium model of the relationship between risk and return. What determines an a

56、ssets expected return? The risk-free rate - the pure time value of money The market risk premium - the reward for bearing systematic risk The beta coefficient - a measure of the amount of systematic risk present in a particular asset The CAPM: E(Ri ) = Rf + E(RM ) - Rf x i,Capital Asset Pricing Mode

57、l (2),Expected return on asset i is a linear function of the risk free rate and the assets marginal risk (beta) times the expected risk premium on the market. E(Ri) = Rf + (E(Rm) - Rf ) i i is the beta of a security. It is derived from the market model and represents the marginal risk of an asset. T

58、he investor is assumed to be rational. As such the investor will know that by holding a diversified portfolio of assets s/he can get rid of all the unsystematic risk. The investor cant however, get rid of the systematic or market risk. In consequence to bear market risk the investor demands compensa

59、tion related to the amount of market risk. All asset returns are related to their risk. In equilibrium all assets will plot on the straight line given representing the CAPM. The straight line is known as the Security Market Line.,The Capital Asset Pricing Model: assumptions (3),Investors select efficient portfolios Investors have the same decision horizon and over this period means and variances exist.