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Errors Found in Study Materials P2.T9. Investment Management (OLD thread)

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Bernardo

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Hi,
Regarding the formula for portfolio risk wiht many assets (p. 6 of Jorion/VaR study notes), the text says the following:

correl_formula.png
"It is evident from the formula above, that the portfolio risk, sigma_ p, tends to zero as N increases".
Shouldn't rhe risk converge to the product of sigma and the square root of the correlation?

Thanks
 

emilioalzamora1

Well-Known Member
Hi @Bernardo,

your statement is partially correct: In the case of correlation coefficient = 0, then the portfolio std dev (sigma p) gets to zero the larger N (no of securities), however, as you can see the correlation coefficient is the critical point here in the portfolio std dev equation.

In any case where correl coefficient, rho (ρ) is greater than 0, portfolio std dev is positive (greater than zero).

In the case of perfect positive corre (rho = 1), portfolio std dev equals the std. deviation of the securities (sigma)

1.) example: N = 50, sigma = 0,14, rho = 1

sigma(p) = 0,14 * sqrt [1/50 + (1-1/50)*1] = 0,14

2.) example: N = 200, sigma = 0,21, rho = 1

sigma(p) = 0,21 * sqrt [1/200 + (1-1/200)*1] = 0,21


1.1) example 1.) example: N = 50, sigma = 0,14, rho = 0.2

sigma(p) = 0,14 * sqrt [1/50 + (1-1/50)*0.2] = 0,14 = 0.065

>>> The smaller rho becomes, the smaller the portfolio std dev.

And yes, portfolio std dev. is the product between the individual securities sigma and the term on the right-hand side involving the no. of assets (n) and rho.

In addition, I have posted this over one year ago in the forum. Perhaps it is helpful:

https://www.bionicturtle.com/forum/...ariance-for-the-entire-stock-portfolio.10153/

 
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Bernardo

New Member
Thanks for your reply, Emilio.
My assertion is:
formula2.png
I really do not understand what is partially corect here. My point is: the study notes mention that this formula is equal to zero, although the limit will be zero only if rho or sigma are equal to zero.
 
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