RE: How to compute SD of portfolio of asset classes?
 From: Jerry W. Lewis <post_a_reply@xxxxxxxxxxxxx>
 Date: Mon, 1 Jan 2007 06:52:00 0800
1. The two approaches are mathematically equivalent, provided the variances
and covariances are estimated compatibly in the two approaches. All else
being equal, your empirical difference is likely due to Excel's COVAR()
function being misnamedin the naming convention of the VAR(), VARP(),
STDEV(), and STDEVP() functions, COVAR() should be called COVARP(); Excel
provides no covariance analog for VAR() and STDEV().
In particular, =COVAR(data,data) gives the same value as =VARP(data), rather
than =VAR(data)
To calculate a covariance analog of VAR() and STDEV(), use either
=COVAR(xdata,ydata)/(11/COUNT(ydata))
or
=CORREL(xdata,ydata)*STDEV(xdata)*STDEV(ydata)
The first approach is simpler for your application, since the correction
factor would be the same for all covariances.
2. A simpler approach would be
=SQRT(SUMPRODUCT(A5:A11*TRANSPOSE(A5:A11)*B15:B21))
It doesn't avoid calculating the variance covariance matrix, but it does
avoid the other intermediates.
Jerry
"nomail1983@xxxxxxxxxxx" wrote:
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 10year
%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?
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