1、1How Are Risk and the Cost of Capital Related?An investment results in a realized returnWhen the investment is made, the investor evaluates the expected returnInvestment Initial0InflowCash Gain Capital01PDPPr001)(PDPPrrE2Opportunity Cost and RiskWhen investments are risky, the opportunity cost must

2、be based on comparable risk. We are concerned with how to measure risk (so we can be specific about the term comparable) how this measure of risk influences the required return.3Sears and HBC are in the Same Industry are they Comparable?SearsSearsHBCTSEPrices051015202530354045Dec-95Apr-96Aug-96Dec-9

3、6Apr-97Aug-97Dec-97Apr-98Aug-98Dec-98Apr-99Aug-99Dec-99Apr-00Aug-00Dec-00Apr-01Aug-01HBCTSESears4One $ invested in each stockHBCTSE012345678Dec-95Apr-96Aug-96Dec-96Apr-97Aug-97Dec-97Apr-98Aug-98Dec-98Apr-99Aug-99Dec-99Apr-00Aug-00Dec-00Apr-01Aug-01SearsHBCTSE5Rate of Return: These Companies Are in t

4、he Same Industry But Are They of Equal Risk? -0.5-0.4-0.3-0.2-0.30.4Jan-96May-96Sep-96Jan-97May-97Sep-97Jan-98May-98Sep-98Jan-99May-99Sep-99Jan-00May-00Sep-00Jan-01May-01Sep-01HBCSearsTSE6Average Returns and Std. Deviation (monthly)HBCSearsTSEAverage Return0.012%-0.40%2.333%1.56%0.800%0.64

5、%StandardDeviation8.86%9.24%12.17%12.53%5.58%5.70%ArithmeticGeometric7Asset Pricing ModelWith future price and dividends given, a return implies a price, P0Issues:How do we measure risk so that we can compare similar investments? Use statistics.How do we evaluate the reward that must be paid to enti

6、ce investors to hold risky securities? Look at investor preferences.premiumrisk )(asset in investingfor reward)(fiirrEirE001)(PDPPrirE8Statistics: A Quick ReviewA random variable takes on a numerical value that depends on chance. A random return could be denoted ri. It is assumed that it will take o

7、n one of n specific values, A distribution assigns a probability to each possible value that a random variable can take on.Probability must be non negative and must sum to one.A very general description of risk.nkr.1for ki)Prob( ikikirr 9Statistics: A Quick ReviewThe mean, expected value, or expecta

8、tion of a random variable is a weighted sum of the possible outcomes.This is a measure of how big, on average, the random variable is.n1kir )(kikiirrE10Statistics: A Quick ReviewThe variance, of a random variable is the expectation of the squared deviation from the mean.A measure of how dispersed th

9、e distribution is.Example: A constant is always equal to its expected value. It has a variance of zero.The standard deviation (s(r) is the square root of the variance.n1k22)()( )(kiikiikiirErrErErVar11An IllustrationExpected Return= .1(50) + .2(30) + .4(10) + .2(-10) + .1(-30)=10%Variance= (50-10)2(

10、.1) + (30-10)2(.2) + (10-10)2(.4) + (-10-10)2(.2) + (-30-10)2(.1) =480% Standard Deviation = 21.9%00.050.4-30%-10%10%30%50%12Multiple Random VariablesA joint distribution for two random variables, ri and rj is a probability, , attached to each possible pair of outcomes. The Cova

11、riance, sij is the expectation of the product of the deviation of each random variable from its meanNote: sii =nklmkmijjmjikijjiiijjirErrErrErrErErrCov11)()()()(),(s2iskmij13Multiple Random VariablesThe Correlation Coefficient, rij is the covariance scaled by the product of the standard deviations.T

12、his value is always between 1 and +1 and tells us how much the two variables move together. Suppose ri depends on rj. The Regression Coefficient, b, is the covariance scaled by the variance of the independent variable.This gives a measure of how much the dependent variable moves when the independent

13、 variable moves.jiijijsssr2jijijssb14Practical IssueWhere do the probabilities come fromAnalyze past behavior or returns.Summarize with distribution function.Assume history is a guide to what is expected.15Risk: Distribution of Possible OutcomesSymmetric about the MeanCompletely described by the mea

14、n and variancerNormal Distribution16Average Annual Returns and Risk Premiums, 1948-1997Average ReturnRisk PremiumCanadian Common13.137.04US common15.189.09Long Bonds7.811.72T bills6.090.0017100 Years of Investment Returns1$ invested in1900Value in 2000US$ReturnUK$ReturnEquities 16,19710.1%16,16010.1

15、%Bonds1194.8%2035.4%Bills574.1%1495.1%Inflation243.2%554.1%18Returns around the world02468101214SwedenUSCanadaUKJapanGermanyItalyReal Nominal19 Real Returnson Equities and Bonds1900-2000-3-2-1012345678SwedenUSCanadaUKJapanGermanyItalyEquitiesBonds20Risk and Return 1973-1997Average ReturnStd. Deviati

16、onCanadian Common 12.24%16.36%US Common16.0917.10Long Bonds11.9512.01Small Stocks15.7523.28Inflation5.793.60Treasury Bills9.053.48If returns are normally distributed, all that is needed to describe investment prospects are mean and standard deviation.21 Std.Dev. Of Real Returnson Equities and Bonds1

17、900-200005101520253035SwedenUSCanadaUKJapanGermanyItalyEquitiesBonds22But, Std. Deviation of the portfolio is less than the components(monthly returns, 1999-2001)AlcanCanforIncoNortelHBCSearsTSEaverage-0.5%-2.4%-1.6%-7.8%-1.2%-3.7%-0.5%std. Dev9.82%13.00%13.19%22.93%9.08%14.18%6.29%23Portfolio Theor

18、y and Risk AdjustmentSteps in the analysisInvestment Opportunities Generated by PortfoliosInvestor Preferences1. Equilibrium Pricing24Investment Opportunities: Risky Assets Only.asset on return expected)(assets. of portfolio aon return expected)(irrErrEiippinvestment portfolio totalin investmentixir

19、eturn portfolio of deviation standardpsA. Expected Returnniiiprxr1B. Risk of a Portfolioninjijjipxx112ssThe sum of n2 terms25Illustration: 2 assets122122222121222221121221111122ssssssssxxxxxxxxxxxxpFrom Basic Statistics:211212sssr2211221222221212ssrsssxxxxpncorrelatio negativeperfect 1tindependen 0n

20、correlatio positiveperfect 1121212rrr26Interpreting Correlationrirj0150.015100.3315200.341050.01-500.33expected 5.00 10.00s8.28.2Cov66.3r0.993rirj0150.355100.1015200.101050.35-500.10expected 5.00 10.00s6.16.1Cov2.5r0.127An Example of a Two Project Firm50.1020121xss50.)1 (10201221xxrr1222002110041400

21、41rsp%15prThereforeIF0.522511.111250.00.152250.1212212212ppppppssrssrssr28In General for Two Assets1)2211222112)(ssssssxxxxpp0 .112rrs1229 2)21212211222112 if 0)()(sssssssssxxxxxpp0 .112rrs1230In General for Two Risky Assets3)0 .1 012rrs12r31In General With Many Assetssrxxxxxx32Limits to Diversifica

22、tion ijijijijiNNNNNNNNiNxsssssssss2p222222p2 As1111)(1covariance average the varianceaverage the1 :Let33Limits to DiversificationAs n gets very large, approaches market risk. 2psMarket, Systematic, or Non diversifiable RiskNumber of AssetsUnique or Diversifiable Risk2ps2ps34The Risk of a Portfolio n

23、injijjipxx112ss 1n2n1 122222n2221 21111111211 121njnpnnjjnnnnnjpjjnjpjjnnxxxxxxxxxxxxxxxsssssssssssssssnxipininjijjipxxx1112sss35Contribution of a Risky Asset to a Risky Portfolionxipipx12ssPortfolio risk is the weighted average of the covariance ofeach asset with the portfolio return.The Proportion

24、al Contribution of Each Assetnxpipix121ssBut, we have to know which portfolio investors will hold36Investment Opportunitiesand PreferencesrsxxxxabcdPreferencesrs37Efficient Portfolios of Risky AssetssrxxxxxEfficient Portfolio:1. Min s for given return2. Max return for given s38Investment Opportuniti

25、es: Risk Free Asset plus Risky Portfoliozero than less 1,an greater th beCan :NOTE21 portfoliorisky in wealth ofpercent )1 ( asset freerisk in wealth ofpercent 1112121011102221212111xxxxxxxrxrxrxxxpfffffpffpffsssssrsss39Opportunity Set with Risky Portfolios and a Risk Free Assetrfamb- The efficient

26、portfolio contains only one risk portfolio, m- m is referred to as the market portfolio, it contains all risky assets. s40Risky Asset Choice is Independent of Preferences.rfm- Both hold the same risky portfolio (m)- Different attitudes towards risk result in different amount of borrowing and lending

27、. s41CAPITAL MARKET LINERisk and Return for a Diversified Portfoliorf- This reflects the risk-return relationship for an individualholding a well diversified portfolio.- It does not tell us how the risk and return vary for an individual asset. rpsmssprmmpfmfprrrrss)(42Capital Asset Pricing Model (CA

28、PM)Central Insight of the CAPM:Risk premium depends only on non-diversifiable risk. The market portfolio is the relevant market.Recall:nxipipx12ssnximimx12market for the Thus,ssniiinimimixx1121bss43Capital Asset Pricing Model (CAPM)Recall:nxipipx12ssnximimx12market for the Thus,ssniiinimimixx1121bss

29、Market Risk is the average of the b risk.The contribution of an asset to market risk isrelated to the b of the asset.44Risk Premium is Proportional to b risk systematic of pricemarket unit per bifirrNote: For the Marketifmfifmmmfmrrrrrrrrbbb)( 1 that know weBut,45An Alternative View of Risk PremiumF

30、or a portfolio withbbaabaxxxxbbbpbin ain Consider a portfolio of the market and a risk free assetfmmmpfmmmpxxrxrxrbbb)1 ()1 (pfmfpfpmppmpfmrrrrrrrxbbbbbb)(or )1 ( 0 , 1But 46This Portfolio is an Alternative to Any Asset iifmfirrrrb)( asset,any for rrsbrfrfririsibiCapital Market Line (CML)Security Ma

31、rket Line (SML)47Estimating bxxxxxxxxxxxxxxb2mimissb0)E(CAPM the torelativereturn excess )(iifmfirrrrb firr fmrr48Implications of the Capital Asset Pricing ModelRequired return depends only on systematic risk. Therefore, for capital budgeting, only systematic risk matters. Corporate diversification

32、does not affect the marginal cost of capital.ifmfirrrrb)(49is the required rate of return on investment of the lenders of a company.ki = kd ( 1 - T )Cost of DebtP0 =Ij + Pj(1 + kd)j nj =150Assume that Basket Wonders (BW) has $1,000 par value zero-coupon bonds outstanding. BW bonds are currently trad

33、ing at $385.54 with 10 years to maturity. BW tax bracket is 40%.Determination of the Cost of Debt$385.54 =$0 + $1,000(1 + kd)1051(1 + kd)10 = $1,000 / $385.54 = 2.5938(1 + kd) = (2.5938) (1/10) = 1.1 kd = .1 or 10% ki = 10% ( 1 - .40 ) = Determination of the Cost of Debt52is the required rate of ret

34、urn on investment of the preferred shareholders of the company.kP = DP / P0Cost of Preferred Stock53Assume that Basket Wonders (BW) has preferred stock outstanding with par value of $100, dividend per share of $6.30, and a current market value of $70 per share.kP = $6.30 / $70. = Determination of th

35、e Cost of Preferred Stock54Cost of Equity Approaches55Dividend Discount Model The , ke, is the discount rate that equates the present value of all expected future dividends with the current market price of the stock. D1 D2 D(1+ke)1 (1+ke)2 (1+ke)+ . . . +P0 = 56Constant Growth Model The reduces the

36、model to:ke = ( D1 / P0 ) + gAssumes that dividends will grow at the constant rate “g” forever.57Assume that Basket Wonders (BW) has common stock outstanding with a current market value of $64.80 per share, current dividend of $3 per share, and a dividend growth rate of 8% forever.ke = ( D1 / P0 ) +

37、 gke = ($3(1.08) / $64.80) + .08 = .05 + .08 = or Determination of the Cost of Equity Capital58Growth Phases Model D0(1+g1)t Da(1+g2)t-a(1+ke)t (1+ke)tP0 = The t=1at=a+1bt=b+1 Db(1+g3)t-b(1+ke)t+ 59Capital Asset Pricing Model The cost of equity capital, ke, is equated to the required rate of return

38、in market equilibrium. The risk-return relationship is described by the Security Market Line (SML).ke = Rj = Rf + (Rm - Rf) j.60Assume that Basket Wonders (BW) has a company beta of 1.25. Research by Julie Miller suggests that the risk-free rate is 4% and the expected return on the market is 11.2% k

39、e = Rf + (Rm - Rf) j = 4% + (11.2% - 4%)1.25 = 4% + 9% = Determination of the Cost of Equity (CAPM)61Before-tax Cost of Debt Plus Risk Premium The cost of equity capital, ke, is the sum of the before-tax cost of debt and a risk premium in expected return for common stock over debt.ke = kd + Risk Pre

40、mium*.* Risk premium is not the same as CAPM risk premium.62Assume that Basket Wonders (BW) typically adds a 3% premium to the before-tax cost of debt. ke = kd + Risk Premium= 10% + 3% Determination of the Cost of Equity (kd + R.P.)63Constant Growth ModelCapital Asset Pricing ModelCost of Debt + Ris

41、k PremiumGenerally, the three methods will not agree.Comparison of the Cost of Equity Methods64Cost of Capital = kx(Wx)WACC = .35(6%) + .15(9%) + .50(13%)WACC = .021 + .0135 + .065 = .0995 or 9.95%Weighted Average Cost of Capital (WACC) nx =165Capital budgeting is the process of identifying, analyzi

42、ng, and selecting investment projects with returns 1 year.Initial cash outflow, 2.) Interim incremental net cash flows and 3.) Terminal-year incremental net cash flowsProject evaluation methods include Payback Period (PBP), Internal Rate of Return (IRR), Net Present Value (NPV) and Profitability Ind

43、ex (PI)The Post-completion Audit is a formal comparison of the actual costs and benefits of a project with original estimatesSummary66SummaryProjected cash-flows on a project will have both an expected return and a standard deviation representing the risk of achieving that cash flow The probability

44、tree is a graphic or tabular approach for organizing the possible cash-flow streams generated by an investment.Cost of Capital is the required rate of return on various types of financing. The overall cost of capital is a weighted average of the individual required rates of return (costs).Cost of eq

45、uity approaches include: Dividend Discount Model, Capital-Asset Pricing Model,Before-Tax Cost of Debt plus Risk Premium67Arbitrage Pricing TheoryArbitrage - arises if an investor can construct a zero investment portfolio with a sure profit.Since no investment is required, an investor can create larg

46、e positions to secure large levels of profit.In efficient markets, profitable arbitrage opportunities will quickly disappear.68Factor Models: Announcements, Surprises, and Expected ReturnsThe return on any security consists of two parts. First the expected returnsSecond is the unexpected or risky re

47、turns.A way to write the return on a stock in the coming month is:return theofpart unexpected theis return theofpart expected theis whereURURR69Factor Models: Announcements, Surprises, and Expected ReturnsAny announcement can be broken down into two parts, the anticipated or expected part and the su

48、rprise or innovation:Announcement = Expected part + Surprise.The expected part of any announcement is part of the information the market uses to form the expectation, R of the return on the stock.The surprise is the news that influences the unanticipated return on the stock, U. 70Risk: Systematic an

49、d UnsystematicA systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree.An unsystematic risk is a risk that specifically affects a single asset or small group of assets.Unsystematic risk can be diversified away.Examples of systematic risk include uncerta

50、inty about general economic conditions, such as GNP, interest rates or inflation. On the other hand, announcements specific to a company, such as a gold mining company striking gold, are examples of unsystematic risk.71Risk: Systematic and UnsystematicSystematic Risk; m Nonsystematic Risk; ns sTotal

51、 risk; UWe can break down the risk, U, of holding a stock into two components: systematic risk and unsystematic risk:risk icunsystemat theis risk systematic theis wherebecomesmmRRURR 72Systematic Risk and BetasThe beta coefficient, b, tells us the response of the stocks return to a systematic risk.I

52、n the CAPM, b measured the responsiveness of a securitys return to a specific risk factor, the return on the market portfolio.)()(2,MMiiRRRCovsb We shall now consider many types of systematic risk.73Systematic Risk and BetasFor example, suppose we have identified three systematic risks on which we w

53、ant to focus:Inflation GDP growthThe dollar-euro spot exchange rate, S($,)1.Our model is:risk icunsystemat theis beta rate exchangespot theis beta GDP theis betainflation theis FFFRRmRRSGDPISSGDPGDPII74Systematic Risk and Betas: ExampleSuppose we have made the following estimates:bI = -2.30bGDP = 1.

54、50bS = 0.50.1.Finally, the firm was able to attract a “superstar” CEO and this unanticipated development contributes 1% to the return.FFFRRSSGDPGDPII%1%150. 050. 130. 2SGDPIFFFRR75Systematic Risk and Betas: ExampleWe must decide what surprises took place in the systematic factors. If it was the case

55、 that the inflation rate was expected to be by 3%, but in fact was 8% during the time period, then FI = Surprise in the inflation rate= actual expected= 8% - 3%= 5%150. 050. 130. 2SGDPIFFFRR%150. 050. 1%530. 2SGDPFFRR76Systematic Risk and Betas: ExampleIf it was the case that the rate of GDP growth

56、was expected to be 4%, but in fact was 1%, then FGDP = Surprise in the rate of GDP growth = actual expected= 1% - 4%= -3%150. 050. 1%530. 2SGDPFFRR%150. 0%)3(50. 1%530. 2SFRR77Systematic Risk and Betas: ExampleIf it was the case that dollar-euro spot exchange rate, S($,), was expected to increase by

57、 10%, but in fact remained stable during the time period, then FS = Surprise in the exchange rate= actual expected= 0% - 10%= -10%150. 0%)3(50. 1%530. 2SFRR%1%)10(50. 0%)3(50. 1%530. 2 RR78Systematic Risk and Betas: ExampleFinally, if it was the case that the expected return on the stock was 8%, the

58、n%150. 0%)3(50. 1%530. 2SFRR%12%1%)10(50. 0%)3(50. 1%530. 2%8RR%8R79Portfolios and Factor ModelsNow let us consider what happens to portfolios of stocks when each of the stocks follows a one-factor model.We will create portfolios from a list of N stocks and will capture the systematic risk with a 1-

59、factor model.The i th stock in the list have returns:iiiiFRR80Relationship Between the Return on the Common Factor & Excess ReturnExcess returnThe return on the factor FiiiiiFRRIf we assume that there is no unsystematic risk, then i = 081Relationship Between the Return on the Common Factor &

60、 Excess ReturnExcess returnThe return on the factor FIf we assume that there is no unsystematic risk, then i = 0FRRiii82Relationship Between the Return on the Common Factor & Excess ReturnExcess returnThe return on the factor FDifferent securities will have different betas0 . 1B50. 0C5 . 1A83Portfolios and Divers