calculations.The distribution used earlier is provided again for ease of use.
8 Measures of Risk E[R]A = 12.5% E[R]B = 20% Probability Distribution: State Probability Return On Return OnStock A Stock B% % %% % %% % %% % %E[R]A = 12.5%E[R]B = 20%
9 Measures of RiskGiven an asset's expected return, its variance can be calculated using the following equation:NVar(R) = s2 = S pi(Ri – E[R])2i=1Where:N = the number of statespi = the probability of state iRi = the return on the stock in state iE[R] = the expected return on the stock
10 Measures of RiskThe standard deviation is calculated as the positive square root of the variance:SD(R) = s = s2 = (s2)1/2 = (s2)0.5
11 Measures of RiskThe variance and standard deviation for stock A is calculated as follows:s2A = .2( )2 + .3( )2 + .3( )2 + .2( )2 =sA = ( )0.5 = = 5.12%Now you try the variance and standard deviation for stock B!If you got .042 and 20.49% you are correct.
12 Measures of RiskIf you didn’t get the correct answer, here is how to get it:s2B = .2( )2 + .3( )2 + .3( )2 + .2( )2 = .042sB = (.042)0.5 = = 20.49%Although Stock B offers a higher expected return than Stock A, it also is riskier since its variance and standard deviation are greater than Stock A's.This, however, is still only part of the picture because most investors choose to hold securities as part of a diversified portfolio.
13 Portfolio Risk and Return Most investors do not hold stocks in isolation.Instead, they choose to hold a portfolio of several stocks.When this is the case, a portion of an individual stock's risk can be eliminated, i.e., diversified away.From our previous calculations, we know that:the expected return on Stock A is 12.5%the expected return on Stock B is 20%the variance on Stock A isthe variance on Stock B isthe standard deviation on Stock A is 5.12%the standard deviation on Stock B is 20.49%
14 Portfolio Risk and Return The Expected Return on a Portfolio is computed as the weighted average of the expected returns on the stocks which comprise the portfolio.The weights reflect the 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 betw