According to literature, the std dev of a portfolio of asset classes is
computed by:
Sqrt(Sum(Sum(w[i]*w[j]*Covar(X[i],X[j]), j=1,...,N), i=1,...,N)), where
w[i] is the allocation weight factor and X[i] is the historical
%returns for each of N asset classes.

But I believe I can also compute a std dev of the balanced porfolio by:
Stdev(Sum(w[i]*X[i,t], i=1,...,N), t=1,...,M), where X[i,t] is the
%return for each of N asset classes in each of M time periods.

The two results are different, at least empirically. Which one is the
correct one to use? Or when should I use each one for the purpose of
determining the std dev of a portfolio?

That is not really an Excel question. But I know there are a few sharp
folks in this forum who are schooled in statistics and financial
mathematics. I hope to hear from them.

And here __is__ a related Excel question: what is the best way to
formulate the first expression, namely Sqrt(Sum,(Sum(...)...))?

Here is what I did....

Assume that X[i,t] is in C5:L11. That is, C5:L5 is the 10-year
%returns for Class 1; C6:L6 for Class 2; etc for each of 7 asset
classes. Also assume that w[i] is in A5:A11 -- the allocation weight
factors for each asset class. (Of course, Sum(w[i]) = 100%.)

First, in B15:H21, I compute the matrix Covar(X[i],X[j]). Thus,
B15:B21 is Covar(X[1],X[j]), the covariance between the 1st class and
each of the classes; C15:C21 is Covar(X[2],X[j]), the covariance among
the 2nd class and all classes; etc. For example, the following
computes Covar(X[1],X[2]):

=COVAR(C5:L5,$C$6:$L$6)

Then in B23:H23, I compute the array Sumproduct(w[j],Covar(X[i],X[j]),
j=1,...,7) for i=1,...,7. For example, the following computes
Sumproduct(w[j],Covar(X[1],X[j])):

=SUMPRODUCT($A$5:$A$11,B15:B21)

Finally, I compute
Sqrt(Sumproduct(w[i],Sumproduct(w[j],Covar(X[i],X[j], j=1,...,7),
i=1,...,7)) with the following array formula:

={SQRT(SUMPRODUCT(A5:A11,TRANSPOSE(B23:H23)))}

Is all that necessary? Or am I missing another way to perform the
computations that would obviate the need for one or more intermediate
maxtrices?