Weighted median of a portfolio

Question

I need to calculate weighted median return for a portfolio (using MATLAB). There is information available online: MATLAB Central/rates&weights and Excel/note 4+4+4+7+7/5 logic

How would you calculate weighted median for following portfolios with given weights and returns?

Much of online help is about returns with frequencies. I think the Matlab code in the solution above does not account for ties correctly.

Eg. A portfolio has 6 investments with the following returns:

            A       B     C     D     E     F
weights:    0.1     0.1   0.1   0.2   0.1   0.4
returns:    10%     20%   30%   1%    2%    1%

weighted mean = *easy!*
weighted median = 1%??

            A      B     C      D      E      F     G     H
weights:    0.15   0.1   0.15   0.09   0.01   0.01  0.14  0.35
returns:    0.05%  1%    1%     1%     1%     2%    2%    5%

weighted mean = *easy!*
weighted median = 1.5%?? <or should it be (1+1+1+1+2+2)/6??>

Answer 1

As detailed in this answer on the math SE, weights in a weighted median are expressed duplicates of the datum. In your top example, replace the weights with their multiples of ten, like so:

            A       B     C     D     E     F
weights:    1       1     1     2     1     4
returns:    10%     20%   30%   1%    2%    1%

Sort these by return and you have 1 1 1 1 1 1 2 10 20 30; the median of this pseudo-dataset is the weighted median, in this case the average of the middle two values, i.e. 1%. If you repeat this exercise for the second set, you'll see that the value is 1.5%.

Also note that the weighted-median is the same for any vector of weights proportional to this one. (This is straightforward to prove, but I leave that to you.)