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Matakuliah Tahun : F0892 - Analisis Kuantitatif : 2009 COVARIANCE DAN CORRELATION PORTFOLIO Pertemuan 11 COVARIANCE • Covariance measures the linear relationship between two variables. For example, how do the asset returns of a high yield hedge fund strategy relate (or compare) to the asset returns of a market neutral strategy. Positive covariance says the variables tend to increase or decrease together; negative covariance says they tend to move in opposite directions. If they negatively covary, that tells us that one asset can be a hedge for the other. If the variables are totally independent, the covariance is zero. Bina Nusantara University 3 Why covariance matters? • Portfolio diversification benefits are realized when assets in the portfolio do not perfectly covary. Or we could also say, when assets are imperfectly correlated. Correlation is the unitless version of covariance: it is covariance translated into a standardized measure, from -1.0 (perfect negative correlation) to 1.0 (perfectly correlated). • Covariance is the linear relationship expressed in situation-specific units; correlation is the same but without units. Bina Nusantara University 4 In the formula above, the covariance between X and Y is shown as sigma-sub-XY • Given a portfolio, we can generally reduce portfolio variance (and standard deviation) by adding uncorrelated assets. • If the variables are (X) and (Y), covariance is equal to the expected product of the variables, E(XY) minus the product of their mean values: In the formula above, the covariance between X and Y is shown as sigma-sub-XY. We could also show this as cov(x,y). In Excel, covariance is given by the function =COVAR(). Bina Nusantara University 5 CORRELATION • The correlation coefficient (often denoted by Greek rho=ρ) is given by the following important equation: • In words, the correlation coefficient is the covariance divided by the product of the standard deviations. If we rearrange this formula, we get the relationship: covariance is equal to the product of (correlation coefficient)(standard deviation of first variable)(standard deviation of second variable). Bina Nusantara University 6 The Covariance Between Two Rates of Return • The covariance is a (statistical) measure of how two random variables (in this case, the returns of two securities) “move together;” • A positive covariance between the returns of two securities indicates that the returns of the two securities tend to move in the same direction, that is, better-than-expected returns for one security are likely to occur when better-than-expected returns occur for the other security; • A negative covariance between the returns of two securities indicates that the returns of the two securities tend to move in opposite directions, that is, better-than-expected returns for one security are likely to occur when worse-than-expected returns occur for the other security; • A relatively small or zero covariance between the returns of two securities indicates that there is little or no relationship between the returns of the two securities; • We denote the covariance between the return of security i and the return of security j by ij (the Greek letter sigma); • Note that ; ij ji • You may use the Excel function “COVAR” to compute the covariance between the returns of two securities. The Correlation Coefficient • The correlation coefficient is a statistical measure closely related with the covariance; • The interpretation of the correlation coefficient is that of a “normalized covariance;” • We denote the correlation coefficient between the return of security i and the return of security j by ij (the Greek letter pho); • The relation between covariance and correlation is given by the following equation: ij ij i ; j • The correlation coefficient between the return of security i and the return of security j lies between -1 and 1; . • If the correlation coefficient between the returns of two securities is positive, then the returns of the two securities tend to move in the same direction, that is, better-than-expected returns for one security are likely to occur when better-thanexpected returns occur for the other security; • If it is negative, then the returns of the two securities tend to move in opposite directions, that is, better-than-expected returns for one security are likely to occur when worse-thanexpected returns occur for the other security; • If it is close to 0, then there is little or no relationship between the returns of the two securities; • You may use the Excel function “CORREL” to compute the correlation coefficient between the returns of two securities. 70% 50% 30% -40% 10% 10% -10% -30% Weekly returns on stock A (y axis) versus Stock B(x axis) in the period October XX-January XY. A & B an internet company 60% Weakly Return 80% 70% 60% 50% 40% 30% 20% 10% 0% -10% -20% -30% Weekly returns on two internet stocks (A=red and B=blue) in the period October XX-January XY 12% 8% 4% -30% -10% 10% 0% 30% -4% Weekly returns on stock C(y axis) and Stock B (x axis) in the period October XX-January XY 50% 70% Weakly Return 80% 70% 60% 50% 40% 30% 20% 10% 0% -10% -20% -30% Weekly returns on stock B (blue) and stock C (red) in the period October XX-January XY • Using the Excel formula “CORREL” the correlation coefficient between the weekly returns of Stock A and Stock B in the period of October XX-January XY turned out to be 0.48; • The correlation coefficient between the weekly returns of Stock C and Stock B in that period was 0.16.