Greetings, allstat.
In my previous e-mail, I wrongly suggested that Allan was mistaken.
> Allan wrote:
> > If the total risk is
> > required for a number of risky assets then, assuming independence
> > between assets, the correct procedure is of course to add the variances.
Sietse replied:
> I'm not sure that that's the correct procedure, though, because the
> variance of the sum is computed as follows:
> var(a+b) = var(a) + var(b) + 2cov(a,b)
Allan was correct; I didn't see that he had clearly specified that
independence was assumed. A covariance of zero indeed leaves
var(a+b) = var(a) + var(b) .
My apologies.
As the rest of my e-mail proceeded from this assumption, the entire
e-mail is irrelevant to the question at hand. Specifically, my
assertion that the industry formula (summing standard deviations,
rather than variances) tends to underestimate variance was wrong. In
fact, the industry formula tends to overestimate risk, unless the
variables are negatively correlated. (Rationale in footnote.)
I apologise again for my mistake.
Regards,
Sietse
Sietse Brouwer
Foonote: the difference between estimating risk by (1) summing the
standard deviations, and (2) summing the variances.
The risk
sum(over i)( sd(x_i) )^2
is higher than the risk estimated by
sum(over i)( sd(x_i)^2 )
by a difference of
sum(over i, j)( 2*sd(x_i)*sd(x_j) ) where i != j
Reasoning:
sum(over i)( sd(x_i) )^2 contains all the variances, and two of every
stdev-product;
sum(over i)( sd(x_i)^2 ) contains all the variances.
--
Sietse Brouwer -- [log in to unmask] -- +31 6 13456848
Wildekamp 32 -- 6721 JD Bennekom -- the Netherlands
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