e proportion of the portfolio invested in the stocks.This can be expressed as follows:NE[Rp] = S wiE[Ri]i=1Where:E[Rp] = the expected return on the portfolioN = the number of stocks in the portfoliowi = the proportion of the portfolio invested in stock iE[Ri] = the expected return on stock i
15 Portfolio Risk and Return For a portfolio consisting of two assets, the above equation can be expressed as:E[Rp] = w1E[R1] + w2E[R2]If we have an equally weighted portfolio of stock A and stock B (50% in each stock), then the expected return of the portfolio is:E[Rp] = .50(.125) + .50(.20) = 16.25%
16 Portfolio Risk and Return The variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together.Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation coefficient.Covariance is a measure that combines the variance of a stock’s returns with the tendency of those returns to move up or down at the same time other stocks move up or down.Since it is difficult to interpret the magnitude of the covariance terms, a related statistic, the correlation coefficient, is often used to measure the degree of co-movement between two variables. The correlation coefficient simply standardizes the covariance.
17 Portfolio Risk and Return The Covariance between the returns on two stocks can be calculated as follows:NCov(RA,RB) = sA,B = S pi(RAi - E[RA])(RBi - E[RB])i=1Where:sA,B = the covariance between the returns on stocks A and BN = the number of statespi = the probability of state iRAi = the return on stock A in state iE[RA] = the expected return on stock ARBi = the return on stock B in state iE[RB] = the expected return on stock B
18 Portfolio Risk and Return The Correlation Coefficient between the returns on two stocks can be calculated as follows:sA,B Cov(RA,RB)Corr(RA,RB) = rA,B = sAsB = SD(RA)SD(RB)Where:rA,B=the correlation coefficient between the returns on stocks A and BsA,B=the covariance between the returns on stocks A and B,sA=the standard deviation on stock A, andsB=the standard deviation on stock B
19 Portfolio Risk and Return The covariance between stock A and stock B is as follows:sA,B = .2( )(.5-.2) + .3( )(.3-.2) +.3( )(.1-.2) +.2( )(-.1-.2) =The correlation coefficient between stock A and stock B is as follows:-.0105rA,B = (.0512)(.2049) =
20 Portfolio Risk and Return Using either the correlation coefficient or the covariance, the Variance on a Two-Asset Portfolio can be calculated as follows:s2p = (wA)2s2A + (wB)2s2B + 2wAwBrA,B sAsBORs2p = (wA)2s2A + (wB)2s2B + 2wAwB sA,BThe Standard Deviation of the Portfolio equals the positive square root of the the variance.
21 Portfolio Risk and Return Let’s calculate the variance and standard deviation of a portfolio comprised of 75% stock A and 25% stock B:s2p =(.75)2(.0512)2+(.25)2(.2049)2+2(.75)(.25)(-1)(.0512)(.2049)=sp = = = 1.28%Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lower variance and standard deviation than either Stocks A or B and the portfolio has a higher expected return than Stock A.This is the purpose of diversification; by forming portfolios, some of the risk inherent in the individual stocks can be eliminated.
22 Capital Asset Pricing Model (CAPM) If investors are mainly concerned with the risk of their portfolio rather than the risk of the individual securities in the portfolio, how should the risk of an individual stock be measured?In important tool is the CAPM.CAPM concludes that the relevant risk of an individual stock is its contribution to the risk of a well-diversified portfolio.CAPM specifies a linear relationship between risk and required return.The equation used for CAPM is as follows:Ki = Krf + bi(Km - Krf)Where:Ki = the required return for the individual securityKrf = the risk-free rate of returnbi = the beta of the individual securityKm = the expected return on the market portfolio(Km - Krf) is called the market risk prem