= (0.60 x 14) + (0.40 x 20) = 8.40 + 8 = 16.40% Measure of variance (risk) of a Portfolio Risk of a portfolio is the variance of returns of the portfolio from its expected return. The variance or standard deviation is a measure of the variation of possible of returns from the expected rate of return. The variance of a portfolio depends not only on the variances of the securities making up the portfolio, but also on the relationship among those securities called and coefficient of variation. (Variance equation) Covariance: Covariance is the statistical measure of the degree to which two securities returns move together relative to their individual means over time. That is the extent to which their returns co-move or co-vary from their respective averages. When the two returns move together in the same direction, they have positive co-variance; when they move in opposite directions they have negative co-variance; when their movements do not affect each other then they have zero co-variance. Thus, if returns of securities in a portfolio exhibit positive covariance then the investment does not reduce overall risk of the investor. On the other hand, if they have negative covariance they reduce the overall risk of the investor. The equation for co-variance between the returns of securities X and Y (covXY) is; Where Rx and Ry = expected returns of securities X and Y and = the average returns of securities X and Y = the probability of occurrence of an event The riskiness or standard deviation of a two–security portfolio is determined by; Portfolio Variance - P= P Where = the weight (proportion) of total funds invested in security X σ2x = Variance of returns of security X = the weight (proportion) of total funds invested in security Y σ2y = variance of returns of security Y
= 10.35% Summary Investment ER ( ) σ (risk) X 10% 7.75% Y 13.4% 17.91% Portfolio XY 11.7% 10.35% An investor should aim at investing in an efficient Portfolio. An efficient Portfolio is the one with lowest level of risk for a given level of return or the one with highest level of return for a given level of risk. Correlation coefficient The correlation coefficient of returns of two securities is the standard measure of their relationships. The equation for correlation coefficient between the returns of securities X and Y is; corrRx,Ry = = = +0.17% Exercise: Gold Coast Securities is planning to invest a client’s investment funds of GH¢10,000 as follows: GH¢5,500 in shares of Tullow Oil Ghana (X) and GH¢4,500 in shares of SIC Insurance (Y). But the expected returns on this plan are predicated on the forecasted likely state of the Ghanaian economy as presented below. Likely state of the economic Probability of occurrence (Pi) Return on Tullow GH shares (Rx) Return on SIC shares (Ry) Boom 0.5 30% 14%
β = beta of the security Thus the expected return on a security increases with increases in its beta. Fore stance, assume a security with a beta of 1.2 is being considered, while the Treasury bill rate (Rf) is 4% and the market (GSE Composite index) return is 12%. The security’s expected rate of return will be: R = Rf + β(Rm – Rf) Where Rf = 4% β = 1.2 Rm = 12% Therefore R = 4% + [1.2 (12% - 4%)] = 4% + [1.2 x 8%] = 4% + 9.6% = 13.60% The investor should therefore require a return of 13.60% on the security as compensation for the non-diversifiable risk assumed. Now if the beta were lower, say 1.00, the R will be; 4% + [1.00 (12% - 4%)] = 12% And if the beta had been higher, say 1.50, R will be; 4% + [1.50 (12% - 4%)] = 16% The Security Market Line When the CAPM is depicted graphically, the resulting straight line is called the Security Market Lime (SML). The security market line provides a benchmark for evaluating investment performance. Given the risk of an investment as measured by its beta, the SML indicates the required rate of return that will compensate investors for the risk of that investment. Risk-Free Return
Application of SML By the application of CAPM equation the SML can be plotted as follows: If the beta is 0, R = 4% + [0 (12% - 4%)] = 4% If the beta is 1, R = 4% + [1 (12% - 4%)] = 12% If the beta is 1.25, R = 4% + [1.25 (12% - 4%)] = 14% If the beta is 1.50, R = 4% + [1.50 (12% - 4%)] = 16% If the beta is 2.00, R = 4% + [2 (12% - 4%)] = 20%