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Deriving the Variance–Covariance for the Entire Stock Portfolio

emilioalzamora1

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Hi All,

I just wanted to reach out to all of you with an update about the Var-Cov matrix derivation. There is one learning question (64.2 I suppose) given in David's notes and I stumbled over this topic again in my work now after some time and would like to update you with some useful piece of information (if not yet discussed in great detail):

I am referring to Joel Bessis book 'Risk Management in Banking' where he says:

'For a portfolio there are as many covariances and correlations as there are pairs of stocks'. The direct measures of the variances and covariances of stock returns would imply measuring N^2 terms with N assets.
Since the matrix is symmetric, we have N(…N-1)†/2 covariance terms plus N variance terms, a
total of: N + N(…N-1)†/2 = ˆ N (…N‡+1†)/2 different terms. The number of cells is proportional to
the square of the number of assets.

The above formula needs a bit of explanation as it is perhaps not immediately crystal-clear how Bessis gets to the final result of having N(N+1)/2 different terms.

1.) Simplyfing N(N-1)/2 yields 0.5N^2 - 0.5N

2.) and then simply adding N will yield 0.5N^2 + 0.5N which is the same as N (…N‡+1†)/2.

Two variance terms (N=2) requires 3 cells and a vector of position weights. 0.5 x 2^2 + 0.5 x 2 = 3

Furthermore, John Teall in 'Quantitative Methods' writes:

'To demonstrate that portfolio variance decreases as n (the number of securities) increases, we simply show that the derivative of σ^2 with respect to n is negative: which will be true whenever the average security variance exceeds the average covariance between different securities. This must hold whenever the correlation coefficient between security returns is less than one. As the number of securities in the portfolio approaches infinity, the portfolio’s risk approaches the average covariance between pairs of securities. Individual security variances are insignificant except to the extent that they affect covariances. Thus, only covariance is significant for large, well-diversified portfolios. If security returns are entirely independent (σi,j = 0), portfolio risk approaches zero as the number of securities included in the portfolio approaches infinity.
 
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