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$ $ Tutorial 13. Covariance and Correlation What you will learn: 1. How to find the covariance of historical data data sets 2. How to find the correlation between two historical data sets Fill out the cells with boxes around them like this => $ $ $ $ $ Enter your name in this box ===> Tutorial 13 - Covariance and Correlation Estimating covariance and correlation, which are used in capital asset pricing models. Problem 1: What is the covariance between large company stocks and risk-free Treasury Bills? Another measure of how they move together is the correlation. The closer the number is to 1 (100%), the greater the two variables track each other. Finance Concepts: Covariance is a statistical calculation that tells us how closely two variables move together. Covarince(x,y) = cov(x,y) = 1(((Xi - Xbar)(Yi - Ybar)) Where n is the number of returns, Xi and Yi are individual returns and Xbar and Ybar are the average of the X and Y returns respectively. Time Series Table of Historical Total Returns: Large Co Stocks -0.0313 0.3053 0.0762 0.1007 0.0127 0.3780 0.2274 0.3343 0.2813 0.2103 Year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Treasury Bills 0.0785 0.0571 0.0357 0.0308 0.0415 0.0564 0.0512 0.0522 0.0506 0.0485 Consumer Price Index 0.0610 0.0306 0.0289 0.0275 0.0268 0.0253 0.0332 0.0170 0.0161 0.0269 Steps: 1. To calculate the covariance between Large Comapany Stocks and Treasury Bills for the 19901999 period, we need to find the average returns. Calulate: Average Return Large Stocks 0.1895 Average Return T-Bills 5.0300 Use the average function: =average(E21:E30) for large stks for example The average return on T-Bills was 5.03% for the years 1990-1999. 2. The next step is to find the difference between the individual returns and the average returns for each year. Subtract the average return from the actual return for each year for both the large company stocks and the T-Bills. In F53 enter: = D53-$G$37. In G53 enter =E53-$G$39. Copy. In I53 we want to multiply the differences. Enter: =F53*G53. Copy. Year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Large Co Stocks -0.0313 0.3053 0.0762 0.1017 0.0127 0.3780 0.2274 0.3343 0.2813 0.2103 Treasury Bills 0.0785 0.0574 0.0357 0.0308 0.0415 0.0564 0.0512 0.0522 0.0506 0.0485 Large minus average -0.22 0.12 -0.11 -0.09 -0.18 0.19 0.04 0.14 0.09 0.02 T-Bills minus average -4.9515 -4.9726 -4.9943 -4.9992 -4.9885 -4.9736 -4.9788 -4.9778 -4.9794 -4.9815 Diff Large Times Diff T-bill 1.0932417 -0.5758768 0.5658042 0.4388798 0.8819169 -0.9375733 -0.1887463 -0.7208352 -0.4571587 -0.1036650 3. The formula: Covarince(x,y) = cov(x,y) = 1(((Xi - Xbar)(Yi - Ybar)) We have found (Xi - Xbar)(Yi - Ybar) in column I. Next, sum the column of the multiplication: Enter =sum(I53:I62) => Then, we divide by the number of years (observations), which is 10. Enter =J69/9 => ### -0.000446 The result is that the covariance between large stock and T-Bills (risk-free rate) is 0.0001074 For practical purposes, the covariance is zero. Note that for historical returns we divide by n (not n-1). 4. We can find the correlation between large stocks and T-Bills. The Formula: Correlation coefficient = cov(x,y) / (sx*sy) where sx and sy are the standard deviations of the Large stocks (x) and T-Bills (y). Correlation is the tendency of two variables to move together, and the correlation coefficient measures this tendency. Standard Deviation is the square root of the variance. A. Covarance of large stocks and T-Bills => -0.000446 from J72 B. We need to find the standard deviations by using the formula: =stdevp(range) standard deviations of Large Co stock ==> standard deviations of T-Bills ===> C. Multiply the two standard deviations=> 0 =H86*H87 Correlation between large stocks and T-Bills is ==> =H83 / H89 (Cov / (stdevStk * stdevT-B)) What meaning does this have? We know that the closer the correlation is to 100% (or 1), the more the two variables track each other. In this case we see that the correlation between large stock returns and Treasury returns is only 5.76%, which is very, very slight. This has portfolio diversification and risk reduction implications. Test Your Skills: The returns for the large stocks and the CPI have been copied to the table below. We can find their covariance and correlation using Excel formulas.. To find their covariance enter: =Covar(D109:D118, E109:E118) ==> -0.0009852 To find their correlation enter: =(Covar(D109:D118,E109:E118))/(stdevp(D109:D118)*(stdevp(E109:E118))) Year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Large Co Stocks -0.0313 0.3053 0.0762 0.1017 0.0127 0.3780 0.2274 0.3343 0.2813 0.2103 Consumer Price Index 0.0610 0.0306 0.0289 0.0275 0.0268 0.0253 0.0332 0.0170 0.0161 0.0269 End of Tutorial 13. End of TVM Tutorial. Thank you very much. ==> -0.624095 MSL