Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 1

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 2 Try these problems l Ch 6  Problem 6  Problem 12  Problems (see p 157) l Ch 7  Problem 1  Problem 5

Portfolio Diversification and the Capital Asset Pricing Model Prof. Ian Giddy New York University New York University/ING Barings

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 5 Equity Risk and Return: Summary l Investors diversify, because you get a better return for a given risk. l There is a fully-diversified “market portfolio” that we should all choose l The risk of an individual asset can be measured by how much risk it adds to the “market portfolio.”

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 6 Capital Allocation Possibilities: Treasuries or an Equity Fund? r f =7% E(r P ) =17%  P =27% 10% P Expected Return Risk 7% THE EQUITY FUND

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 7 We Can Buy Some T-bills and Some of the Risky Fund... C.A.L. SLOPE=0.37 E(R) SD 17% 14% 18.9%27% ONE PORTFOLIO: 30% Bills, 70% Fund E(R)=.3X7+.7X17=14% SD=.7X27=18.9% r f =7%

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 8...Or Buy Two Risky Assets A E(r) B

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 9 Diversification Asset FAsset G Portfolio of Assets F and G ReturnReturn Time ReturnReturn ReturnReturn kkk

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 10 Portfolio Return... To compute the return of a portfolio: use the weighted average of the returns of all assets in the portfolio, with the weight given each asset calculated as (value of asset)/(value of portfolio). The portfolio return E(R p ) is: E(R p) = (w 1 k 1 )+(w 2 k 2 )+... (w n k n ) =   w j k j where w j = weight of asset j, k j = return on asset j

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 11...and Risk (Standard Deviation) l Portfolio return is the weighted average of all assets’ returns, l But portfolio standard deviation is normally less than the weighted average of all assets’ standard deviations! l The reason: asset returns are imperfectly correlated.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 12 Measuring Portfolio Risk The variance of a 2-asset portfolio is: where w A and w B are the weights of A and B in the portfolio.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 13 Return and Risk, Generalized Portfolio return: where w i are the weights of each asset in the portfolio. (Expected return is simply the weighted sum of the individual asset returns.) Portfolio variance: When i = j, the term w i w j F i F j D ij becomes w i 2 F i 2.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 14 Covariance and Correlation Covariance and correlation measure the relationship between two series of numbers  Positively Correlated means the series move in the same direction  Negatively Correlated means the series move in opposite directions A correlation coefficient of  +1 indicates perfect positive correlation  -1 indicates perfect negative correlation  0 indicates uncorrelated

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 15 Covariance and Correlation The correlation coefficient scales the covariance to a value between -1 and +1:

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 16 Diversification Diversification is a method of reducing the risk of a portfolio by combining assets that have negative (or low positive) correlation. Combinations of assets that are: 1.Negatively Correlated can reduce  k below that of the least risky asset 2.Uncorrelated can reduce  k, below that of the least risky asset, but not as effectively as with negatively correlated assets 3.Positively Correlated can reduce  k below that of the least risky asset but not as effectively as with uncorrelated assets

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 17 Correlation, Diversification, Risk, and Return l In general, the lower the correlation between asset returns, the greater the potential diversification of risk l Only in the case of perfect negative correlation can risk be reduced to zero l The amount of risk reduction achieved through diversification is also dependent upon the proportions in which the assets are combined

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 18 International Diversification l Including foreign stocks and bonds in a portfolio of U.S. corporate and government securities enhances risk reduction. l Over a longtime horizon international diversification strategies tend to yield returns superior to those yielded by domestic strategies, for a given level of risk.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 19 Risk and Return of Stocks, Bonds and a Diversified Portfolio

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 20 The Correlation Between Stock and Bond Returns l Covariance = (-7-11)(17-7) (12-11)(7-7) (28-11)(-3-7) = l Correlation = / 14.3(8.2) = -0.99

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 21 Portfolio Return and Standard Deviation

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 22 Attainable Set of Risk/Return Combinations A E(r) B

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 23 The Minimum-Variance Frontier of Risky Assets Efficient frontier Individual assets Global minimum- variance portfolio E(r)

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 24 The Efficient Frontier of Risky Assets with the Optimal CAL Efficient frontier CAL(P) E(r)

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 25 Optimal Overall Portfolio Indifference curve Opportunity set CALE(r) P Optimal complete portfolio

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 26 The Investment Opportunity Set with Differential Rates for Borrowing and Lending CAL1 E(r) CAL2 Efficent Frontier

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 27 The Optimal Portfolio of Defensive and Aggressive Investors with Differential Borrowing and Lending Rates CAL1 E(r) CAL2 Defensive Investor Aggressive Investor

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 28 The Capital Asset Pricing Model (CAPM) CAPM Says:  The total risk of a financial asset is made up of two components. A. Diversifiable (unsystematic) risk B. Nondiversifiable (systematic) risk  The only relevant risk is nondiversifiable risk. CAL(P) E(r)

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 29 CAPM Assumptions/1 l Investors are price takers l All investors have an identical one- period investment horizon l Investors are limited to publicly traded financial assets, with unlimited borrowing/lending opportunity l Investors pay neither taxes nor transactions costs on investments

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 30 CAPM Assumptions/2 l All investors are mean-variance optimizers l All investors have homogenous expectations. l The investment universe is a single-factor security market; the return on the market portfolio is that factor.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 31 Types of Risk P o r t f o l i o R i s k  kp Number of Securities (Assets) in Portfolio } } { TOTAL RISK NONDIVERSIFIABLE RISK DIVERSIFIABLE RISK

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 32 The Model: CAPM The CAPM (Capital Asset Pricing Model) links together nondiversifiable risk and return for all assets: A.Beta Coefficient (b) is a relative measure of nondiversifiable risk; an index of the change of an asset's return in response to a change in the market return B.Market Return (k m ) is the return on the market portfolio of all traded securities

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 33 Security Market Line Nondiversifiable Risk,  SML } Market Risk Premium: 4% } Asset Z’s Risk Premium: 6% R z = R m = R F = Required Return, R(%)  RF RF  m m  z z

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 34 The Equation for the CAPM R j = R F +  j (R m - R F ) where: R j = Required return on asset j; R F = Risk-free rate of return  j = Beta Coefficient for asset j; R m = Market return The term [  j (R m - R F )] is called the risk premium and (R m -R F ) is called the market risk premium

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 35 Derivation of the CAPM Risk Premium for the Market Portfolio: S = E[(R M ) - R f ] /  M Risk premium for security’s risk per unit of contribution to the market portfolio risk: [E(R s ) - R f ] /  S  M Setting the two values equal to each other: [E(R M ) - R f ] /  M = [E(R S ) - R f ] /  S  M From which one derives the CAPM’s expected return-beta relationship: E(R S ) = R f +  S [E(R M ) - R f ]

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 36 Silicon Graphics Consider an investment in Silicon Graphics. It has a Beta of 2.0 (riskier than the average stock). If the T-bill rate is 5% and the S&P return is 10%, what is the required return for Silicon Graphics stock? k j =.05 + [2.0 x ( )] =.05 + [2.0 x (.05)] = =.15 or 15%

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 37 Graphic Depiction of CAPM REQUIRED RETURN 15% R j with b j = % R m 5% R F Security Market Line } Market Risk Premium: 5% } Stock’s Risk Premium: 10% Beta (Nondiversifiable Risk) SML = R j =.05 +  j ( ) Given: R F = 5%; R m = 10%

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 38 Interpreting Beta l Market Beta = 1.0 = average level of risk  A Beta of.5 is half as risky as average  A Beta of 2.0 is twice as risky as average  A negative Beta asset moves in opposite direction to market

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 39 Interpreting Beta l Beta Coefficients are estimated from historical data by regression analysis. l Betas are easily obtained from published sources, such as Value Line Investment Survey, and through brokerage firms. l Portfolio Betas are determined by calculating the weighted average of the Betas of all assets included in the portfolio, using each asset's proportion of the total dollar value of the portfolio as its weight.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 40 Summary: CAPM Equilibrium l Investors, in the aggregate, hold the market portfolio. l The market porfolio will be on the efficient frontier and will be the optimal risky portfolio. All investors hold the same risky portfolio (M), adding T-bills to their portfolios to obtain desired risk levels l The risk premium on the market portfolio is proportional to the variance of the market portfolio and the degree of risk aversion of investors.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 41 CAPM Equilibrium (Cont.) The risk premium on individual assets is proportional to the risk premium on the market portfolio and to the  of the security.  measures the extent to which the stock returns respond to the market returns. R i = R f +  i (R M - R f )

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 42 Some Comments on CAPM l Since Beta coefficients are derived from historical data, they are best viewed as approximations of future expectations of actual risk-return behavior. l CAPM is based upon an assumed efficient market which, although seemingly unrealistic, is supported empirically in active markets such as the New York Stock Exchange. l While CAPM is not applicable to all assets, it does provide a conceptual framework that is useful in linking risk and return in financial decisions.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 43 Finding Beta: Example You have found the following data relative to the stock of the Telmex Corp. and current conditions: Required/expected return = 20% Market portfolio return= 11% Risk premium for market portfolio= 6% What is the Beta of Telmex stock?

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 44 Determine the Risk-Free Rate Algebraic Solution Graphic Solution R m - R F = R F =.06 R F =.05 R m =11% R f = 5% } 6% } 5% 1.0 Beta SML R i = R f +  i (R M - R f )

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 45 Plug into SML Formula.20 =.05 + [Beta x ( )].15 = Beta x (.06).15 = Telmex Beta

Portfolio Theory Assignment Prof. Ian Giddy New York University New York University/ING Barings

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 47 Try these problems l Ch 6  Problem 6  Problem 12  Problems (see p 157) l Ch 7  Problem 1  Problem 5

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 48 BKM Chapter 6, Problem 6 Gold Stocks Optimal CAL E(r) P A. If  G,S <+1, gold is still an attractive asset to hold as part of a portfolio. E(r) Optimal CAL Gold Stocks P B. If  G,S =+1, a portfolio of stocks and bills only dominates a portfolio with gold in all instances

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 49 BKM Chapter 6, Problem 12 R A -R f Since B’s error is small, diversification effect is less than for A, which has large unsystematic risk. R M -R f R B -R f R M -R f Stock A has a large error term so would be very risky if all funds were in this one basket. A B

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 50 BKM, Chapter 6, Problem 14 n Most diversification achieved with 1st 20 stocks n By choosing low-correlated assets in the portfolio, risk may not be affected significantly. But would these be the best-return stocks? P o r t f o l i o R i s k  kp Number of Securities (Assets) in Portfolio

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 51 BKM, Chapter 6, Problem 15 n The risk/number of stocks relationship is nonlinear, so risk increases as number of stock is further reduced P o r t f o l i o R i s k  kp Number of Securities (Assets) in Portfolio

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 52 BKM, Chapter 6, Problem 16 Hennessy’s portfolio E(r) n Limiting Hennessy’s holdings may have little impact on the risk of the total portfolio

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 53 BKM, Chapter 7, Problem 1 E(R P ) = R f +  [E(R M ) - R f ] 20 = 5 +  (15-5)  =15/10 = 1.5

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 54 BKM, Chapter 7, Problem 5

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 55 BKM, Chapter 7, Problem 5  ) The beta is the change in the stock return per change in the market return. Therefore:  Aggressive = (2-32)/(5-20) = 2.00  Defensive = (3.5-14)/(5-20) =.70 B ) The expected return is an average of the two possible outcomes: E(R Agg. ) =.5(2+32) = 17% E(R Def. ) =.5(3.5+14) = 8.75%

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 56 BKM, Chapter 7, Problem 5  / The SML is determined by the market expected return of.5(20+5) = 12.5%, with a beta of 1, and the bill return of 8%. Therefore, the equation for the security market line is: E(R) = 8 +  ( )

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 57 BKM, Chapter 7, Problem 5 SML E(r) M D A 8% 12.5%  17%

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 58 BKM, Chapter 7, Problem 5 D/D/ SML E(r) M D A 8% 12.5% DD

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 59 BKM, Chapter 7, Problem 5 E / The hurdle rate is determined by the project beta.7. The correct discount rate is 11.15%, the fair return on the defensive stock.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 60 Equity Risk and Return: Summary l Investors diversify, because you get a better return for a given risk. l There is a fully-diversified “market portfolio” that we should all choose l The risk of an individual asset can be measured by how much risk it adds to the “market portfolio” l The CAPM tells us how the required return relates to the relevant risk.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 61

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 62

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 63

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 64 Ian Giddy NYU Stern School of Business Tel ; Fax

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 65 BMK, Chapter 7, Problem 11 CAL1 E(r) Defensive Investor Aggressive Investor FALSE! With Differential Borrowing and Lending Rates, Defensive and Aggressive Investors have different optimal portfolios. CAL2

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 66 BKM, Chapter 7, Problem 12 FALSE! The portfolio standard deviation will be equal to the weighted average of the component-asset standard deviation if and only if all assets are perfectly positively correlated. In all other instances, the portfolio standard deviation will be less than the weighted average of the component-asset standard deviation

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 67 BKM, Chapter 8, Problem 3 The appropriate discount rate for the project is:R f +  [E(R M ) - R f ] = (19-9) = 26% l Using this discount rate, The internal rate of return on the project is 49.55%. The highest value that the beta can take before the hurdle rate exceeds the IRR is determined by: = 9 +  (19-9)  = / 10 = NPV t10      t1