where A, B, C, ... are relative weights of securities a, b, c, ... in portfolio and r(a), r(b), r(c) , ... are corresponding expected returns of securities. Relative weight of each security in portfolio is a fraction of this security’s present value in the total present value of the portfolio:

where "val" means present value.

Let’s consider numeric example. First we compute relative weights of securities in portfolio:

Security: Present Value: Relative Weight: a $400 $400/$2500 = 0.16 b $900 $900/$2500 = 0.36 c $700 $700/$2500 = 0.28 d $500 $500/$2500 = 0.20 --------------------------------------------------------- Total: $2500 1.0

Now we can calculate expected return of portfolio R:

Security: Weight: Expected Ret: a 0.16 5% 0.16 * 5% = 0.80% b 0.36 3% 0.36 * 3% = 1.08% c 0.28 7% 0.28 * 7% = 1.96% d 0.20 11% 0.20 * 11% = 2.20% ------------------------------------------------------------ Portfolio Expected Return = 6.04%

where A and B are returns and r(a) and r(b) are expected values of a and b securities. If random variables A and B are discrete (i.e. their values can be listed), the above formula will take the form:

Covariance(a,b) = [x-r(a)]*[y-r(b)]*P(x,y) + [y-r(a)]*[z-r(b)]*P(y,z) + + [x-r(a)]*[z-r(b)]*P(x,z) + ...where x, y, z, ... are particular returns for securities a and b, P(x,y) is the joint probability of x and y, etc. A large positive covariance implies that the returns move together, i.e. if one security’s return is high then other security’s return is likely to be high. Large negative covariance signifies the opposite trend in movements of returns of two securities, i.e. if one security’s return is high, the other security’s return is likely to be low. Finally, if covariance is close to zero then two securities have very weak relationship, or in other words, movements of returns of two securities are not related to each other. But what does large mean, what is the comparison criteria? Given the computed value of covariance, it’s difficult to say whether it is large or small. This is where the correlation is useful. The correlation is computed by dividing the covariance by the standard deviations of the two securities:

Since standard deviations are positive values, the correlation always has the same sign as covariance. The division of covariance by standard deviations forces values of correlation to be between -1 and 1. -1 means perfect negative relationship between securities, 1 means perfect positive relationship, and values close to zero means weak relationship between movements of returns of two securities.

where A, B, C, D, ... are corresponding relative weights of a,b,c,d, ... securities in portfolio and "cv()" symbol means covariance. Standard deviation of portfolio is a square root from variance value:

where "sqrt" means square root. For simplest case of the portfolio consisting of two securities the variance will be:

The covariance of security a with itself cv(a,a) equals to variance of the security, and similarly cv(b,b) equals to variance of security b. Then above formula for portfolio variance takes the form:

Where "sq" means square, "var" means, variance, "cv" means covariance,
A and B are relative weights of securities a an b in portfolio.

Copyright 2001, George N. Sharia. All Rights Reserved.